nmi tr 19 uncertainty in measurement for biochemists: differential
TRANSCRIPT
Uncertainty in Measurement for Biochemists:
Differential Calculus for calculating Sensitivity Coefficients
Robert B Frenkel
June 2015
Page 2 1. Overview
Page 10 2. Background
Page 12 3. The differential calculus, sensitivity to errors and uncertainty
Page 20 4. The notion of ‘differentials’ and ratio of differentials
Page 27 5. The notion of a function
Page 34 6. The exponential function
Page 39 7. The natural logarithm
Page 41 8. The derivative of the natural logarithm
Page 41 9. The derivatives of some simple functions
Page 51 10. Derivative of one function of x multiplied by another function of x
Page 53 11. Derivative of a ratio: one function of x divided by another function of x
Page 56 12. Derivatives of the trigonometric functions
Page 62 13. Derivatives of the hyperbolic functions
Page 63 14. From errors to uncertainties and bias, using the derivative and differentials
Page 65 15. Functions of more than one variable, and partial differentiation
Page 72 16. Propagation of uncertainties from inputs to output
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1 Overview
Measurements in biochemistry, and in particular clinical biochemistry, can indicate a person’sstate of health, but in common with almost all measurements they have uncertainties. Theseuncertainties must be known at least approximately, since high uncertainties will imply anunreliable indication. The quantitative estimation of these uncertainties is the subject of thisdocument.
The method of estimation is that recommended by the Guide to the Expression of Uncertaintyin Measurement [1] (GUM), which has had a wide impact: practitioners in physical, chemical,biochemical, legal and trade metrology use it or are increasingly aware of it.
In general, the measurements are taken of one or more so-called ‘input’ variables – there areusually more than one – and the measurements of these inputs have uncertainties. In all casesthe input variables are related to the single output variable – the ‘measurand’ – by means of anequation or formula. The goal is, firstly, to determine the value of the ‘output’ or measurand,given the values of the inputs, and, secondly, to estimate the uncertainty in the value of the outputthat the output ‘inherits’ from the uncertainties in the inputs. One may imagine the uncertaintiesin the inputs as ‘propagating’ through the formula to create a corresponding uncertainty in theoutput. The GUM method indeed involves the so-called ‘propagation of uncertainties’ formula, tobe discussed in section 16 of this document.
For example, in the measurement of a patient’s anion gap – the output or measurand – theinputs are the concentrations of sodium, potassium, bicarbonate and chlorine ions as determinedfrom a serum sample. A simple formula relates the anion gap to the concentrations of thesefour chemical species. Uncertainties in each of these four measurements propagate to create anuncertainty in the anion gap. The value of the anion gap can reveal such conditions as metabolicacidosis. The uncertainty in the value of the anion gap is reflected in the range of possible valuesin which the anion gap can be located with reasonably high confidence. Thus the anion gap maybe measured as 14.5 mmol/L, with a range of 5 mmol/L on either side of this value.
As another example, an index of kidney function is the glomerular filtration rate (GFR).This is calculated as the output using a formula known as CKD-EPI (Chronic Kidney DiseaseEpidemiology Collaboration). The inputs to the formula are the serum creatinine level, in µmol/L,and the age of the subject. There are empirical ‘constants’ in the formula, obtained from extensiveexperimental tests and that therefore have their own uncertainties and can also be regarded as‘inputs’.
The degree of methylation of specific regions of a DNA molecule may be linked to some formsof cancer. A reference methylation ratio is therefore needed as the output of a measurement inwhich one of the inputs is the proportion of particular sites that are methylated, and another inputis the concentration ratio of methylated to unmethylated material. The uncertainty in the output,the reference methylation ratio, is therefore of interest.
As a fourth example, in patients with atrial fibrillation, the clotting time of a blood sample isone of the input variables in determining the output variable, namely the patient’s internationalnormalised ratio (INR). A value of INR that is too low may indicate the need for a blood-thinningmedication such as warfarin. Again, a formula connects the output INR with the input clottingtime and other input quantities such as the normal clotting time and the international sensitivityindex (ISI). Uncertainties in these input quantities will propagate into an uncertainty in the valueof INR. Evidently high uncertainties in the input quantities will result in a high uncertainty inINR, and this in turn will create an uncertainty in how much warfarin can safely be prescribed tothe patient.
In all cases, the influence of the uncertainties in the inputs on the uncertainty in the outputis determined using so-called ‘sensitivities’ or ‘sensitivity coefficients’. The notion of sensitivity isclear: if a proportional uncertainty of, say 1% in an input creates a proportional uncertainty of3% in the output, then the output is highly sensitive to the input, and extra care must be takenin measuring that input. A lesser sensitivity would be present if, on the other hand, the 1% inputuncertainty creates a similar 1% output uncertainty.
Sensitivity coefficients are determined by using the methods of the differential calculus. Most
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of this document is devoted to explaining these methods, starting from very elementary firstprinciples. The notion of sensitivity is introduced and discussed in terms of a simple exampleof a circle and then of a sphere, and this leads on in a natural way to some basic differentialcalculus. However, ahead of this and immediately below, the above four biochemical examplesare worked through in full, in order to illustrate the complete procedure in real-world cases. Thispresentation of real-world cases at the outset is done on the basis that a detailed view of thedestination can also indicate the route.
This document discusses several basic concepts in sections 3, 4 and 5, such as differentials, ratioof differentials, derivatives and functions. Derivatives are essential for calculating sensitivities; thisis the main material of sections 3 and 4. Several functional forms are discussed in sections 6through 9 and sections 12 and 13. General theoretical results, as for example in sections 10 and 11,are presented in order to enable the reader to evaluate the derivative of almost any function thatmay be pertinent to a particular case. The GUM propagation formula is approached by means ofelementary steps in sections 14 through 16.
Some of the functions discussed – for example, the hyperbolic functions in section 13 – mayappear at present to be remote from the needs of biochemists. But there is no way of foretellingwhich functional forms may become highly relevant in the future. Thus the logarithmic functionenters, perhaps unexpectedly, into uncertainty calculations for GFR and INR measurements.
Nevertheless, if a quick application of the differential calculus is needed, it may be helpful tostate here the most important results and their location in the text:
(a) The relation y = xn implies the derivative dydx = nxn−1. The quantity n may be integer or
fractional, positive or negative. See equations (5.16) and (5.17).
(b) The exponential function y = ex is its own derivative: dex
dx = ex. The quantity e is themathematical constant e ≈ 2.71828.... See the discussion about the more common form y = e−x
and ‘decay processes’ in section 6.
(c) The logarithm (‘natural’ logarithm to base e, symbol ln) is the ‘inverse’ form of theexponential: if y = ex, then x = ln y. For manipulating logarithms, see rules (1) to (12) in section7. The relationship between the natural logarithm and the perhaps more familiar ‘logarithm tobase 10’ is stated in equations (7.4) and (7.5). For the derivatives of logarithms, see section 8,equations (8.1). The useful result (and a generalisation of y = ex) for the derivative of y = ax,where a is any constant, is given in (8.2): dax
dx = ax ln a.
(d) The so-called ‘chain rule’ is stated in (5.25): dydx = dy
dw× dwdx . The variable w is a conveniently
defined ‘intermediate’ variable. In section 9, which shows how the derivatives of functions can beobtained, the chain rule is used several times.
(e) If y is the product of two functions of x, the formula for dydx is given in equation (10.8).
Similarly, if y is the ratio of two functions of x, the formula for dydx is given in equation (11.5).
These formulas are useful for obtaining the derivatives of more complicated functions. An exampleof the use of (11.5) occurs in the determination of the standard uncertainty of a DNA methylationratio (see the third real-world example below).
(f) In (a) to (e) above, the output y is a function of a single input x. In practice, there arealmost always several inputs. The derivatives of y are now ‘partial derivatives’, denoted by thesymbol ∂ instead of d, but these partial derivatives are simply ordinary derivatives taken withrespect to one input at a time, during which all other inputs are considered to be constant. Seesection 15, equations (15.1), (15.2) and (15.6).
(g) The uncertainty of an input or of an output is invariably stated numerically as a standarduncertainty, which is a standard deviation specifically intended to describe and summarise errors inmeasurements. (The more general notion of a ‘standard deviation’ includes cases where there is anatural variation in a quantity, for example people’s heights, but where the variation is not regardedas the result of errors). The propagation formula that links input standard uncertainties to theoutput standard uncertainty, and that makes essential use of derivatives, is stated in (16.14) and(16.16) for two and three inputs respectively. Correlations are included for greater generality, butthe correlation terms can be set equal to zero if the situation permits assuming zero correlations.
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The generalisation to more than three inputs is obvious. The propagation formula is approachedby means of elementary steps; this is the general subject matter of sections 15 and 16.
(h) The Gaussian or ‘normal’ distribution is depicted in Fig. 32, with the 68% (‘one standarddeviation’), 95% (‘two standard deviations’ )and 99% (‘2.58 standard deviations’) points as marked.Its mathematical form is stated in equation (16.16).
In much of biochemistry, and in particular clinical biochemistry, it is common to state inputand output standard uncertainties using the notion of a ‘coefficient of variation’ or ‘coefficient ofvariability’, CV. This is a statement of proportional uncertainty; thus a standard uncertainty of 2.3mmol/L of an anion gap measured as 14.5 mmol/L can be expressed as a CV of (2.3/14.5)× 100or 16%. Expressing the final result, namely the standard uncertainty of a measurand, as a CV inmany cases simplifies the mathematical expression linking the input standard uncertainties to theoutput standard uncertainty. However, it is strongly recommended that absolute, not proportionaluncertainties, should be used in the intermediate stages of determining the propagation ofuncertainty from inputs to output. Confusion may arise, and an incorrect formula obtained,by premature use of proportional standard uncertainties. Converting the absolute standarduncertainty of the output to a proportional standard uncertainty or CV should be the final stageof the calculation.
Some examples from the real world
Until otherwise stated, all inputs in these examples are assumed to be mutually uncorrelated.This means, roughly speaking, that the inputs are independent of one another. Correlation will bediscussed further in section 15.
The general form of the propagation equation is as follows. We have, in general, a functionalrelationship between the output (the ‘measurand’) y and the various inputs x1, x2, x3 ....
y = f(x1, x2, x3...) (1.1)
The standard uncertainties of each of the inputs must be known, at least approximately. If theseare u(x1), u(x2), u(x3)...then these uncertainties propagate into y according to the rule
u2(y) =(
∂y
∂x1
)2
u2(x1) +(
∂y
∂x2
)2
u2(x2) +(
∂y
∂x3
)2
u2(x3) + ... (1.2)
The partial derivatives ∂y∂x1
, ∂y∂x2
, ∂y∂x3
... are the sensitivities or the sensitivity coefficients.
The standard uncertainty u(y) is therefore obtained by taking the square root of (1.2). Theproportional standard uncertainty can then be easily obtained as u(y)
y . However, as stated above,the propagation equation involves not proportional, but absolute standard uncertainties for theinputs on the right side of (1.2).
The procedure is illustrated in the following four examples.
1. Standard uncertainty of an anion gap.
In clinical tests, the anion gap AG [9] of a sample of plasma or serum is calculated as:
AG =[Na+]
+[K+]− [
HCO−3]− [
Cl−]. (1.3)
where the square brackets denote concentrations. The positive ions (cations) are sodium andpotassium, while the negative ions (anions) are bicarbonate and chlorine. The concentrations ofthe four ions in (1.3) are measured using standard clinical tests. Since the total electrical chargeof the sample must be zero, an AG represents ‘unmeasured’ ions. A normal value of AG is about14 millimols/litre (mmols/L). The AG is useful in revealing disorders of acid-base metabolism orwhere there may be an imbalance in other ionic species.
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The standard uncertainty u(AG) of the anion gap is given in terms of partial derivatives ofthe AG with respect to each of the ion concentrations, and the standard uncertainties of the ionconcentrations. The propagation formula corresponding to (1.2) is:
u2(AG) =
(∂AG
∂[Na+
])2
u2([Na+]
)+(
∂AG∂ [K+]
)2
u2([K+]
)+
(∂AG
∂[HCO−3
])2
u2([HCO−3
])+
(∂AG
∂[Cl−
])2
u2([Cl−
]).
(1.4)Because of the straightforward linear relationship in (1.3), the partial derivatives of the anion gapAG with respect to each of the ion concentrations is given simply by
∂AG∂
[Na+
] = 1,
∂AG∂ [K+]
= 1,
∂AG∂
[HCO−3
] = −1,
∂AG∂
[Cl−
] = −1. (1.5)
Thus all four (absolute) sensitivities are numerically equal to 1 in this simple case. The plus orminus signs on the right-hand sides of (1.5) indicate whether the AG increases or decreases inresponse to a change in a particular ion concentration. Looking at (1.3), with the minus sign infront of the
[Cl−
](for example), it is obvious that if
[Cl−
]increases, then AG must decrease,
hence the minus sign for the last equation in (1.5).
Since the squares of these partial derivatives are all +1, u2(AG) is simply given by:
u2(AG) = u2 [Na+]
+ u2 [K+]
+ u2 [HCO−3
]+ u2 [
Cl−]. (1.6)
and the square root of both sides of this equation gives u(AG).
Typical values, in healthy people, are:[Na+
]= 140 mmol/L, with u(
[Na+
]) = 1.2 mmol/L
[K+
]= 4.5 mmol/L, with u(
[K+
]) = 0.10 mmol/L
[HCO−3
]= 25 mmol/L, with u(
[HCO−3
]) = 1.2 mmol/L
[Cl−
]= 105 mmol/L, with u(
[Cl−
]) = 1.5 mmol/L.
These values give an AG of 14.5 mmol/L, with a standard uncertainty of√1.22 + 0.102 + 1.22 + 1.52 mmol/L, or 2.27 mmol/L. The ratio of this standard uncertainty to
the AG itself is therefore a fairly high 16%. The reason for this high proportional uncertaintyis that the AG is relatively small compared with two of its component inputs, the sodium andchlorine, both of which have a concentration about one order of magnitude higher than the AGitself. As a general metrological feature, a small difference between two much larger numbers willhave a relatively high proportional standard uncertainty, so extra accuracy is demanded in themeasurement of the larger numbers.
In this example, the standard uncertainty u(AG) of the output AG is the root-sum-square ofthe standard uncertainties of the inputs. This follows from the linear relationship (1.3) betweenthe output and inputs, and is by no means a general feature linking standard uncertainties.
The ratio of a standard uncertainty to a mean or actual value is often termed the coefficient ofvariation or variability (CV). In this example, the CV is about 16%.
2. Standard uncertainty of glomerular filtration rate.
The glomerular filtration rate (GFR) is generally considered to be the best overall index ofkidney function. Impaired kidney function can be the effect of diabetes, chronic high blood pressure
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or determined by family history. However, an accurate GFR cannot be easily measured. As aconsequence, it is current laboratory practice to estimate GFR (estimated GFR or eGFR) using oneof several empirical prediction equations that rely on measurements of serum (plasma) creatinineand body size and that adjust for gender, race and age. One such set of equations originated in theModification of Diet in Renal Disease (MDRD) study [10]. However, it has been largely supersededby the CKD-EPI equations (Chronic Kidney Disease Epidemiology study) [11]. For male subjectswith a creatinine level exceeding 80 µmol/L (micromoles per litre) one of the equations is:
eGFR = 141(SCr× 0.0113/0.9)−1.209 × 0.993age in years, (1.7)
where SCr is measured in µmol/L. Equ. (1.7) then yields the eGFR in millilitres/min (mL/min)per 1.73 square metres of body surface area.
The ‘constants’ 141, −1.209, 0.9 and 0.993 are empirically derived and so must have their ownstandard uncertainties. The quantity 0.01131137 is a conversion factor from µmol/L to milligramsper decilitre. Its proportional standard uncertainty is very low but nevertheless will be taken intoaccount below.
For brevity in the following analysis (1.7) is written as
G = A(BK/C)DJT , (1.8)
where G = eGFR, A = 141, B = SCr level in µmol/L, K = 0.0113, C = 0.9, D = −1.209,J = 0.993 and T = age in years.
The standard uncertainty u(G) of eGFR is given in terms of partial derivatives of G withrespect to each of the inputs A, B, K, C, D, J and T , and the standard uncertainties of theseinputs. The propagation formula is:
u2(G) =(
∂G
∂A
)2
u2(A) +(
∂G
∂B
)2
u2(B) +(
∂G
∂K
)2
u2(K) +(
∂G
∂C
)2
u2(C)+
(∂G
∂D
)2
u2(D) +(
∂G
∂J
)2
u2(J) +(
∂G
∂T
)2
u2(T ). (1.9)
Thus for estimating the standard uncertainty of the output G we require the following sevenpartial derivatives:
∂G
∂A= (BK/C)DJT ,
∂G
∂B= A(K/C)DDBD−1JT ,
∂G
∂C= A(BK)D(−D)C−D−1JT ,
∂G
∂D= A(BK/C)D(ln BK/C)JT ,
∂G
∂J= A(BK/C)DTJT−1,
∂G
∂K= A(B/C)DDKD−1JT ,
∂G
∂T= A(BK/C)DJT ln J. (1.10)
So (1.10) gives the (absolute) sensitivities of G with regard to the various input quantities.
The required standard uncertainty u(G) in G is then given by:
u2(G) = (BK/C)2DJ2T u2(A) + A2(K/C)2DD2B2D−2J2T u2(B)
+A2(BK)2DD2C−2D−2J2T u2(C)+A2(BK/C)2D(ln BK/C)2J2T u2(D)+A2(BK/C)2DT 2J2T−2u2(J)
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+A2(B/C)2DD2K2D−2J2T u2(K) + A2(BK/C)2DJ2T (ln J)2u2(T ). (1.11)
Equ. (1.9) can be simplified by dividing both sides by G2:
(u(G)
G
)2
=u2(A)
A2+D2
(u2(B)
B2+
u2(C)C2
+u2(K)
K2
)+T 2 u2(J)
J2+(ln BK/C)2u2(D)+(lnJ)2u2(T )
(1.12)and taking the square root of both sides of (1.12) gives the proportional standard uncertainty u(G)
G .
The standard uncertainties of the ‘constants’ are assumed to be as follows. The quantityA = 141 is assigned a standard uncertainty u(A) = 1.0, the conversion factor K = 0.01131137with u(K) = 3×10−7, C = 0.9 with u(C) = 0.001, D = −1.209 with u(D) = 0.001, and J = 0.993with u(J) = 0.001. For a particular patient, B (the SCr level) may have the value B = 150 µmol/Lwith u(B) = 5.0 µmol/L, and T may be T = 60 years with u(T ) = 0.0014 years. Plugging all thesevalues into (1.8) and (1.12) yields G = 42.98 mL/min per 1.73 square metres of body surface area,u(G) = 3.137 mL/min per 1.73 square metres of body surface area and u(G)
G ≈ 3.13742.98 ≈ 7.3%.
3. Standard uncertainty of DNA methylation ratio.
The DNA molecule includes two helically coiled strands and four nucleotides: cytosine (C),guanine (G), adenine (A) and thymine (T). They are paired on opposing strands as the base-pairs CG and AT. The base-pair CG is commonly written CpG, where ‘p’ is a phosphodiesterbond. Some of the CpG sites can exist in a so-called methylated state, whether as an effect of aninternal process that may be implicated in some cancers, or as the deliberate result of treatmentfor the purposes of research. A reference methylation ratio is therefore needed, and its variability,measured as a standard uncertainty, is of interest [12].
The approximate equation for the reference methylation ratio MR is
MR =PM
1 + mU/FmM, (1.13)
where PM is the proportion of CDKN2A 550 bp (base-pair) CpG sites that were methylatedin the methylated reference material. CDKN2A is a human tumour suppressor gene associatedwith loss of expression in many cancer types. The quantity mM is the mass of the methylatedpreparation used to prepare the mixture (in mg), mU is the mass of the unmethylated preparationused to prepare the mixture (in mg),and F is the concentration ratio of methylated material tounmethylated material.
Equ. (1.13) is approximate because it is assumed that the proportion of methylated sites inthe unmethylated reference material is negligible.
Since mM and mU occur only as a mutual ratio, it is convenient to give this ratio a symbol,say α = mU/mM . Then:
MR =PM
1 + α/F. (1.14)
The quantities with non-negligible uncertainties are the inputs PM and F . The propagationequation for this case is
u2(MR) =(
∂MR
∂PM
)2
u2(PM ) +(
∂MR
∂F
)2
u2(F ). (1.15)
The two required partial derivatives are:
∂MR
∂PM=
11 + α/F
, (1.16)
and∂MR
∂F=
−PM
(1 + α/F )2
(−α
F 2
)=
αPM
(F + α)2. (1.17)
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Equ. (1.17) can be obtained by differentiating the following expression that is the same as (1.14)but rearranged:
MR =PMF
F + α
This follows from (1.14) when both top and bottom of (1.14) are multiplied by F . Then we usethe rule for the derivative of one function u(x) divided by another function v(x), as described insection 11. Here u(x) = u(F ) = PMF and v(x) = v(F ) = F + α.
Therefore, (1.16) and (1.17) give:
u2(MR) =u2(PM )
(1 + α/F )2+
α2P 2Mu2(F )
(F + α)4. (1.18)
For obtaining proportional standard uncertainties, both sides of (1.18) are divided by M2R, giving
(u(MR)
MR
)2
=(
u(PM )PM
)2
+(
u(F )F
11 + F/α
)2
. (1.19)
For numerical values PM = 0.9818, u(PM ) = 0.012, F = 0.9845, u(F ) = 0.021, α = 3.8310,
(1.19) gives u(MR)MR
≈ 0.0209, namely a CV of about 2.1%.
4. Standard uncertainty of an International Normalised Ratio (INR).
The INR [13,14] is an index of the clotting time of blood. Denoting the INR by R, it is definedas
R =(
P
N
)c
, (1.20)
where P is the prothrombin clotting time of a particular patient using a particular thromboplastinreagent, and N is the geometric mean of a group of normal subjects whose prothrombin clottingtimes have been previously measured (the geometric mean will be discussed in section 16.2). Thequantity c is the international sensitivity index or ISI, which is determined by the thromboplastinreagent and instrument combination used by a pathology laboratory in its assay. The INR is acommonly requested test in patients who are taking warfarin or other anticoagulation medication.The desired INR value for patients depends on individual circumstances and on the type ofanticoagulant which the patient is receiving. For patients taking warfarin, the target value isusually between 2.0 and 3.5.
The propagation equation for this case is:
u2(R) =(
∂R
∂P
)2
u2(P ) +(
∂R
∂N
)2
u2(N) +(
∂R
∂c
)2
u2(c). (1.21)
To obtain the standard uncertainty u(R) in R, the following partial derivatives are needed:
∂R
∂P=
cP c−1
Nc ,
∂R
∂N= −cP cN−c−1 = −cP c 1
Nc+1,
∂R
∂c=
(P
N
)c
ln(
P
N
). (1.22)
For the first two derivatives above, we used (5.16) and (5.17): if y = xn, then dydx = nxn−1, which
holds for n a positive or negative integer or a positive or negative fraction. For the last derivative,we used (8.2) for the derivative of ax with respect to x: dax
dx = ax ln a.
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Then for the standard uncertainty u(R) of R, the standard uncertainties u(P ) of P , u(N) ofN and u(c) of c propagate into u(R) as follows:
u2(R) =(
cP c−1
Nc
)2
u2(P ) +(−cP c 1
Nc+1
)2
u2(N) +(
P
N
)2c (ln
P
N
)2
u2(c). (1.23)
This equation can be considerably simplified if we now express this final result as proportional
standard uncertainties. To do this, we divide u2(R) by R2, namely by(
PN
)2cfrom (1.20) giving:
(u(R)
R
)2
= c2
[(u(P )
P
)2
+(
u(N)N
)2]
+ u2(c)(
lnP
N
)2
. (1.24)
Numerical values might be: P = 26 seconds, u(P ) = 0.5 second, N = 13 seconds, u(N) = 0.1second, c = 1.2, u(c) = 0.024. Then (1.20) gives R = 2.30 and (1.24) gives R a CV u(R)
R of about0.03 or 3%.
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2 Background
In 1993 two documents, the Guide to the Expression of Uncertainty in Measurement (GUM)[1], and the International Vocabulary of Basic and General terms in Metrology (VIM) [2] werecreated by a group of international metrological organisations under the chairmanship of theBureau International des Poids et Mesures (BIPM) in Paris. Until the dissemination of these twodocuments, there had been no definitive and consistent application of statistical procedures in thesciences and in particular in metrology – the ‘infrastructure’ of the sciences – and no uniform andunambiguous terminology to accompany these procedures. The concepts of ‘error’ and ‘uncertainty’were often jumbled, and the now-obsolete term ‘standard error’ was often misused. There was noset method of determining the uncertainty of a result in terms of the uncertainties of its contributingcomponents, and moreover the uncertainty was sometimes conflated with an estimated forecast ofany future change in the instrument or system in question. So the two documents, and in particularthe GUM, have been forces for clarity and consistency in the expression of uncertainties.
This is not to say that prior to the GUM there was no known rigorous method of estimatinguncertainties. On the contrary, a procedure identical to that in the GUM, but called the ‘calculusof errors’, was described by R. Courant in his ‘Differential and Integral Calculus’, first published in1936. In fact many of the statistical tools were available by the late eighteenth and early nineteenthcenturies. An ambitious seven-year project took place over the period 1792-1798 to establish astandard metre. The metre would be defined as one ten-millionth of the pole-equator meridian thatpasses through a landmark near Dunkirk, then south through Paris to a landmark near Barcelona.It was therefore necessary to measure accurately the distance of roughly 1100 kilometres betweenthese landmarks. Ken Alder in ‘The Measure of All Things’ tells the fascinating story of thisproject. Led by Jean Delambre and Pierre Mechain, it demanded expertise in astronomy and inthe triangulation of distances, using an ingenious instrument known as the Borda repeating circle,and alertness to errors caused by instrument wear, atmospheric refraction, the Earth’s oblatenessand even any gravitational deflection of plumb lines by mountains. Then as now – because errorscan appear from unexpected directions – good metrology depends not only on depth of knowledgein one or two scientific areas, but also on breadth of knowledge across several scientific areas. Butas observed by Alder, no firm distinction was made at the time between random and systematicerrors, and moreover committing errors was regarded as a ‘moral failing’, an attitude which isnow thankfully extinct! Nevertheless the estimate of the length of the meridian differed from thepresent-day satellite-aided estimate by a mere 0.02%.
So the GUM makes no radically novel contribution to metrology. Its value is to be found inits strict definition of various statistical terms, in disseminating to a wide audience an awarenessof the close link between metrology and good statistical practice, and in its recommendations inthe interests of clarity and consistency as mentioned above. However, these welcome attributeshave come at a price. The procedure for calculating the uncertainty of a result, in terms ofthe uncertainties of its contributing components that ‘propagate’ into it – to use an apt termpopularised by the GUM – must use the methods of the differential calculus. The reason is thatderivatives, calculated by these methods, are the ‘sensitivity coefficients’ that govern the natureand extent of the propagation, and hence the uncertainty, of the result. There is therefore a needfor a widespread familiarity with the differential calculus.
The procedure just described is generally referred to as the ‘bottom-up’ procedure for evaluatinguncertainties, in contrast to the ‘top-down’ procedure common in the biological sciences whereprominence is given to such matters as internal and external quality control, traceability ofmeasurements and treatment of bias. However, the bottom-up procedure is frequently neededhere as well [3],[4]. But the degree of familiarity with the differential calculus is probably betterin the physical than in the biological sciences.
This document is an attempt to explain, by means of elementary steps and with numerousexamples, the concepts of the differential calculus that are needed for using the GUM. In itslater sections 14, 15 and 16, the document attempts to explain, also by means of elementarysteps, the propagation equation which is a prominent feature of the GUM and which involvesthe differential calculus. So the document is written for those who may wish to brush up on theircalculus. (Yes, that sounds like something a dentist might say!) There is no particular bias towards
10
readers trained in the biological sciences, except as in section 1 above with the real-world examples.Clinical biochemistry in particular has lately seen some increased emphasis on the estimation ofuncertainties, as recommended by the GUM. The content of the document is heavily mathematical,and a reader may well feel that he or she has been invited to step out into a mathematical blizzard.However, it should be emphasised that the material in this document has been broken down intoelementary steps, and so to be adequately understood needs scarcely more mathematical trainingthan is offered in middle high school.
For example, an understanding of powers is assumed: thus 32 × 34 = 36; when the base is thesame (here it is 3), one adds up the powers when multiplying (and subtracts them when dividing).This is easily checked: 9 × 81 = 729 = 36. On the other hand, powers are multiplied in anexpression such as (32)4. This is the same as 32×4 = 38, because 94 = 38 = 6561.
A negative sign outside brackets changes the sign of every term inside: thus −(6−5) = −6+5 =−1. When two algebraic symbols, say x and y, are multiplied together, this multiplication isdenoted by x×y or (with one basic exception, as will be explained in section 3.1) as xy. Attentionis also given, again through elementary steps, to the number e ≈ 2.71828... (or ‘exp’), which playsan important part in decay processes with a time-constant. (In a typical decay process, the amountof some quantity decreases in time, and the rate of decrease itself decreases in time). This leadson to logarithms. The notion of a function, and the key notions of differentials, differentiationand derivatives are, again, easily understood by means of graphs which illustrate that a derivative,obtained by the process known as differentiation, is simply a precise description of a ‘rate of change’.Several examples of derivatives of functions are given. The propagation equation is approachedin small, and it is hoped, easy steps. In the interest of rapid understanding without interruption,mathematical rigour is nowhere attempted and, instead, an intuitive approach is offered whereverappropriate.
It is of course true that there is software, for example Mathematica, that will give the derivativeof a function. There is no question that software is an essential resource. However, softwarecombined with an inadequate theoretical understanding can yield erroneous results. I have takenthe view that a good grasp of a subject such as the differential calculus is an asset, and I hopethat this document provides it and that the time spent on acquiring this grasp is time well spent.
11
3 The differential calculus, sensitivity to errors and uncer-tainty
It often happens that the value of a quantity is not directly measured but must be calculated, interms of components that can be directly measured. As an example, we might wish to calculatethe area of a circle, having first measured its diameter. Then the question arises: how sensitiveis the area to any error in measuring the diameter? This is where the differential calculus comesin: for statistical purposes and specifically for calculating uncertainties, the differential calculusprovides a rigorous method for determining the sensitivity of a quantity to small changes in otherquantities.
So it is natural to introduce the notion of input and output quantities. The input quantitiesare the quantities directly measured; the output quantities are calculated in terms of the inputquantities. This phrase, ‘in terms of’, in mathematical parlance can be restated: ‘as a functionof’. Often there is a single output quantity and several input quantities. The name usually givento the output quantity is the measurand, but this is a rather fluid term, because an input quantitycan also be a measurand if this input quantity is itself calculated as a function of ‘sub-inputs’.As a simplified example, suppose the measurand is the pressure exerted by a force, for examplea weight, applied to a circular area. Pressure is, quite generally, defined as a force divided bythe area subjected to that force. The input quantities are then the area of that circle and theweight. However, the area of the circle is itself a measurand when we consider its diameter as the‘sub-input’.
In analytical chemistry and related areas, the term ‘analyte’ has much the same meaning as‘measurand’.
It is the existence of error that creates uncertainty. But ‘error’ and ‘uncertainty’ are differentnotions. An error is an unwanted change in a quantity. If the error is a single error, the directionand magnitude of this change are sometimes known or can be deduced from knowledge of anylimitations in the measuring procedure or in the measuring equipment. It may then be possibleto correct for the bias due to the single error, which is then a so-called ‘systematic’ error. Thebias will of course have its own associated uncertainty. An example of such an error occurs whenthe concentration of an analyte is determined by transferring it from its original complex matrixinto a simpler solution for analysis: some analyte may be left behind during the transfer, and its‘recovery’ will then be less than 100%. The analysis will yield a value for the concentration that isconsequently too low. There are guidelines for estimating numerical values for the recovery and forits uncertainty [18]. Other examples of errors are those caused by an out-of-calibration instrument,by instrument wear or by observer bias (this last cause being less common nowadays with digitalinstruments). All these errors are, therefore, notionally ‘single errors’ in the sense that they canbe expressed as ‘the error’ or ‘the bias’. In principle such an error can be corrected for. There isfurther discussion of bias in section 14.
The error may on the other hand be one of many ‘random’ errors. They allow of no suchcorrection but their collective effect is the uncertainty of the measurement. There are, broadlyspeaking, four such sources of uncertainty, and these are of course not mutually exclusive:
(1) limited resolution of an instrument;
(2) inherent fluctuation in the measurand, for example the voltage across a resistor that ismeasured using a high-resolution digital voltmeter;
(3) insufficient detail in specifying exactly what is to be measured, which amounts to inadequatedefinition of the measurand. In physical metrology, a simple example is the measurement using amicrometer of the diameter of a cylindrical steel rod: exactly where along the rod is the diameterto be measured?
(4) insufficient detail in describing how the measurement is to be carried out, which can beclassed as incomplete standardisation of method.
Cases (3) and (4) often introduce an unexpected and confusing variability, for example if air-temperature is measured without specifying the height above ground or how the thermometer isto be shielded against wind and direct sunlight.
12
Uncertainty attributable to the measuring equipment can be likened – although this is not anexact analogy – to the blurring in an optical system that is caused by imperfect focus. Such blurringcan also be usefully visualised as the effect of the other three sources. Uncertainty blurs, to a greateror lesser extent, the numerical value of the quantity under measurement. Just as blurring makesresolving the fine details of an object difficult or impossible, so uncertainty limits the accuracy of ameasurement and in particular limits the number of significant digits that we are entitled to claimin the numerical value resulting from the measurement. Uncertainty is almost always present ina measurement (the exception being when the measurement is simply a count), and a bias willalways have its own associated uncertainty. The effect of uncertainty can be quantified as a blurredregion where the numerical value can be correctly located with high probability (often 95%). Thelower the uncertainty, the narrower that region for a given probability. Uncertainty therefore hasa ‘plus-or-minus’, ±, attribute, relative to the centre of the region, implying that some of thecontributing errors are in the positive direction and some (often a presumed equal number) in thenegative direction.
Further discussion on errors and uncertainty and those high-confidence regions – so-called‘confidence ranges’ and associated ‘levels of confidence’ – , and related topics, can be found in [5].
It is often – but not always – the case that an ‘input’ quantity may be regarded as a ‘cause’,with the output quantity playing the role of the corresponding ‘effect’.
So the key notion of the differential calculus, in the context of the calculation of uncertainties,is the sensitivity of the measurand to small changes in the inputs. Such sensitivity to small changes– in other words, to errors in an input and therefore to the uncertainty of an input – governs theextent to which such an uncertainty is propagated to the output. Here, then, is the link between thedifferential calculus and the formula for the propagation of uncertainties as stated in the ‘Guide tothe Expression of Uncertainties in Measurement’ (GUM). This formula will be discussed in section14.
Intuitively, if the sensitivity is high, then the small changes in the input are ‘amplified’ so thatthe measurand undergoes larger changes. An immediate question arises: how do we compare asmall change with a larger change? For such a comparison to be meaningful, both changes shouldhave the same dimensions and should be measured in the same units. In fact it is usually thecase that they have different dimensions: thus the input may be a temperature and the output alength (as in a mercury-in-glass thermometer, where the cause-effect relationship mentioned aboveis clear: we regard the temperature as influencing the length of the mercury column, and not viceversa).
The answer to the question in the paragraph immediately above is that we compare relativeerrors and therefore proportional sensitivities, which are dimensionless. In section 3.1 we illustratethis point.
3.1 A few sensitivity examples: a circle and a sphere
We consider a circle of diameter D = 1 metre, and we calculate the circumference C = πDand the surface area A = πD2/4 of this circle. (When a quantity x is divided by a quantity y,we shall denote this as x/y or as a fraction x
y ). The symbol π = 3.14159... is the ratio of thecircumference of a circle to its diameter. We also calculate C and A for small measurement errorsin D, that is, for measured values of D that differ slightly (in both negative and positive directions)from the ‘central’ value of 1 metre. (Note: The ‘circularity’ of the circle is not affected by thesemeasurement errors; it remains a perfect circle. The same consideration applies to the sphere,whose sphericity remains unaffected by measurement errors). These small deviations are denotedby the prefix δ (the Greek letter delta). Thus δD is a small error in D. (Note: it does not mean:‘delta multiplied by D’. Otherwise, we use the standard mathematical notation where the productof two quantities a and b is written ab, meaning a× b).
So for the circumference C = πD, we have results as in Table 1. The (m) in brackets means thatthe units are metres. The ratios δD/D and δC/C are of course dimensionless and are calculatedwith D and C having their ‘central’ values. In Table 1 and the following three tables, suchdimensionless ratios are stated as percentages. For simplicity, the percentages are given to onlyone place after the decimal point, although this occasionally creates small round-off errors.
13
Table 1
D (m) C (m) δD (m) δD/D (%) δC (m) δC/C (%)0.950 2.985 −0.050 −5.0 −0.157 −5.00.990 3.110 −0.010 −1.0 −0.031 −1.00.998 3.135 −0.002 −0.2 −0.006 −0.20.999 3.138 −0.001 −0.1 −0.003 −0.11.000 3.142 0.000 0.0 0.000 0.01.001 3.145 +0.001 +0.1 +0.003 +0.11.002 3.148 +0.002 +0.2 +0.006 +0.21.010 3.173 +0.010 +1.0 +0.031 +1.01.050 3.299 +0.050 +5.0 +0.157 +5.0
For the area A = πD2/4 of the same circle, we have results as in Table 2. The units for A and forδA are square metres, m2.
Table 2
D (m) A (m2) δD (m) δD/D (%) δA (m2) δA/A (%)0.950 0.709 −0.050 −5.0 −0.077 −9.80.990 0.770 −0.010 −1.0 −0.016 −2.00.998 0.782 −0.002 −0.2 −0.003 −0.40.999 0.784 −0.001 −0.1 −0.002 −0.21.000 0.785 0.000 0.0 0.000 0.01.001 0.787 +0.001 +0.1 +0.002 +0.21.002 0.789 +0.002 +0.2 +0.003 +0.41.010 0.801 +0.010 +1.0 +0.016 +2.01.050 0.866 +0.050 +5.0 +0.081 +10.2
We now consider a sphere, also of diameter D =1 metre, and consider how sensitive its surfacearea As = πD2 might be to small changes in D. (The suffix s on As stands for ‘sphere’). Theresults are in Table 3.
Table 3
D (m) As (m2) δD (m) δD/D (%) δAs (m2) δAs/As (%)0.950 2.835 −0.050 −5.0 −0.306 −9.80.990 3.079 −0.010 −1.0 −0.063 −2.00.998 3.129 −0.002 −0.2 −0.013 −0.40.999 3.135 −0.001 −0.1 −0.006 −0.21.000 3.142 0.000 0.0 0.000 0.01.001 3.148 +0.001 +0.1 +0.006 +0.21.002 3.154 +0.002 +0.2 +0.013 +0.41.010 3.205 +0.010 +1.0 +0.063 +2.01.050 3.464 +0.050 +5.0 +0.322 +10.2
Finally, we consider the volume V = πD3/6 of this sphere, and the sensitivity of V to small changesin D is described in Table 4. The units of V and of δV are cubic metres, m3.
14
Table 4
D (m) V (m3) δD (m) δD/D (%) δV (m3) δV/V (%)0.950 0.449 −0.050 −5.0 −0.075 −14.30.990 0.508 −0.010 −1.0 −0.016 −3.00.998 0.520 −0.002 −0.2 −0.003 −0.60.999 0.522 −0.001 −0.1 −0.002 −0.31.000 0.524 0.000 0.0 0.000 0.01.001 0.525 +0.001 +0.1 +0.002 +0.31.002 0.527 +0.002 +0.2 +0.003 +0.61.010 0.539 +0.010 +1.0 +0.016 +3.01.050 0.606 +0.050 +5.0 +0.083 +15.8
3.2 Discussion of the tables
In all four tables, a diameter D is given a ‘central’ value of 1 metre, and is assigned smallmeasurement errors that deviate it from this central value. For example, the largest positive erroris 50 millimetres, giving D = 1.050 m, or a proportional error of +5%. The smallest absolute erroris 1 millimetre, giving D = 0.999 m or D = 1.001 m, so this proportional error is only −0.1% or+0.1% respectively. These percentage errors are listed in boldface in the fourth column of eachtable.
The diameter D can be considered to be an ‘input’ quantity, and in Table 1 the correspondingoutput quantity is the circumference C, the measurand. In Tables 2, 3 and 4 the same inputquantity D has corresponding outputs A (the area of the circle), As (the surface area of thesphere) and V (the volume of the sphere) respectively. The proportional errors are given in thesixth column of each table, also as percentages and also in boldface. The question of interest ishow these output percentage errors are related to the input percentage errors.
For the circumference of the circle (Table 1), it is seen that the output percentage errors are thesame as the input percentage errors. But for the area of the circle (Table 2) and area of the sphere(Table 3), the output percentage errors are twice the input percentage errors. For the volume ofthe sphere (Table 4), the output percentage errors are three times the input percentage errors. Wealso see that for a large percentage error such as 5% in the input, the output percentage error isperturbed slightly in the positive direction from twice the input percentage error (for the areas)and from three times the input percentage error (for the volume).
So we can say that the calculated circumference of the circle is sensitive to errors in measuringthe diameter. This is obviously so; how could it be otherwise? However, what is interesting is thatthe area of the circle, and the area of the sphere, are more sensitive, by a factor of two, to errorsin the diameter. The volume of the sphere is much more sensitive, by a factor of three, to errorsin the diameter.
Where does this extra sensitivity come from? We might guess that the relative factors of 1 (forthe circumference), 2 (for the areas) and 3 (for the volume) have something to do with the powersof 1, 2 and 3 respectively in the formulas for these quantities (the formula for the circumferenceC = πD can be written as C = πD1). That would be the correct guess! The following sectionexplains these factors in more detail.
3.3 The explanation for the increases in sensitivity
We first draw attention to two matters, the first one being an algebraic detail.
(1) Suppose a and b are two quantities that can be added (implying that they have the samedimensions and are measured in the same units). We wish to ‘expand’ the expressions (a+ b)2 and(a + b)3.
(a + b)2 = (a + b)× (a + b) = (a + b)(a + b) = a2 + ab + ba + b2 = a2 + 2ab + b2. (3.1)
(a+b)3 = (a+b)2×(a+b) = (a+b)2(a+b) = (a2+2ab+b2)(a+b) = a3+a2b+2a2b+2ab2+b2a+b3
= a3 + 3a2b + 3ab2 + b3. (3.2)
15
Note the numerical coefficients in front of the various terms. The original sum a+ b can be written1a + 1b. The square of this, (a + b)2 = a2 + 2ab + b2, can be written 1a2 + 2ab + 1b2. The cube,(a + b)3 = a3 + 3a2b + 3ab2 + b3, can be written 1a3 + 3a2b + 3ab2 + 1b3. So the coefficients followthe pattern 1, 1, then 1, 2, 1, then 1, 3, 3, 1 and so on: these are the ‘Pascal’s triangle’ numbers,which can be set out as in Table 5 where they appear in boldface. Each number (whether zero ornonzero) is the sum of the two numbers immediately above it diagonally to the left and diagonallyto the right. Evidently the table can be continued indefinitely with an increasing number of rowsand columns. We can use it to read off immediately what would be the expansion of, say, (a+ b)4.This is a4 + 4a3b + 6a2b2 + 4ab3 + b4. We also note in every case the sum of powers of the a’s andb’s. For the case (a + b)3, the sum of powers is 3: thus a3, a2b, ab2 and b3. For the case (a + b)4,the sum of powers is 4: a4, a3b, a2b2, ab3 and b4; and so on.
Table 5
0 0 0 0 1 0 1 0 0 0 00 0 0 1 0 2 0 1 0 0 00 0 1 0 3 0 3 0 1 0 00 1 0 4 0 6 0 4 0 1 01 0 5 0 10 0 10 0 5 0 1
(2) A proportional error such as δD/D, or δC/C, etc. as above is always quite small, meaningless than, say, 10%. These are ‘first-order’ errors. But the squares and higher powers of theseproportional errors are therefore very much smaller still. For example, if δD/D = 5% = 0.05,then (δD/D)2 = 0.0025 = 0.25%, and (δD/D)3 = 0.000125 = 0.0125%. In what follows, variousexpressions will be obtained for proportional errors, and squares and higher powers of proportionalerrors will be explicitly written out but then considered negligible. It might appear that thedifferential calculus should then be regarded as only an approximate procedure, but in fact one ofits key features, the notion of a ‘derivative’, is exact, as will be shown later.
The proportional errors for circumference of a circle, areas of a circle and sphere, and volumeof sphere, will be now formulated in terms of the proportional errors in the diameter.
(a) We haveC = πD (3.3)
as the general formula for the circumference C of a circle as a function of its diameter D. (This isof course also a definition of π). Then if D has an error δD (positive or negative) that takes it toa value D + δD, then C will have a corresponding error δC taking it to a value C + δC. However,the relationship between the diameter and circumference of a circle must still hold, and thereforewe have also
C + δC = π(D + δD). (3.4)
But since C = πD, inserting this into (3.4) gives
πD + δC = π(D + δD) = πD + π δD. (3.5)
The term πD cancels from both sides of (3.5), so this leaves
δC = π δD. (3.6)
To obtain proportional errors, we divide both sides of (3.6) by C, getting
δC
C= π
δD
C= π
δD
πD=
δD
D. (3.7)
In (3.7), C in the denominator has been replaced by its equivalent πD, and then the π cancelsfrom numerator and denominator. So we have
δC
C=
δD
D. (3.8)
This is consistent with the results of Table 1: the proportional errors in the circumference areequal to the proportional errors in the diameter.
16
(b) We haveA =
π
4D2 (3.9)
as the general formula for the area A of a circle as a function of its diameter D. Then if D has anerror δD that takes it to a value D + δD, A will have a corresponding error δA taking it to a newvalue A + δA. Since the relationship between diameter and area must still hold, we have
A + δA =π
4(D + δD)2. (3.10)
By means of a very similar procedure to that in (a), A in (3.10) is replaced by its equivalent π4 D2,
so that when we expand (D + δD)2 using (3.1), (3.10) gives
π
4D2 + δA =
π
4(D2 + 2DδD + (δD)2) (3.11)
and the term π4 D2 cancels from both sides of (3.11), leaving
δA =π
4(2DδD + (δD)2). (3.12)
To obtain proportional errors, we divide both sides of (3.12) by A = π4 D2, getting
δA
A=
π4 (2DδD + (δD)2)
π4 D2
, (3.13)
The term π4 cancels from the top and bottom of (3.13), leaving
δA
A=
2DδD + (δD)2
D2, (3.14)
soδA
A= 2
δD
D+
(δD
D
)2
. (3.15)
As stated in (2) above in this section, the term ( δDD )2 is much smaller than δD
D , and so ignoring itgives
δA
A= 2
δD
D. (3.16)
This is consistent with the results of Table 2: because of the factor 2 in (3.16), the proportionalerrors in the area are twice the proportional errors in the diameter. But it may be noted, further,that for the relatively high proportional error of +5% in Table 2, the proportional error in the areais not exactly 10% but more like 10.2%. Equ. (3.15) confirms this, because for δD
D = 0.05, (3.15)gives when we do not ignore the second-order term:
δA
A= 2× 0.05 + (0.05)2 = 0.10 + 0.0025 = 0.1025 = 10.25% ≈ 10.2%. (3.17)
The same calculation for an error −5% in the diameter gives a similar perturbation, again in thepositive direction, of the proportional error in the area from −10% to −9.8% (allowing for round-offerrors).
(c) We haveAs = πD2 (3.18)
as the general formula for the surface area As of a sphere as a function of its diameter D. Then ifD has an error δD that takes it to a value D + δD, As will have a corresponding error δAs takingit to a new value As + δAs. Since the relationship between diameter and area must still hold, wehave
As + δAs = π(D + δD)2. (3.19)
But this is obviously the same calculation as for the area of the circle in (b) above (see (3.10)),except for the factor 1
4 . So without further ado we can conclude that the proportional error in As
17
is twice the proportional error in D, just as in case (b), and indeed the third, fourth and sixthcolumns in Table 3 for this case (c) are identical to those in Table 2 for case (b).
(d) We have
V =π
6D3 (3.20)
as the general formula for the volume V of a sphere as a function of its diameter D. Then if D hasan error δD that takes it to a value D + δD, V will have a corresponding error δV taking it to anew value V + δV . Since the relationship between diameter and volume must still hold, we have
V + δV =π
6(D + δD)3. (3.21)
By means of a very similar procedure to the previous cases, V in (3.21) is replaced by its equivalentπ6 D3, so that when we expand (D + δD)3 using (3.2), (3.21) gives
π
6D3 + δV =
π
6
(D3 + 3D2δD + 3D(δD)2 + (δD)3
)(3.22)
and the term π6 D3 cancels from both sides of (3.22), leaving
δV =π
6
(3D2δD + 3D(δD)2 + (δD)3
). (3.23)
To obtain proportional errors, we divide both sides of (3.23) by V = π6 D3, getting
δV
V=
π6 (3D2δD + 3D(δD)2 + (δD)3)
π6 D3
, (3.24)
The term π6 cancels from the top and bottom of (3.24), leaving
δV
V=
3D2δD + 3D(δD)2 + (δD)3
D3, (3.25)
soδV
V= 3
δD
D+ 3
(δD
D
)2
+(
δD
D
)3
. (3.26)
As stated in (2) above, the term ( δDD )2 is much smaller than δD
D , and ( δDD )3 is much smaller still,
and so ignoring these two higher-order terms gives
δV
V= 3
δD
D. (3.27)
This is consistent with the results of Table 4: because of the factor 3 in equ. (3.27), the proportionalerrors in the volume are three times the proportional errors in the diameter. As previously it maybe noted, further, that for the relatively high proportional error of +5% in Table 4, the proportionalerror in the volume is not exactly 15% but more like 15.8%. Equ. (3.26) confirms this, because forδDD = 0.05, (3.26) gives when we do not ignore the higher-order terms:
δA
A= 3× 0.05 + 3× (0.05)2 + (0.053) = 0.15 + 0.0075 + 0.000125 = 0.1576 ≈ 15.8%. (3.28)
The same calculation for an error −5% in the diameter gives a similar perturbation, again inthe positive direction, of the proportional error in the area from −15% to −14.3% (allowing forround-off errors).
As might be expected, the relationships just illustrated between powers and sensitivities carryover into fractional powers: square roots, cube roots and so on. Here we note that the square rootis the same as the ‘one-half’th power (and, similarly, a cube root is the same as the ‘one-third’power). This can be seen from the rule that if w is any variable, then wm × wn = wm+n, for anynumbers m and n (the case w = m = n = 0 is excluded). For example, 42 × 43 = 42+3 = 45, or
18
16 × 64 = 1024. So w12 × w
12 = w
12+1
2 = w1 = w, so w12 multiplied by itself gives w; in other
words, w12 is the same as
√w.
Thus for example, suppose that we were given the area A of a circle, and wished to inferits diameter D. Now we have the ‘inverse’ problem: A is the input, and D is the output. The
relationship A = π4 D2 must now be rewritten D =
√4Aπ , or, equivalently, D =
(4π
)12 × A
12 . The
dependence of D on the square root of A implies that a proportional error of, say, 0.4% in thearea creates a proportional error of only 0.2% in the diameter. This is exactly what we see in theseventh row (below the heading row) of Table 2.
In sections 3.1, 3.2 and 3.3 above, single assigned errors, (denoted by δ) of various magnitudesand of either sign, were assigned to the diameter. There was increasing sensitivity of circumference,area and volume, in that order, to these errors in the diameter. Uncertainty in measurement isthe effect of a swarm of errors, and so the uncertainty in, say, the measurement of the volume ofa sphere in terms of its measured diameter is highly sensitive to the uncertainty in the diameter.By contrast, the uncertainty in the circumference of a circle is much less sensitive to uncertaintyin its diameter. These contrasts in sensitivity were determined by the methods of the differentialcalculus.
19
4 The notion of ‘differentials’ and ratio of differentials
In the description above, the small quantities with δ in front of the symbol are known as‘differentials’, because they represent a small difference in a quantity. They are also sometimescalled ‘increments’, a term which strictly implies a small difference that moves the quantity inquestion to a slightly more positive value. (‘Decrement’ is the equivalent term for a negativechange). Thus if T denotes a temperature of 20oC, δT might be, say, 0.1oC. We now introducea new notion, namely the ratio of such differentials, in other words one differential divided byanother differential. (In section 3 above, no such ratio was explicitly stated). The key insight hereis that while the differentials themselves are small quantities, their ratio need not be. For example,if one differential is, say, one-thousandth or 0.001 (a small number relative to, say, 1), and anotherdifferential is 0.002, still a small number (although twice as large), their ratio 0.001
0.002 or 0.001/0.002is the much larger number 1
2 .
Equ. (3.6) above stated:δC = π δD.
Dividing both sides by the differential δD gives
δC
δD= π. (4.1)
Equ. (3.12) above stated:δA =
π
4(2DδD + (δD)2).
Dividing both sides by the differential δD gives
δA
δD=
π
4(2D + δD). (4.2)
Similarly, as shown in (3.18) and (3.19), for the area of the sphere we have the same equation as(4.2), basically, but without the factor 1
4 :
δA
δD= π (2D + δD). (4.3)
Finally, for the volume of the sphere, (3.23) stated:
δV =π
6(3D2δD + 3D(δD)2 + (δD)3).
Again, dividing both sides by the differential δD gives:
δV
δD=
π
6(3D2 + 3DδD + (δD)2). (4.4)
We now look at some graphs of the above equations.
Fig. 1(a) illustrates the relationship between the circumference C and diameter D of a circle,as in (3.3) above:
C = πD.
On a graph of C on the vertical axis against D on the horizontal axis, this relationship is thestraight line shown in Fig. 1(a), with a slope of π = 3.14159.... This is a steep angle of about72.34o denoted by the Greek letter θ (theta) in Fig. 1(a). Fig. 1(a) has approximately equalscales on the horizontal and vertical axes, and it may be checked that the straight line indeed hasapproximately that slope. The line goes through the origin (0, 0), which it must do since a circlewith zero diameter has zero circumference! An arbitrary point P is shown on the line, at diameterD (the distance of P horizontally from the vertical axis), and at corresponding circumference C(the distance of P vertically from the horizontal axis). Now a small deviation δD from P in thehorizontal direction is made and is shown in Fig. 1(a). This is the point Q, which is not on thestraight line and which is at a distance D + δD horizontally from the vertical axis. So PQ = δD.But since the same relationship between diameter D and circumference C must hold (namely,
20
0 2 4 6 8 100
2
4
6
8
10
Cir
cu
mfe
ren
ce
C
Diameter D
P Q
R
D
C
C
D
Fig. 1(a): Circumference plotted against diameter of circle
C = πD), the point on the line corresponding to D + δD must be the point R, which is at adistance C + δC vertically from the horizontal axis. So RQ = δC. Consider the small trianglePQR, where the angle RQP is a right angle. To summarise the above, the base of this triangleis PQ = δD and its height is RQ = δC. The angle RPQ must be the same angle θ = 72.34o,basically because the straight line denoting C = πD has constant slope (otherwise it would not bea straight line!). The tangent of the angle θ is δC
δD and, as we have seen (equation (4.1)), has thevalue π.
The tangent as a trigonometrical function will be discussed in section 12, but we can digressfor a moment and describe the tangent of any angle very generally as the ratio ‘opposing sidedivided by adjacent side’ when these two sides meet in a right angle. For example, in the largertriangle in Fig. 1(a) the opposing side to the angle θ is C and the adjacent side is D, and in thesmaller triangle – an exact scaled copy of the larger, so a similar triangle in the technical sense –the opposing side is RQ = δC and the adjacent side is PQ = δD. The ratios are the same in thelarger as in the smaller triangle, therefore the tangents are the same and this is obviously so sinceas indicated in Fig. 1(a) the same angle θ is present in both triangles. We see moreover that if thetwo sides happen to be equal, the angle must be 45o and so tan 45o = 1. Also we see that tan 90o
must be infinite, since then the opposing side will be infinitely long. Fig. 1(b) shows the samesituation but with bigger increments δD and δC. But the new and larger triangle PQR is similarto the triangle PQR in Fig. 1(a). So it is still true that δC
δD has the value π.
Now imagine that, instead of Fig. 1(b) showing larger increments, it showed very much smallerincrements. (No attempt is made to depict this). The triangle PQR would become tiny but –another key insight, but a fairly obvious one – the relationship δC
δD would still have the value π.(Remember the remark made previously that the ratio of two tiny quantities need not itself betiny; in fact here it has the value π, or a bit more than three!) With very small increments, tendingin the limit to zero increments, the relationship δC
δD is denoted by the ordinary Roman letter ‘d’,instead of by δ. So, in the limit, if
C = πD (4.5)
21
0 2 4 6 8 100
2
4
6
8
10
Cir
cu
mfe
ren
ce
C
Diameter D
P Q
R
D
C
C
D
Fig. 1(b) Fig. 1(a) with larger increments
then we have:dC
dD= π. (4.6)
(Note that just as in the case of δ, dC does not mean ‘d multiplied by C). The mathematicalstatement corresponding to (4.5) and (4.6) is that, if C = πD, then the derivative of C with respectto D is π. This derivative is a measure of the ‘rate of change of C with D’, at the point P.
The relationship C = πD is a linear relationship (because its graph is a straight line, as inFigs. 1(a) and (b)). From this, it follows that dC/dD = π. But obviously any linear relationshipbetween two quantities will have a similar derivative form. For example, suppose an object ismoving so that the distance y it covers in a time t is given by the relationship
y = Kt, (4.7)
where K is a constant. Thendy/dt = K. (4.8)
K is none other than the (constant) speed of the object. Fig. 2(a) is a graph of y against t withtwo values of speed K, a constant low speed K = 1 and a constant higher speed K = 2. Themotion of the object is shown by the straight lines through the origin, with a low slope equal to thelower speed K = 1, and a higher slope equal to the higher speed K = 2. A high slope representsa high speed since, for a given distance t along the horizontal axis, the high-slope line reaches ahigher value of y along the vertical axis. Thus, at time t = 4 units (for example), the line y = Ktwith K = 1 has reached the point y = 4, whereas the line y = Kt with K = 2 has reached thepoint y = 8. Moreover, if the relationship between y and t involved another constant L in the formy = L + Kt (so that the graph of x against t would now not pass through the origin (0, 0)), thederivative dy/dt would still be equal to K; the derivative of the constant L (as of any constant) iszero. The presence of the constant L does not affect the speed of the object, but indicates that attime t = 0 the position of the object was not 0 but L. Fig. 2(a) also shows the line y = L + Ktfor L = 3, K = 2, and this line is parallel to the line y = Kt for K = 2, indicating that the objecthas the same speed in both cases.
22
0 2 4 6 8 100
2
4
6
8
10
Dis
tan
ce
y
Time t
y = Kt, K=1
y = Kt, K=2
y = Kt + L
K = 2L=3
Fig. 2(a): Distance-time relationship for constant speed
Fig. 2(b) is a graph of the speed, dydt = K, against time t for K = 1. The speed being constant,
the graph is a horizontal line parallel to the time axis.
Fig. 3 illustrates the relationship between the area A of a circle and its diameter D, as in (1.9)above:
A =π
4D2.
On a graph of A on the vertical axis against D on the horizontal axis, this relationship is the curveshown in Fig. 3, again passing through the origin (0, 0). The relationship of A to D is not linear,because D occurs as a squared term; hence we do not expect a straight line as in Figs. 1. Anarbitrary point P is shown on the curve, at diameter D (the distance of P horizontally from thevertical axis), and at corresponding area A (the distance of P vertically from the horizontal axis).A small deviation δD from the horizontal direction is made and is shown in Fig. 3 as the pointQ. The point Q is not on the curve and is at a distance D + δD horizontally from the verticalaxis. The same relationship A = π
4 D2 between area A and diameter D must hold, so the point onthe curve corresponding to D + δD must be the point R, which is at a distance A + δA verticallyfrom the horizontal axis. Now in contrast to Fig. 1(a), the shape PQR is not an exact triangle,because one side (from P to R) is slightly curved. However, we can still work out the ratio δA
δD ,the counterpart to the ratio δC
δD in Fig. 1(a). This ratio was given in equ. (4.2), repeated here:
δA
δD=
π
4(2D + δD).
In contrast to (4.1) for the circumference of the circle, we now have, instead of a constant and novariables on the right-hand side (π in 4.1), the variables D and D + δD. This is why the side PRof the ‘triangle’ PQR on Fig. 2(a) is curved. Increasing δD, as previously in going from Fig. 1(a)to Fig. 1(b), will not give a similar but enlarged ‘triangle’. However, if δD is made smaller andstill smaller, then PQR approaches an exact triangle, because the curve joining P to R becomeseffectively a short but straight line. (For any ‘well-behaved’ curve, no matter how ‘curved’, a shortenough section of it will be indistinguishable from a straight line). But as δD becomes shorter and
23
0 2 4 6 8 100
2
4
6
8
10
Sp
ee
d d
y/dt
Time t
Fig. 2(b) Speed-time relationship for constant speed K = 1
shorter, it becomes negligible relative to D on the right-hand side of (4.2), and so (4.2) becomes:
δA
δDapproaches
dA
dD=
π
42D =
π
2D, (4.9)
so the derivative of A = π4 D2 with respect to D is π
2 D. The derivative is a measure of the ‘rate ofchange of A with D’ at the point P.
Fig. 4 illustrates the relationship between the area As of a sphere and its diameter D, as in(3.18):
As = πD2. (4.10)
Except for the factor 14 , the situation in Fig. 4 is just like that in Fig. 3. We now have instead of
(4.9):δAs
δDapproaches
dAs
dD= π2D = 2πD, (4.11)
so the derivative of As = πD2 with respect to D is 2πD.Fig. 5 illustrates the relationship between the volume V of a sphere and its diameter D, as in
(3.20):V =
π
6D3. (4.12)
This is again a curve, and once again we can measure the rate of change of the volume withdiameter by the ratio of δV to δD at the point P on the curve. This ratio, δV
δD , was given in (4.4)above:
δV
δD=
π
6(3D2 + 3DδD + (δD)2). (4.13)
Again, we allow the δ’s to become smaller and smaller, implying both a smaller δD and aconsequently smaller δV . Then, on the right side of (4.13), we are left with
δV
δDapproaches
dV
dD=
π
63D2 =
12πD2, (4.14)
so the derivative of V = π6 D3 with respect to D is 1
2πD2.
24
0 2 4 6 8 100
2
4
6
8
10
Are
a A
Diameter D
PD
A
Q
R
D
A
Fig. 3: Area plotted against diameter of circle
0 1 2 3 4 50
2
4
6
8
10
Sur
face
are
a A
s
Diameter D
PQ
R
D
As
As
D
Fig. 4: Surface area plotted against diameter of sphere
25
0.0 0.5 1.0 1.5 2.0 2.5 3.00
2
4
6
8
10
Vol
ume V
Diameter D
P Q
R
D V
V
D
Fig. 5: Volume plotted against diameter of sphere
26
5 The notion of a function
A ‘function’ links a measurand (or output quantity) to an input quantity or several such inputquantities. The measurand is usually written on the left, and the input(s) on the right. We denotethe function generally by y, the measurand, or equivalently by f(x), where x is the input. (Formore than one input, the general form would of course be f(x1, x2, ...etc).
For example, suppose a particular functional form was: f(x) = 1 − x + 2x2. Then f(0) = 1:we simply plug in the value 0 for x. Similarly, f(1) = 2, f(2) = 7, f(−1
2 ) = 2, and so on.
At a given point (x, y) on a function, a small change δx and its corresponding δy have a ratioδyδx that approaches the derivative dy
dx as δx and δy decrease and become tiny. At any point ona curve, the ‘tangent-line’ is that line that ‘touches’ the curve at only that point. Obviously theslope of the tangent-line is a measure of the ‘steepness’ of the curve at that point. In (say) Fig.3, where A plays the part of y and D of x, the decrease of δD and, therefore, of δA, imply that Rmoves closer and closer to P. Then the tangent-line becomes practically indistinguishable from thevery short stretch of curve joining P to R. The slope of the tangent-line at P is dA
dD . The measureof the slope is the angle θ as shown in Fig. 3, and like any angle, θ has a tangent, tan θ, whichowes its name ‘tangent’ to the very reason that tan θ = tan angle of slope = dA
dD . The tangentfunction (along with the sine and cosine functions) will be discussed further in section 12.
An important relationship between the derivative of y with respect to x, namely dydx , and the
differentials δy and δx, is as follows. As mentioned earlier (for example, in (4.9)), the ratio ofdifferentials δy
δx approaches the derivative dydx when the differentials become very small. But the
ratio of differentials is just an ordinary fraction; therefore
δy
δx≈ dy
dx(5.1)
implies that we can say, upon multiplying both sides of (5.1) by δx:
δy ≈ dy
dxδx. (5.2)
This is an approximate formula: the error in the output y is approximately the derivative multipliedby the error in the input x. The GUM in its presentation of the propagation formula (to beintroduced and discussed in sections 15 and 16) makes use of just this relationship (5.2). Thelarger the derivative, the more the output is sensitive to the input, and it makes sense to call thederivative the sensitivity or the sensitivity coefficient. For example, in the case of the sphere asconsidered in the previous section, equ. (4.14) can be written in the form corresponding to (5.2):
δV =12πD2δD, (5.3)
This indicates a high sensitivity of δV to δD, since the derivative involves the square of the diameterD. (Note that (5.3) is consistent with the previously determined (3.27), which was δV
V = 3 δDD .
This consistency can be checked by substituting for V the volume 16πD3 of a sphere of diameter
D).
It is important to note that a derivative must be evaluated at the particular relevant value ofthe input x. If the derivative does not itself involve x, then of course there is no particular relevantvalue: this happens when the relationship between input x and outut y is linear, as in y = Kx+L
with K, L constants, giving dydx = K. In all other cases, the value of the derivative will depend
on the actual value of the input x, and this must be known and inserted into the value of thederivative.
In sections 3.3 and 4, we had the following functions, all of which are functions of only oneinput, the diameter D:
C = πD, where the measurand is the circumference C of a circle. We obtained for the derivativeof C with respect to D: dC
dD = π (see 4.1).
A = π4 D2, where the measurand is the area A of a circle. We obtained for the derivative of A
with respect to D: dAdD = π
2 D (see 4.9).
27
As = πD2, where the measurand is the surface area As of a sphere. We obtained for thederivative of As with respect to D: dAs
dD = 2πD (see 4.11).
V = π6 D3, where the measurand is the volume V of a sphere. We obtained for the derivative
of V with respect to D: dVdD = π
2 D2 (see 4.14).
Is there a pattern that would enable us, given any one of the functions above, immediately towrite down its derivative? Yes: in each case we observe that the power of D comes down to thefront as a multiplying factor, and then the power is reduced by 1. For example, if A = π
4 D2, thepower 2 to which D is raised becomes a multiplying factor 2, and the power becomes 2 − 1 = 1:thus dA
dD = 2× π4 D1 = π
2 D. In the case of the circumference C = πD, this can be written C = πD1,and the same rule applies here, giving dC
dD = 1× πD0 = π (since any quantity, except zero, raisedto the power zero gives 1).
This rule is obviously much more general than the case of the circle and sphere.
5.1 Derivative of y = xn when n is a positive integer
For any functiony = xn, (5.4)
where x is the input giving an output y, and n is an integer, we have, as just stated above, for thederivative dy
dx of y with respect to x:dy
dx= nxn−1. (5.5)
To understand how this comes about, we do the following operation (by now a routine one!). Wechange x by a small amount δx, so that x moves to a new value x+ δx. The output y will respondby changing by a corresponding amount δy and so moving to a new value y + δy. Since the samerelationship as (5.4) must still apply to the input and output, we now have:
y + δy = (x + δx)n. (5.6)
The next step is to ‘expand’ the right-hand side of (5.6), (x + δx)n. This is where we can use thetriangular numbers in Table 5. These numbers are: 1,1; then 1,2,1; then 1,3,3,1; then 1,4,6,4,1,then 1,5,10,10,5,1 etc.
Suppose for example that n = 4; then we can write down immediately (x+δx)4 = x4 +4x3δx+6x2(δx)2 + 4x(δx)3 + (δx)4. The coefficients are 1,4,6,4,1. It is also worth noting that the sum ofpowers of x and of δx for each of the five terms must be 4.
Before proceeding any further, we can check that this expansion of (x + δx)4 is indeed correct.We can start by expanding (x+ δx)2 = x2 +2xδx+(δx)2 (the coefficients being 1,2,1, and we notethat in each term the sum of powers of x and of δx is 2). Then (x + δx)3 = (x + δx)2(x + δx) =(x2 + 2xδx + (δx)2)(x + δx). Multiplying this out gives:
(x2 + 2xδx + (δx)2)(x + δx) = x3 + x2δx + 2x2δx + 2x(δx)2 + x(δx)2 + (δx)3, (5.7)
so collecting all the like terms gives
(x2 + 2xδx + (δx)2)(x + δx) = x3 + 3x2δx + 3x(δx)2 + (δx)3. (5.8)
The coefficients on the right side of (5.8) are 1,3,3,1 and we note that in each of the four terms thesum of powers of x and of δx is 3. Finally (for the case n = 4) we have
(x + δx)4 = (x3 + 3x2δx + 3x(δx)2 + (δx)3)(x + δx) = x4 + x3δx + 3x3δx + 3x2(δx)2+
+3x2(δx)2 + 3x(δx)3 + x(δx)3 + (δx)4 (5.9)
so collecting all the like terms gives:
(x + δx)4 = x4 + 4x3δx + 6x2(δx)2 + 4x(δx)3 + (δx)4. (5.10)
The coefficients on the right side of (5.10) are 1,4,6,4,1, and the sum of powers of x and of δx is 4for each of the five terms in (5.10). So, for the particular case n = 4, (5.6) may be written
y + δy = (x + δx)4 = x4 + 4x3δx + 6x2(δx)2 + 4x(δx)3 + (δx)4. (5.11)
28
Putting y = x4 in (5.11) (the particular case of (5.6) with n = 4) gives
x4 + δy = (x + δx)4 = x4 + 4x3δx + 6x2(δx)2 + 4x(δx)3 + (δx)4 (5.12)
and x4 cancels from the left and right sides of (5.12). This leaves:
δy = 4x3δx + 6x2(δx)2 + 4x(δx)3 + (δx)4, (5.13)
so the ratio of differentials that we are after, namely δyδx , is available from (5.13) simply by dividing
both sides by δx. This gives:
δy
δx= 4x3 + 6x2(δx) + 4x(δx)2 + (δx)3. (5.14)
Now we make δx smaller and still smaller. The second, third and fourth terms on the right-handside of (5.14) start to vanish, compared with the first term on the right-hand side, which does notat all reduce in size. Moreover, although there is a δx in the bottom of the fraction on the left-handside of (5.14), and this δx of course also becomes smaller and smaller, so does δy in the top of thefraction, since δy is the response to a change δx. If this change δx in x is very small, so is thechange δy in y. We remember that, as stated earlier (first paragraph of section 2), the ratio of twosmall quantities need not itself be a small quantity. As δx approaches zero, (5.14) approaches thederivative dy
dx of y with respect to x:dy
dx= 4x3, (5.15)
a particular case of (5.5) with n = 4.
It is worth taking a ‘bird’s eye-view’ of the procedure that leads from y = xn to the derivativedydx = nxn−1. In the set of triangular numbers in Table 5, the first non-zero number in any row is1 and the second non-zero number is n where n is the row number. This is the key fact that givesus the coefficient n in front of xn−1 in the expression for the derivative of y = xn with respect tox. The expansion of the expression (x + δx)n has coefficients which follow the series of non-zeronumbers in row n in the table. For example, if n = 4 the coefficients are 1,4,6,4,1 in the terms1x4, 4x3δx, 6x2(δx)2...etc. But the first term 1x4 cancels (as seen above in going from (5.12) to(5.13)), and then the second term 4x3δx becomes just 4x3 when δy on the left is divided by δx.So this second term is, in effect, the required derivative. The third, fourth ....etc terms still involvea δx or higher powers of δx and so vanish when δx is made very small and when ‘in the limit’ δxbecomes zero.
In summary, ify = xn, (5.16)
thendy
dx= nxn−1 (5.17)
and this expression in (5.17) for the derivative is exact. The route towards this expression involvedsmall changes denoted by δ, giving this procedure a kind of ‘approximate’ or even ‘rough-and-ready’ feel. But ultimately the small changes are allowed to go to zero, while the ratio of smallchanges does not go to zero (in general), and ‘in the limit’ of zero changes, the exact expressionfor the derivative is obtained.
The derivative at a given point on a curve is the slope of the line that ‘just touches’ the curve atthat point. That line, as previously noted (third paragraph in section 5), is called the tangent-lineto the curve at that point. If the curve happens to be a straight line, then of course the derivativeis the slope of the line regardless of the point at which the derivative is calculated. It may be worthrepeating that the slope, and equivalently the derivative, is also the tangent of the angle that thetangent-line makes with the horizontal.
Three further remarks should be made about the derivative of a function:
(a) If the function linking y and x is of the form y = Bxn, where B is a constant, thendydx = Bnxn−1. The B remains unaltered as a multiplying factor for both the original function and
29
0.0 0.4 0.8 1.2 1.6 2.00
2
4
6
8
10
Dis
tanc
e D
Time t
Fig. 6(a): Distance-time relationship for constant acceleration g = 10
its derivative. This rule was of course implicitly followed in the descriptions above for the circleand sphere, where (for example) for the circle B = π and for the volume of the sphere B = π
6 .
(b) If y is the sum of two or more functions, for example y = 3x4 + 7x2, the derivative istaken of each component function and the derivatives themselves are summed. In this example,dydx = 12x3 + 14x, where of course rule (a) has also been used.
(c) The derivative of a constant is zero: obvious since a constant has zero rate of change. So ifone (or more) of the summed functions in (b) above is actually a constant, it disappears when thederivative is taken. For example, if y = 6 + 5x3, then dy
dx = 15x2.
Figs. 2(a) and (b) illustrated the relation y = Kt for the distance y travelled by an object ata constant speed K. Fig. 6(a) illustrates the relation y = 1
2gt2, with g a constant, for an objectwhose speed is not constant: this is because the time t occurs as a squared, instead of as a linear,term. The derivative of y with respect to t is now, following (5.17) and rule (a) above, dy
dt = gt.This is a straight line as illustrated in Fig. 6(b): the speed is not constant, but increases at auniform rate. We can take the derivative of dy
dt = gt itself with respect to t. This is
ddt
(dy
dt
)= g, (5.18)
which we recognise as the constant acceleration of the object. Because it is constant, itsrepresentation in Fig. 6(c) is a straight line parallel to the time-axis. In Figs. 6(a), (b) and(c), g is given the value 10, because if distance is measured in metres and time in seconds, theng = 10 is close to the acceleration due to gravity at the Earth’s surface. As a matter of notation,the quantity on the left side of (5.18) is usually written as:
d2y
dt2
and is called the second derivative of y with respect to t. We note the superscripts: on the top itis d (not y) that has the superscript 2, while on the bottom it is t that has the power 2, denotingthe square of t. The positioning of the ‘2’ reflects the fact that the units of the second derivativeare metres per second per second (or ‘metres per second squared’) (assuming distance is measuredin metres and time in seconds), so distance occurs in linear fashion, but time as a squared term.
At the moment, n is a positive integer in the relation y = xn. This requirement will be relaxedin what follows.
5.2 Derivative of y = xn when n is a positive fraction
Suppose we wish to find dydx when y =
√x. This relationship can be written y = x
12 , because, as
shown earlier (third last paragraph in section 3.3), the square root is the same as the ‘one-half’th
30
0.0 0.4 0.8 1.2 1.6 2.00
4
8
12
16
20
Spee
d
Time t
Fig. 6(b) Speed-time relationship for constant acceleration g = 10
0 2 4 6 8 100
4
8
12
16
20
Acc
eleration
Time t
Fig. 6(c) Acceleration-time relationship for constant acceleration g = 10
power. But if y = x12 , then applying (5.17), ignoring the fact that n is non-integer, would surely
givedy
dx= (1/2)x−
12 . (5.19)
Is this correct? Yes, in fact equ. (5.17) applies to non-integer n as well as to integer n. This can
be illustrated as follows for the case y = x12 , because, squaring both sides, this relationship can
equally well be written y2 = x or x = y2.
A rule to note here, since it will be used below, is that for any quantities w and m (exceptw = m = 0), we have the identity w−m = 1
wm . For example, 5−2 = 152 = 1
25 . This can beseen using the same addition rule for powers stated in the first paragraph of this section. Thus5−2 × 52 = 5−2+2 = 50 = 1, so 5−2 must be the reciprocal of 52 = 25, namely, 5−2 = 1
25 .
Now we can apply (5.17) (nothing mathematically illegal here, although the roles of ‘input’ and‘output’ have been reversed). We get from x = y2:
dx
dy= 2y, (5.20)
and it is permissible to invert both left and right sides of (5.20), which gives:
dy
dx=
12y
=1
2x12
= (1/2)x−12 . (5.21)
31
So the rule dxn
dx = nxn−1 still holds true. We can check it again using another example: what
is the derivative of y with respect to x if y is the cube root of x? So now y = x13 . We expect
that dydx = 1
3x−23 = 1
3
(1/x
23
). This is correct, and can be shown in a similar manner to the case
y =√
x. We now havex = y3, (5.22)
whencedx
dy= 3y2 = 3x
23 , (5.23)
since if y = x13 , then taking squares, y2 = x
23 . Inverting (5.23) verifies what we expect:
dy
dx=
13x−
23 . (5.24)
We next consider a function such as y = x32 , so y is the cube of the square root of x. Again
applying (5.17) would give us dydx = 3
2x12 . This is indeed correct, and can be shown to be so by
defining a new variable w = x3. Then y = w12 , and so dy
dw = 12w−
12 . Also dw
dx = 3x2. We now usea form of what is called the ‘chain rule’: it goes:
dy
dx=
dy
dw
dw
dx. (5.25)
Equ. (5.25) can be regarded, for our purposes, as simply a case of the familiar manipulation offractions that, for example, yields the identity: 5
8 = 53 × 3
8 . The 3 cancels from the bottom of thefirst fraction and the top of the second. Then using (5.25) and the expressions for dy
dw and dwdx :
dy
dx=
12w−
12 × 3x2 =
32
x2
w12
=32
x2
x32
= (3/2)x2−(3/2) = (3/2)x12 . (5.26)
5.3 Derivative of y = xn when n is a negative fraction
Again we take a particular case, say y = x−13 . Equ. (5.17) would imply that dy
dx = −(1/3)x−43 ,
and again, this is the correct result. To see this, we note that y = x−13 is equivalent to y = 1
x13
or y3 = 1x . Then x = 1
y3 = y−3, so anticipating the result of section 5.4 (the n being a negative
integer), we obtain dxdy = −3y−4 = −3(x−
13 )−4 = −3x
43 . Inverting this gives dy
dx = −(1/3)x−43 . So
assuming that this particular case of a minus one-third power also works for other negative fractions– and this is easily checked – we conclude that (5.17) is applicable to n a negative fraction.
5.4 Derivative of y = xn when n is a negative integer
We take the particular case y = x−3, expecting (5.17) to deliver dydx = −3x−4. This again is
correct. If y = x−3, then inverting both sides gives x3 = 1y , so taking cube roots gives x =
(1y
)13 =
y−13 . This is now the case in section 5.3 above, with ‘n’ a negative fraction, and we have seen that
(5.17) is still valid. So dxdy = −(1/3)y−
43 , and substituting y = x−3 gives dx
dy = −(1/3)(x−3)−43 =
−(1/3)x+4, so inverting again gives dydx = −3x−4 as expected.
The particular case n = −1 is interesting since it can be examined in a simple and straightfor-ward way using differentials. With y = x−1 = 1
x , we increase x by a small amount δx, so that it
32
goes to x + δx, and y accordingly goes to y + δy. Then since the relationship (a reciprocal one)between y and x must still hold, we have
y + δy =1
x + δx. (5.27)
Subtracting y (by now a routine follow-up operation!) from both sides of (5.27) gives
δy =1
x + δx− y =
1x + δx
− 1x
. (5.28)
The two fractions on the right-hand side of (5.28) can be rearranged over a common ‘denominator’x(x + δx), so that
δy =x− (x + δx)x(x + δx)
=−δx
x(x + δx). (5.29)
Thereforeδy
δx=
−1x(x + δx)
(5.30)
and as δx goes to zero, the right-hand side of (5.30) goes to −1x2 and the left-hand side goes to the
derivative dydx . So
dy
dx=−1x2
= −x−2 if y = x−1, (5.31)
and so the case n = −1 obeys the rule dydx = nxn−1 if y = xn.
To summarise sections 5.1 to 5.4: if y = xn, then dydx = nxn−1 holds for n a positive or negative
integer or a positive or negative fraction. The rule also applies to the case n = 0, since then y = 1(unless x = 0), and the derivative of a constant is zero. So the rule is pretty general.
Taking the derivative of a function with respect to a variable is also known as differentiatingthat function with respect to the variable.
33
6 The exponential function
At this point an interesting question may be asked. Is there a function whose derivative is thesame as the function? The functional dependence of an output y on an input x can be denotedvery generally as:
y = f(x). (6.1)
So to answer that question, we search for a function f(x) such that
dy
dx= y, (6.2)
or of course equivalentlydf(x)
dx= f(x), (6.3)
which is to hold for all values of x.
We assume that f(x) can be written as a ‘power series’ in the variable x:
y = f(x) = a0 + a1x + a2x2 + a3x
3 + a4x4 + a5x
5 + ...... (6.4)
where all the a’s are constants, whose values are to be determined by the condition expressed in(6.3).
Differentiating (6.4) with respect to x, using (5.17) and rules (a) and (b) above (just after(5.17)) gives:
df(x)dx
= a1 + 2a2x + 3a3x2 + 4a4x
3 + 5a5x4 + ... (6.5)
If (6.3) is to hold for all values of x, then the coefficient of each power of x in (6.5) must be thesame as the coefficient of the same power of x in (6.4). So we have:
a1 = a0, 2a2 = a1, 3a3 = a2, , 4a4 = a3, 5a5 = a4, .... (6.6)
where the left-hand side in each of these equalities is taken from (6.5), and the right-hand sidefrom (6.4). But if 2a2 = a1 and a1 = a0, then a2 = (1/2)a0. If 3a3 = a2 and a2 = (1/2)a0, thena3 = (1/3)a2 = (1/6)a0. If 4a4 = a3 and a3 = (1/6)a0, then a4 = (1/24)a0.
So all the a’s can be expressed as a fraction of a0. These fractions are, successively, (1/1) = 1(for a1), (1/2) (for a2), (1/6) (for a3), (1/24) (for a4), and so on. The numbers 1, 2, 6, 24... maybe recognised as the ‘factorials’, denoted by the exclamation mark ! The factorial of the numbern!, where n is a positive integer, is defined as the product n× (n− 1)× (n− 2)× (n− 3)....× 2× 1.So 2! = 2× 1 = 2, 3! = 3× 2× 1 = 6, 4! = 4× 3× 2× 1 = 24, and so on.
So the required function looks like this:
f(x) = a0
(1 + x +
x2
2!+
x3
3!+
x4
4!+ ....
)(6.7)
It is easily checked that differentiating (6.7) term by term and again using (5.17) yields the samefunction f(x).
Is there a short-hand way of referring to this function? In a trivial way, yes. Simply call it, say,H(x), where by universal agreement H(x) is given by the right-hand side of (6.7)! But in fact theshort-hand has an actual specific functional form, which we can discover as follows.
We express the function of x in (6.7) as the same function of another variable y, and then wemultiply the two functions together:
f(x)f(y) = a0
(1 + x +
x2
2!+
x3
3!+
x4
4!+ ....
)×
×a0
(1 + y +
y2
2!+
y3
3!+
y4
4!+ ....
)(6.8)
34
To see where this multiplication leads, let us simply take the first few terms, say up to and includingthe quadratic terms such as x2, y2 or xy. (xy is also considered a quadratic term since the sumof the powers of x and y is 1 + 1 = 2, just like the power 2 of x2 and the power 2 of y2). So weignore all cubic and higher-order terms and get:
f(x)f(y) ≈ a20
(1 + x +
x2
2!+ y + xy +
y2
2!
), (6.9)
but (remembering that 2! = 2) this is the same as
f(x)f(y) = a20
[1 + (x + y) + ((1/2)x2 + xy + (1/2)y2)
]. (6.10)
and the term on the right-hand side of (6.10) can be written
f(x)f(y) = a20
[1 + (x + y) +
(x + y)2
2!
], (6.11)
and the sum (x+ y) appears to be playing the same role as x or y did in the original function f(x)(see (6.7)) or f(y) respectively. This is borne out if we continue the multiplication by includingcubic and higher-order terms:
f(x)f(y) = a20
(1 + (x + y) +
(x + y)2
2!+
(x + y)3
3!+
(x + y)4
4!+ ...
)= a0f(x + y). (6.12)
So the function f(x) must be such that the same function of y, f(y), when multiplied by f(x) givesagain basically the same function (multiplied by a0) but with (x + y) as the argument, namelyf(x + y). (‘Argument’ is used in the mathematical sense: as the variable(s) inside the bracketsimmediately after the function symbol. It does not denote a dispute!)
What kind of function has this property? It cannot be, say, f(x) = x2, because then we wouldhave f(x)f(y) = x2y2, whereas a0f(x+y) = a0(x+y)2, which is different from x2y2. The clue liesin the third-last paragraph in section 3.3, where the example was given of 42 × 43 = 42+3. Withthe same ‘base’, namely 4 in this case, we can add up the powers when multiplying: just what isrequired in the present case. So it looks as if f(x) must have the form:
f(x) = a0bx, (6.13)
where
bx =(
1 + x +x2
2!+
x3
3!+
x4
4!+ ....
), (6.14)
and b itself is a number whose value will be determined in a moment. Meanwhile, if f(x) = a0bx,
then f(y) = a0by and f(x)f(y) = a2
0 bx+y as required.
To find the number b, we note that for the particular case x = 1, we have f(x) = f(1) = a0b1 =
a0b and now we define (by convention) a0 = 1. So f(1) = b and putting x = 1 in (6.14) gives:
b =(
1 + 1 +12!
+13!
+14!
+ ...
)
= 2 + (1/2) + (1/6) + (1/24).... (6.15)
and after adding a few more terms, such as (1/5!) = (1/120), (1/6!) = (1/720), etc, we can checkthat this series converges rapidly to the value 2.71828.... This number is universally known ase, the ‘exponential’ number (sometimes referred to as Euler’s number). (Up to this point weused b instead of e in order to describe the process of discovery of this number, without using itsshort-hand symbol that might already be familiar to readers).
So
e =(
1 + 1 +12!
+13!
+14!
+ ....
)(6.16)
= 2 + (1/2) + (1/6) + (1/24) + .... (6.17)
35
and moreover, following (6.14), raising e to any power x gives:
ex =(
1 + x +x2
2!+
x3
3!+
x4
4!+ ....
), (6.18)
So ex, defined by this infinite series in (6.18), is its own derivative: dex
dx = ex. The short-hand wayof referring to the function f(x) in (6.7) is a specific form: it is actually a certain number (e) thatis precisely calculable, raised to the power x.
In (6.18), x is often called the ‘exponent’ of e. Thus e5 = 1+5+(25/2)+(125/6)+(625/24)+....It may appear that the terms are all large and increasing numbers, giving an infinite result for e5.This is not so; in fact, if we take a few more terms, using 5! = 120, 6! = 720, 7! = 5040, we get:
e5 = 1 + 5 + (25/2) + (125/6) + (625/24) + (3125/120) + (15625/720) + (78125/5040) + ...
= 6 + 12.5 + 20.833 + 26.042 + 26.042 + 21.701 + 15.501 + ..., so the numbers start to decrease,and in fact if we continue the series further the sum converges to e5 ≈ 148.413. For any valueof x, no matter how large, ex sums to a finite quantity, basically because the factorials on thebottom of the fractions eventually increase faster than the powers of x on top. Thus e10 starts as1 + 10 + (100/2) + (1000/6) + (10000/24) + .... + 1030
30! + ... and this term 1030
30! is less than 1. Thisis obviously so if this fraction is written out in full: on the top there are thirty 10’s multipliedtogether, but on the bottom 30× 29× 28...× 2× 1, which is larger. In fact e10 ≈ 22026.4.
We note that (like any number, except zero, raised to the power zero) e0 = 1. Moreover itis possible to have a negative exponent, e−x (x being positive). (Sometimes ex or e−x is writtenon one line as exp(x) or exp(−x) respectively, exp standing for ‘exponent’). Then we have inplace of (6.18) above, the following equation where, in each term, x in (6.18) is replaced by −xand, consequently, in each term an odd power of x must take on a negative sign. (For example,(−3)4 = +81, but (−3)5 = −243). So the companion equation to (6.18) is:
e−x =(
1− x +x2
2!− x3
3!+
x4
4!− ...
), (6.19)
For example e−8 ≈ 0.000335463 (which is of course the reciprocal of e8 ≈ 2980.96). It is interestingthat, if we put (say) x = 8 in (6.19), we obtain:
e−8 =(
1− 8 +642− 512
6+
409624
+ ...
), (6.20)
and it appears very unlikely that this series could sum to such a small amount as 0.000335463. Butin fact it does; however, the series converges to this limit so slowly that one must sum no fewerthan about thirty-five terms! As x becomes larger (more positive), e−x becomes tiny and e−infinity
is zero.
The form e−x arises more commonly in the sciences than does ex. This is because ex representsan ‘explosive’ process leading to an infinite amount of some quantity, which is impossible, implyingthat another process must intervene. By contrast, as just noted e−x represents a ‘decay’ processwhich leads eventually to a zero amount, requiring no such intervention. This decay processfrequently evolves in such a manner that the amount of the quantity decreases and the rate ofdecrease itself decreases. Equivalently, the rate of decrease is proportional to the existing undecayedamount at any instant. So x is often a time, t. Because the power (x or t or whatever) to whiche is raised must be dimensionless, then if the power is a time it always occurs in a form such ase−t/t0 , where t0 is a fixed time known in many contexts as the ‘time-constant’. Here are threeexamples:
(1) The charge Q on a capacitor of capacitance C across which a resistor R is connected attime t = 0 will decay according to the law Q = Q0 exp(−t/RC), where t is a later time, Q0 is theinitial charge and the product RC is the time-constant. Initially, the rate of reduction of chargeis high because there is a relatively large voltage ‘pushing’ it; but this rate decreases since, as thecharge decreases, so does the pushing voltage. If R is a resistor of resistance R = 1000 Ω (‘ohms’)and C is a capacitor of capacitance C = 100 pF (picofarads) (that is, C = 10−10 farad), then thetime-constant RC = 0.1 µs (a tenth of a microsecond or a tenth of a millionth of a second).
36
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Qua
ntity
Time t
Half-life
Time-constant
Fig. 7 Half-life and time-constant for exponential decay
(2) In the decay of a radioactive species to a more stable form, the process is governed by thelaw N = N0 exp(−t/t0), where N0 is the initial number of undecayed radioactive particles, N isthe number of undecayed particles at time t and t0 is the time-constant. In this context, it is morecommon to refer to the ‘half-life’ t1/2 of the species, and the relationship is t1/2 = t0 ln 2 ≈ 0.693t0where ln 2 denotes the natural logarithm of 2. (Logarithms will be described in the next section).The half-life is that time required for the number of undecayed particles to have decreased to one-half the original number N0. (The ‘full-life’ is mathematically infinite and in practice is indefinitelylong). The time-constant, on the other hand, is the time required for the number of undecayedparticles to have decreased to 1/e = 1/2.718 ≈ 0.368 of the original number. So the half-life isobviously shorter than the time-constant.
For example, radioactive fluorine as used in PET (positron-emission tomography) decays to anisotope of oxygen with a half-life of about 109.8 minutes. So the time for radioactive fluorine todecay until it is a tenth of its original amount is about 6 hours.
The notion of a half-life also occurs in clinical biochemistry. Thus the metabolic half-life of theanticlotting agent warfarin is about 40 hours, varying roughly from 20 hours to 60 hours.
(3) Hot water initially at, say, 80oC exposed to a room temperature of 20oC will cool accordingto the law T = 60 exp(−t/t0) + 20, where T is the temperature in degrees Celsius (an upper-caseT usually denotes the absolute temperature in degrees Kelvin, but the lower-case t often used forthe Celsius temperature is here needed to denote the time). Thus, initially, at time t = 0, thewater is at 60 + 20 = 80 degrees Celsius (because exp (0) = 1), but the temperature eventually(that is, at ‘infinite time’) decays to 20oC (because exp (−infinity) = 0). The quantity t0 is the‘time-constant’ for this process. For water in an average cup, t0 ≈ 35 minutes.
Fig. 7 is a graph of e−t/t0 against t for a time-constant t0 = 1, showing t0 and the half-lifet12. It is worth noting that x in exp (x) can be positive or negative, integer or non-integer, or
indeed can involve the so-called ‘irrationals’ such as√
2 or ‘transcendentals’ such as π. Moreover,expressions such as exp
[(−x2)/2
]are common, particularly in statistics, because this specifies the
so-called Gaussian or ‘normal’ probability distribution, to be discussed briefly in section 16.
To measure the time-constant t0 for exponential decay, it is convenient to set t = t0 in thedefining equation, since this will indicate the measurable amount of quantity left at this time. Forexample, if the decay is of the form A = A0 exp(−t/t0), where A is the quantity and A0 is its initialvalue at t = 0 (cases (1) and (2) above), then at time t = t0, the amount of the quantity left isAe−1 = A
e = A/2.718 = 0.368A. Similarly in case (3) the temperature at time t0 is ((60/2.718) +20)oC = 42.1oC. Exponential decay in time has the interesting and simplifying property that the
37
initial time t = 0 can be chosen arbitrarily when the time-constant (or, equivalently, the half-life)is to be determined. This is directly so for cases (1) and (2), and is also true for case (3) providedthat the temperature is measured relative to the final temperature 20oC. In practice, of course, foran accurate measurement of time-constant or half-life the time t = 0 would be set at some timeduring the initial quite rapid rate of decrease of the quantity.
What is the derivative of y = exp (−x) with respect to x? The chain rule, stated previously in(5.25), can be used here. We define an ‘intermediate’ variable w as w = −x; the chain rule states
dy
dx=
dy
dw
dw
dx. (6.21)
Now if y = exp (w) (because w = −x), then dydw = exp (w), the fundamental property of the
exponential function. Also dwdx = −1, since w = −x. Plugging these values into (6.21) gives
dy
dx= − exp (−x) (6.22)
when y = exp (−x).
38
7 The natural logarithm
One use of logarithms is to convert very large or very small (positive) numbers to a more manageablesize. Logarithms are commonly expressed ‘to base 10’, meaning that, for example, because 102 =100, therefore log10 100 = 2. Because 10−2 = 1/102 = 0.01, therefore log10 0.01 = −2. Again,because 108.5 ≈ 316, 227, 766, therefore log10 316, 227, 766 ≈ 8.5, and so forth. In this thirdexample, the large number of over three hundred million is ‘reduced’ to 8.5. The power 8.5 towhich 10 was raised and which was written as a superscript on the 10 has been, almost literally,‘brought down to earth’ as the simple number 8.5. Similarly, a quantity that varies over a verylarge range, say by a factor of 10000 from its smallest value, say 0.01, to its largest value, say100, and would therefore be difficult to plot with the required detail on an ordinary ‘linear’ graph,can be more easily plotted on a ‘logarithmic’ graph where its range would be −2 to +2. (This isbecause log10(0.01) = log10
(1
100
)= log10 1− log10 100 = 0−2 = −2 (see notes (2) and (8) below),
and similarly log10 100 = +2). On such a logarithmic graph, the tic marks along the relevant axiswould be equispaced but labeled: 0.01, 0.1, 1, 10 and 100. More tic marks could of course beplaced on the axis; the label (say) 50 would lie between 10 and 100 but would lie closer to 100than to 10.
Exactly the same considerations apply to the use of logarithms not to base 10, but to base e.The logarithms are then called ‘natural logarithms’ and are commonly given the symbol ln ratherthan log. The choice of e as base reflects the fact, discussed above, that many processes evolveaccording to y = exp (−x) (and x often represents an elapsed time). Going back to the superscriptnotation for the exponent, if we have in general:
y = ex, (7.1)
then the natural logarithm ln y of y is given by:
loge y = ln y = x. (7.2)
This is exactly analogous to base 10 logarithmic relationships: for example, if y is 1000, so thaty = 103, then log10 y = 3.
It is worth revising some simple facts about logarithms. These are as follows, where thelogarithms are natural logarithms. However, all the rules apply equally to logarithms to base 10,provided that where e occurs it is replaced by 10 and ln by log10:
(1) If y = ex,then ln y = x or x = ln y. Putting this value of x into y = ex gives y = eln y. Theoperation eln ‘self-cancels’; we get back whatever quantity had its logarithm taken. For example,(to base 10), 10log10 100 = 102 = 100.
(2) Because e0 = 1, therefore ln 1 = 0.
(3) Because e(−infinity) = 0, therefore ln 0 = −infinity.
(4) Because e1 = e, therefore ln e = 1.
(5) The logarithm of a positive number less than 1 is negative.
(6) There is no logarithm of a negative number (among ‘real’, that is, ‘non-complex’ numbers:such complex numbers are based on the ‘imaginary’ number
√−1).
(7) For any two numbers a and b, ln a + ln b = ln(a× b) = ln ab.
This can be shown by introducing numbers f and g where ln a = f , so that a = ef , andsimilarly ln b = g, so that b = eg. Then ab = efeg = ef+g, so that going back to logs (see rule(1)): ln ab = f + g, and substituting back for f and g gives ln ab = ln a + ln b.
This rule applies equally well to summing more than two logarithms: thus ln a + ln b + ln c =ln abc.
A cautionary note: ln a + ln b is not in general the same as ln(a + b). The equality holds onlyfor particular cases, for example when a = b = 2, or a = 3
2 and b = 3, and so on. In all such cases,the sum of the two numbers equals their product.
(8) As might well be expected, a similar rule holds when logarithms are subtracted: thusln a− ln b = ln(a/b). As in (7) above, let ln a = f , so that a = ef , and let ln b = g, so that b = eg.
39
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0-10
-8
-6
-4
-2
0
2
4
6
8
10
ln x
x
Fig. 8 Natural logarithm ln x as a function of x
Then a/b = ef/eg = ef−g, so that going back to logs: ln(a/b) = f − g, and substituting back forf and g gives ln(a/b) = ln a− ln b.
(9) From (7), a useful particular case can be stated: for any numbers a, b, c, if c = ab (in words:b a’s multiplied together), then ln c = b ln a since we can then add b ln a’s together. So ln ab = b ln a.Putting a = e then gives ln eb = b ln e = b × 1 = b. So the operation ln e also ‘self-cancels’, justlike the operation eln (see rule 1).
(10) As a particular case of (2) and (8), we note that for any number a, ln(1/a) = ln 1− ln a =0− ln a = − ln a.
(11) As a particular case of (9), ln(√
a) = ln(a12 ) = (1/2) ln a.
(12) It is of interest to relate the natural logarithm of a number to the logarithm to base 10 ofthe same number. This relationship can be found as follows:
We note first the rule that, if a, b and c are any numbers, then (ab)c = abc. For example,(32)4 = 94 = 6561 = 38. Consider y = ln 10, so that ey = 10. Now consider z = log10 e, sothat 10z = e. But 10 = ey, so if 10z = e, then we can say: (ey)z = e. By the rule just quoted,eyz = e, so we must have: y = 1/z. So y is the reciprocal of z, and the natural logarithm of 10 (thedefinition of y) is the reciprocal of the logarithm to base 10 of e (the definition of z). Numerically(this requires a calculator or a short computer program):
ln 10 ≈ 2.3026, and log10 e ≈ 1/2.3026 ≈ 0.43429.
Returning to y = ex, we know that this implies ln y = x and also implies log10 y = x log10 e(see rules (1) and (9)). Dividing one equation by the other gives, since the x cancels:
ln y
log10 y=
1log10 e
=1
0.43429= 2.3026, (7.3)
so for any number y,ln y = 2.3026 log10 y, (7.4)
andlog10 y = 0.43429 ln y. (7.5)
Fig. 8 is a graph of ln x against x, showing the ‘run-away’ nature of the logarithm deep into thenegative region when x becomes very small, and the value ln x = 0 when x = 1.
40
8 The derivative of the natural logarithm
Let y = ln x; what then is dydx? We can obtain the result by observing that if y = ln x, then ey = x.
Then, since an exponent is its own derivative, we have dxdy = ey. Inverting this gives dy
dx = e−y.
But y = ln x; therefore dydx = e− ln x. From rule 10, − ln x = ln(1/x); so dy
dx = eln(1/x). But this isjust 1/x (see rule 1). So, since y = ln x, we have:
d ln x
dx=
1x
. (8.1)
Two closely related derivatives are the derivative of ln(Kx) with respect to x, and the derivativeof ln(xa) with respect to x. The quantities K and a are constants.
Since ln(Kx) = ln K + ln x (see rule (7)), d ln(Kx)dx = d(ln K+ln x)
dx = 0 + 1x = 1
x .
Since ln(xa) = a ln x, d ln(xa)dx = a
x .
We now ask: knowing that dex
dx = ex (because ex is its own derivative), what is the derivativeof ax, where a is a constant, not necessarily equal to e? This can be answered by appeal torule (9) above: ln ax = x ln a. So let y = ax; then ln y = x ln a, and taking the derivative withrespect to y on each side, using (8.1), gives: 1
y = dxdy ln a. Then dx
dy = 1y ln a . Inverting gives:
dydx = y ln a = ax ln a since y = ax. Summarising this:
dax
dx= ax ln a. (8.2)
As a check, putting a = e in (8.2) gives dex
dx = ex ln e = ex × 1 = ex (rule (4) above), which isalready known to be correct. Equ. (8.2) was used in the eGFR and INR real-world examples insection 1.
9 The derivatives of some simple functions
The symbol a denotes a constant in the following examples, and all derivatives are understood tobe taken with respect to the input variable x. The output variable is y.
(1) y = 1a+x
In this and the next example, the derivative dydx will be evaluated in two ways: the first using a
direct differential method as in (5.27) and the equations immediately following it, and the secondusing the chain rule as in (5.25).
If y = 1a+x , let us increase x to x + δx, so that y must change to y + δy, but the relationship
between the new input and new output remains unaltered; therefore
y + δy =1
a + x + δx. (9.1)
Thenδy =
1a + x + δx
− y =1
a + x + δx− 1
a + x. (9.2)
Putting these two fractions on the right-hand side of (9.2) over a common denominator gives
δy =(a + x)− (a + x + δx)(a + x + δx)(a + x)
, (9.3)
=−δx
(a + x + δx)(a + x), (9.4)
41
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
1/(1+x)
x
P
Q
Fig. 9 The function 1/(a + x) with a = 1
which means that, dividing both sides of (9.4) by δx:
δy
δx=
−1(a + x + δx)(a + x)
. (9.5)
Now δx is imagined to decrease in size until it is tiny and keeps on decreasing. So δy must alsoreduce in size. On the right-hand side of (9.5), (a + x + δx) is dominated by a + x and so tends to(a + x). The right-hand side therefore tends to −1
(a+x)2.
Fig. 9 is a graph of the function y = 1a+x for the particular case a = 1 and shows general points
P and Q where derivatives is taken. Whether x is positive (at P) or negative (at Q), it can be seenthat a small increase in x must be accompanied by a small decrease in y; this is obviously becausethe slope is negative (hence the minus sign on the right-hand side of (9.5) for δy
δx ). In (9.5), as δx
and δy both tend to zero, the ratio of the differentials approaches the required derivative dydx :
dy
dx= − 1
(a + x)2when y =
1a + x
. (9.6)
The same result can be obtained using the chain rule, using an ‘intermediate’ variable w:
dy
dx=
dy
dw× dw
dx. (9.7)
A good choice for w is w = a + x. So y = 1w = w−1, so that using (5.17):
dy
dw= −w−2 =
−1w2
=−1
(a + x)2. (9.8)
Also, for the second term on the right of (9.7), we have since w = a + x, dwdx = 1, because a is a
constant. Thereforedy
dx=
dy
dw× dw
dx=
−1(a + x)2
× 1 =−1
(a + x)2, (9.9)
agreeing with (9.6).
42
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
1/(1-x)
x
P
Q
Fig. 10 The function 1/(a− x) with a = 1
(2) y = 1a−x
The difference from (1) is a minus instead of a plus sign in front of the x. Fig. 10 is a graphof this function for the particular case a = 1, and in contrast to (1) the function 1
a−x increases asx increases. This is plainly the case: as x in the bottom of the fraction becomes more positive,the quantity a− x in the bottom decreases, and the fraction itself therefore increases (just as, forexample, 1
7 is an increase over 18 ). (All this applies equally well to the case where a is negative).
So the slope of y = 1a−x is positive, and working out the derivative gives results that agree with
this. Again the derivative will be obtained in two ways, directly using differentials and then usingthe chain rule.
If y = 1a−x , let us increase x to x + δx, so that y must change to y + δy, but the relationship
between the new input and new output remains unaltered; therefore
y + δy =1
a− (x + δx). (9.10)
Thenδy =
1a− (x + δx)
− y =1
a− x− δx− 1
a− x. (9.11)
Putting these two fractions on the right-hand side of (9.11) over a common denominator gives
δy =(a− x)− (a− x− δx)(a− x− δx)(a− x)
, (9.12)
=δx
(a− x− δx)(a− x), (9.13)
Dividing both sides of (9.13) by δx gives:
δy
δx=
1(a− x− δx)(a− x)
. (9.14)
Now δx is imagined to reduce in size until it is tiny and keeps on decreasing. So δy must alsoreduce in size. On the right-hand side of (9.14), (a− x− δx) is dominated by a− x and so tendsto (a− x). The right-hand side therefore tends to 1
(a−x)2.
43
Fig. 10 shows the general points P (positive x) and Q (negative x). At both points, it canbe seen that a small increase in x is now accompanied by a small increase in y; this is obviouslybecause the slope is positive. In (9.14), as δx and δy both tend to zero, the ratio of the differentialsapproaches the required derivative dy
dx :
dy
dx=
1(a− x)2
when y =1
a− x. (9.15)
The same result can be obtained using the chain rule, using an ‘intermediate’ variable w:
dy
dx=
dy
dw× dw
dx. (9.16)
A good choice for w is w = a− x. So y = 1w = w−1, so that using (5.17):
dy
dw= −w−2 =
−1w2
=−1
(a− x)2. (9.17)
Also, for the second term on the right of (9.16), we have since w = a− x, dwdx = −1. Therefore
dy
dx=
dy
dw× dw
dx=
−1(a− x)2
×−1 =1
(a− x)2, (9.18)
agreeing with (9.15).
(3) y = 1a2+x2
We write the constant term as a2 (rather than as a), since a and x will then have the samedimensions, and the form a2 + x2 is commonly found in mathematical analysis (as is a2 − x2).
Here a good choice for w is w = a2 + x2, so that dwdx = 2x and y = 1
w = w−1, so thatdydw = −w−2 = −1
w2 = −1(a2+x2)2
. The chain rule as in (5.25) above then gives:
dy
dx=
−1(a2 + x2)2
× 2x =−2x
(a2 + x2)2. (9.19)
When x = 0, dydx = 0 because of the x in the top of the fraction in (9.19). So when x = 0, the
slope of the curve is zero. Fig. 11 is a graph of y = 1a2+x2 for a = 1 and indeed shows zero slope
at x = 0, in contrast to the cases in (1) and (2) above. Moreover, we also see that the function issymmetric about the vertical line x = 0, because putting −x instead of x in y = 1
a2+x2 does not
change y (for any quantity x, (−x)2 = x2).
(4) y = 1a2−x2
Here a good choice for w is w = a2 − x2, so that dwdx = −2x and y = 1
w = w−1, so thatdydw = −w−2 = −1
w2 = −1(a2−x2)2
. The chain rule as in (5.25) above then gives:
dy
dx=
−1(a2 − x2)2
×−2x =2x
(a2 − x2)2. (9.20)
As shown in Fig. 12 (for a = 1) when x = 0, dydx = 0 and the function has zero slope at x = 0. The
function is symmetric about the y-axis and is an increasing function of x up to when x = ±a. Herethe bottom line of the function is zero and the function shoots off to infinity. When x is positivebut larger than a the function is negative, slowly approaching zero again, and this is mirrored onthe negative side of the x-axis.
44
-10 -8 -6 -4 -2 0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
1/(1+x2)
x
Fig. 11 The function 1/(a2 + x2) with a = 1
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0-10
-8
-6
-4
-2
0
2
4
6
8
10
1/(1-x2)
x
Fig. 12 The function 1/(a2 − x2) with a = 1
45
-5 -4 -3 -2 -1 0 1 2 3 4 50
2
4
6
8
10
(1+x2)0.5
x
Fig. 13 The function +√
a2 + x2 with a = 1
(5) y =√
a2 + x2
In this and several forthcoming examples, y is defined as a square root of an expression involvingx. A square root can always have either sign, positive or negative (for example,
√9 = +3 or −3).
So the graph of y versus x in such cases will always be symmetric about the x-axis; for a givenvalue of x, there will be two corresponding values of y, of equal magnitude but opposing sign.
The particular function y =√
a2 + x2 can of course be written y = (a2 + x2)12 . We choose
w = a2 + x2, so that y = w12 and dy
dw = 12w−
12 and dw
dx = 2x. Hence
dy
dx=
12w−
12 × 2x
= x1√
a2 + x2. (9.21)
The function y =√
a2 + x2 is graphed in Fig. 13 for a = 1 and taking only the positive squareroot. Again we see a zero slope at x = 0.
(6) y =√
a2 − x2
We choose w = a2 − x2, so that y = w12 and dy
dw = 12w−
12 and dw
dx = −2x. Hence
dy
dx=
12w−
12 ×−2x
= −x1√
a2 − x2. (9.22)
The function y =√
a2 − x2 is graphed in Fig. 14 for a = 1. It is a circle, of radius a = 1, andappears on the page as a circle if equal scales are used for the x and y axes. When x = 0, dy
dx = 0from (9.22), and these are the points P and R on the circle in Fig. 14: the tangents to the circlehave zero slope. On the other hand, at x = +1 (point S) and x = −1 (point Q), dy
dx is infinite,since the bottom line in (9.22) is zero. Infinite dy
dx means an infinite slope, namely the slope of a
46
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
(1/2)1/2
(1-x2)0.5
x
(1/2)1/2P
Q
R
S
T
UO
1
j
Fig. 14 The function√
a2 − x2 with a = 1
line parallel to the y-axis. At the point T in Fig. 14, where OT makes an angle of 45o with the xaxis (O being the centre of the circle), and if TU is the perpendicular from T down to the x-axis,then TU = 1/
√2 and OU = 1/
√2. This follows from symmetry and the Pythagoras theorem:
OU2 + TU2 = OT2. Then T has the x-coordinate x = 1/√
2 and inserting this into (9.22), witha = 1, gives dy
dx = −1, so as expected the tangent-line to the circle at T has a slope of −45o (sincetan(−45)o = −1). There is more about the tangent in section 12.
Equ. (9.22) can be written dydx = −x
y . So when both x and y are positive, the slope is negative.When x is negative but y is positive, the slope is positive. When x and y are both negative, theslope is negative. Finally, when x is positive but y negative, the slope is positive. All four casesare evidently correctly illustrated in Fig. 14.
(7) y = 1√a2+x2
We choose w = a2+x2 as in example (5), but now y = w−12 , so that dy
dw = −12w−
32 = −1
21
w32
=
−12
1
(a2+x2)32
. Also dwdx = 2x. Therefore
dy
dx=
−x
(a2 + x2)32
. (9.23)
Fig. 15 shows the graph of y = + 1√a2+x2
for a = 1. There is zero slope at x = 0.
(8) y = 1√a2−x2
We choose w = a2 − x2 as in example (6), and as in example (7) y = w−12 , so that dy
dw =
−12w−
32 = −1
21
w32
= −12
1
(a2+x2)32
. Also dwdx = −2x. Therefore
dy
dx=
x
(a2 − x2)32
. (9.24)
47
-10 -8 -6 -4 -2 0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
1/(1+x2)1/2
x
Fig. 15 The function + 1√a2+x2
with a = 1
Fig. 16 shows the graph of y = + 1√a2−x2
for a = 1. There is zero slope at x = 0, and infinite
slope at x = +a = +1 and at x = −a = −1.
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00
2
4
6
8
10
1/(1-x2)1/2
x
Fig. 16 The function + 1√a2−x2
with a = 1
48
(9) y = 1(a+x)2
Defining w = a + x gives y = 1w2 = w−2, so that dy
dw = −2w−3 = −2(a+x)3
. Also dwdx = 1, so we
obtaindy
dx=
−2(a + x)3
. (9.25)
. Fig. 17 shows the graph for a = 1, and of course when x = −a = −1 the graph shoots off toinfinity.
-10 -8 -6 -4 -2 0 2 4 6 8 100
2
4
6
8
10
1/(1+x)2
x
Fig. 17 The function 1(a+x)2
with a = 1
(10) y = 1(a−x)2
Defining w = a − x gives y = 1w2 = w−2, so that dy
dw = −2w−3 = −2(a−x)3
. Also dwdx = −1, so
we obtaindy
dx=
2(a− x)3
. (9.26)
. Fig. 18 shows the graph for a = 1, and again when x = +a = +1 the graph shoots off to infinity.
49
-10 -8 -6 -4 -2 0 2 4 6 8 100
2
4
6
8
10
1/(1-x)2
x
Fig. 18 The function 1(a−x)2
with a = 1
50
10 Derivative of one function of x multiplied by anotherfunction of x
All the functions y = f(x) so far considered have two things in common: they cannot be writtenas a product of two other functions of x. Moreover, either they can be written on one line, like√
a2 + x2, or as a fraction with 1 on the top, like 1a+x . This second example is of course essentially
equivalent to having any other constant on the top; thus if this constant is 2, then the derivativeis simply twice as large as when the constant is 1.
In this and the following section, we consider the cases where y is the product of two otherfunctions, for example y = x
√(a2 + x2), and also where the fraction has a function of x on the
top line and another function of x on the bottom line; an example would be y = xa+x .
More generally, let us write:y = f(x) = u(x)v(x) (10.1)
So f(x) is the product of u(x) and v(x). As a particular example, if y = f(x) = x√
(a2 + x2),then u(x) = x and v(x) =
√(a2 + x2). In general, then, what is the derivative dy
dx?
Fig. 19 is a general graph of u(x) (along the vertical axis) against x (along the horizontal axis).The point P on the graph has coordinates x and u(x), meaning that P is at a horizontal distancex from the vertical axis, and at a vertical distance PS = QT = u(x) from the horizontal axis. Theneighbouring point R on the graph has, similarly, coordinates x + δx and u(x + δx). For smallchanges, the shape PQR is approximately a triangle whose base PQ has length δx, and whoseheight RQ is δ [u(x)]. But R is at a distance RT = u(x + δx) above the horizontal axis, since the‘x’-coordinate of R is x + δx. Since RT = RQ + QT, we have
u(x + δx) = u(x) + δ [u(x)] . (10.2)
We examine the second term in (10.2): δ [u(x)] =RQ can be written as
δ [u(x)] = PQ× RQPQ
= δx × δ [u(x)]δx
. (10.3)
The reason for writing δ [u(x)] in this way is that in the limit of tiny changes, δ[u(x)]δx becomes
du(x)dx . Equ. (10.2) then becomes
u(x + δx) = u(x) +du(x)
dxδx. (10.4)
Equ. (10.4) is effectively the so-called ‘first-order Taylor’ expansion of the function u(x).Similarly, the companion function v(x) obeys the equation
v(x + δx) = v(x) +dv(x)dx
δx. (10.5)
We had, originally, y = f(x) = u(x)v(x). Then, increasing x to x + δx,
y + δy = f(x + δx) = u(x + δx)v(x + δx) =(
u(x) +du(x)
dxδx
)(v(x) +
dv(x)dx
δx
)=
= u(x)v(x) + u(x)dv(x)dx
δx + v(x)du(x)
dxδx +
du(x)dx
dv(x)dx
(δx)2. (10.6)
Subtracting y = f(x) = u(x)v(x) from both sides of (10.6), and then dividing by δx and going tothe limit where differentials become derivatives, in what is by now a routine procedure, gives
dy
dx=
d (u(x)v(x))dx
= u(x)dv(x)dx
+ v(x)du(x)
dx+
du(x)dx
dv(x)dx
δx, (10.7)
51
P QR
S Tx x+ x
u(x)
x
u(x+ x)
(u(x))
Fig. 19 Illustrating u(x) and δ [u(x)].
and the last term in (10.7), which contains δx instead of (δx)2 (as at the end of (10.6)) since therehas been a division by δx, goes to zero as δx becomes very small. We therefore have, finally,
dy
dx=
d(u(x)v(x))dx
= u(x)dv(x)dx
+ v(x)du(x)
dx. (10.8)
In words, then, the derivative of a product of two functions is very simply obtained by first takingthe derivative of one of them, keeping the other as a ‘temporary constant’, and adding to this thederivative of the other, keeping the first one as a temporary constant.
For example, if y = f(x) = 4x(a2 + x2), so that u(x) = 4x and v(x) = a2 + x2, then du(x)dx = 4
and dv(x)dx = 2x. Applying (10.8) then gives:
dydx = 4x×2x+(a2 +x2)×4 = 8x2 +4a2 +4x2 = 4a2 +12x2. We can check this by multiplying
out y = f(x). This is y = 4xa2 + 4x3, whence dydx = 4a2 + 12x2, which is the same quantity.
52
11 Derivative of a ratio: one function of x divided byanother function of x
The form for y = f(x) is now
y = f(x) =u(x)v(x)
= u(x)× 1v(x)
. (11.1)
Then, as shown in section 10,
dy
dx= u(x)
d 1v(x)
dx+
1v(x)
du(x)dx
. (11.2)
Using a form of the chain rule,d 1
v(x)
dx=
d 1v(x)
dv(x)× dv(x)
dx=
−1[v(x)]2
× dv(x)dx
. (11.3)
In the last part of (11.3), we have used the previously established fact (from (5.17)) that, ify = 1
z = z−1, then dydz = −1
z2 . The quantity z here represents v(x) in (11.3). Hence (11.2) becomes
dy
dx= u(x)× −1
[v(x)]2× dv(x)
dx+
1v(x)
du(x)dx
(11.4)
and putting everything as a fraction with [v(x)]2 on the bottom,
dy
dx=
v(x)du(x)dx − u(x)dv(x)
dx
[v(x)]2. (11.5)
To show how (11.5) works in a very simple case, where we know the result in advance, suppose
y = f(x) = x2
x . This of course simplifies to y = f(x) = x, since the common factor x cancels fromboth top and bottom. Therefore we know that dy
dx = 1 (from the general rule in (5.17)).
Now let us identify u(x) = x2 and v(x) = x. Then dudx = 2x and dv
dx = 1. Plugging these valuesinto (11.5) gives:
dy
dx=
x× 2x− x2 × 1x2
=2x2 − x2
x2=
x2
x2= 1, (11.6)
agreeing with what we know in advance.
As a more complicated example, suppose that
y = f(x) =a− x
a + x(11.7)
with a a constant, so that u(x) = a − x and v(x) = a + x. Then du(x)dx = −1, and dv(x)
dx = +1.Consequently,
dy
dx=
(a + x)× (−1)− (a− x)× (+1)(a + x)2
=
=−a− x− a + x
(a + x)2=
−2a
(a + x)2. (11.8)
Fig. 20 is a plot of y = a−xa+x for a = 1.
As a final example, suppose that
y = f(x) =1− e−x
x. (11.9)
53
-10 -8 -6 -4 -2 0 2 4 6 8 10-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
x
(1-x)/(1+x)
Fig. 20 The function a−xa+x
with a = 1.
We define u(x) = 1− e−x and v(x) = x.
Then du(x)dx = +e−x, and dv(x)
dx = 1. Therefore
dy
dx=
xe−x − (1− e−x)x2
=e−x(1 + x)− 1
x2. (11.10)
Fig. 21 is a graph of y = 1−e−x
x . When x = 0, it might appear that y goes to infinity (a nonzeronumber divided by zero gives infinity). But in fact when x is very small and approaches zero,the top line, 1 − e−x, is also very small and approaches zero. So y stays finite at x = 0, and hasthe value y = 1. This can be checked using (6.19), which indicates that when x is very small,e−x ≈ 1− x, so the top of the fraction defining y becomes 1− (1− x) = x and so dividing by thebottom of the fraction, which is also x, gives x
x = 1.
It is instructive to approximate e−x not simply as 1− x for small x, as we just did, but to goto the second-order term in (6.19): so let us put e−x = 1−x+ 1
2x2 for x small, but not that small.
Then the function is y =1−(1−x+1
2x2)
x = 1 − 12x, so this verifies that at x = 0, y indeed has the
value 1. So it might be thought that going to the second-order term is a needless precaution againstobtaining an incorrect approximation for small x. However, when we calculate the derivative, itturns out that staying with the first-order approximation e−x = 1−x will give an incorrect result.This is demonstrated as follows. We include the second-order term e−x = 1 − x + 1
2x2 in theexpression for the derivative in (11.10):
dydx = e−x(1+x)−1
x2 =(1−x+1
2x2)(1+x)−1
x2 =−x2+1
2x2(1+x)
x2 = −1 + 12 (1 + x). So when x = 0, the
slope of the function is dydx = −1 + 1
2 = −12 .
The slope therefore has a tangent −12 or about −26.6o. If we had stayed with only the first-
order term, we would have obtained dydx = −1, or −45o. But in Fig. 21, which has approximately
equal scales along the x and y axes, it is plain that the tangent-line at x = 0 has an incline thatis much less negative than −45o (or, in plainer language, is much less steep than 45o). Going to athird-order approximation, e−x = 1 − x + 1
2x2 − 16x3 (see (6.19)), still gives the same result: the
tangent-line at x = 0 is at −26.6o to the horizontal. So it is often wise, when doing approximationsof this nature, to include higher-order approximations for checking what seems to be a valid result.
54
-5 -4 -3 -2 -1 0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
2
3
4
5
(1-e-x)/x
x
Fig. 21 The function 1−e−x
x.
55
12 Derivatives of the trigonometric functions
The basic trigonometric functions are the sine of an angle, the cosine of an angle and the tangentof an angle. Their mathematical abbreviations are respectively sin, cos and tan. The tangenttan is defined as the ratio sin
cos . These three functions have reciprocals, namely the cosecant,csc = 1/ sin, the secant, sec = 1/ cos and the cotangent, cot = 1/ tan. As noted earlier (section3, second paragraph), when the slope of a line or curve is calculated (by finding the derivative ofthe function in question), the numerical value of the slope is the tangent of the angle between thetangent-line to the line or curve (at the point of interest) and the horizontal axis.
We normally talk of angles in degrees, o, but this is an arbitrary measure and for use in thedifferential calculus they should be measured in radians. The measure of an angle in radians isthe length of the arc of a circle ‘covered’ by the angle, divided by the radius of that circle. In Fig.22(a), the value of the angle, θ, in radians is the arc AB divided by the radius OA or OB; thusθ = (arc AB)/OA. We note that the length of the arc is the actual ‘curved length’ along the arc;it is not the length of the straight line joining A to B.
If θ in Fig. 22(a) is increased to a right angle, the length of the arc obviously increases toone-quarter of the total circumference of the circle, that is, 1
4 × 2πa = πa2 where a is the radius of
the circle. (In section 1 we used the diameter D of the circle, where of course D = 2a, but now theradius a is the more convenient measure). So a right angle, 90o, is equivalent to πa
2a = π2 radians.
This number is approximately 3.14159.../2 ≈ 1.5708 radians. So 90o = 1.5708 radians, or 1 radianis about 90/1.5708 ≈ 57.296o. Conversely, 1 degree is about 0.01745 radians. Of course 2π radiansis the entire 360o degrees around the circle.
In Fig. 22(b), the right-angle triangle OAC, with the angle ACO as the right angle, serves todefine the sine, cosine and tangent of a general angle θ, as follows:
Fig. 22(a) Definition of an angle θ. (b) The trigonometrical functions: sin θ = ACOA
, cos θ = OCOA
,
tan θ = ACOC
.
sin θ =ACOA
; cos θ =OCOA
; tan θ =ACOC
. (12.1)
Using these definitions, and the Pythagoras theorem OA2 = OC2 + AC2 (OA is the ‘hypotenuse’of the right-angle triangle), we have the following identity, meaning that the equation is valid forany angle θ:
cos2 θ + sin2 θ = 1. (12.2)
Fig. 23 and 24 show how sin θ (continuous curve in Fig. 23(a)), cos θ (dotted curve) and tan θvary with θ; the first two are of course the familiar ‘sinusoidal’ patterns that, among other things,describe wave motion. The sin and cos functions each have period of 2π = 360o, meaning that, forexample, sin(θ+2nπ) = sin θ for any value of the integer n. It can be seen that the sine and cosineare mutually ‘out of phase’ by 90o; when θ = 0, sin θ = 0 and cos θ = 1, but when θ = π
2 = 90o,sin θ = 1 and cos θ = 0. It can also be seen that sin(−θ) = − sin θ, so the sine is an ‘odd’ function
56
of θ (this does not mean ‘peculiar’, but simply a function that changes sign when its argumentchanges sign!). By contrast, cos(−θ) = cos θ, so the cosine is an ‘even’ function of θ.
Fig. 24 shows that tan θ has a periodicity of π, namely half that of sin θ and cos θ. Like sin θ,tan θ is an odd function of θ. Fig. 24 also shows how the tangent shoots off to infinity at 90o and−90o and likewise at ±270o. At 45o the tangent has the value 1, and at −45o the value −1.
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
value
0 /2 3 /2Radians:
Degrees: 090
180270
360
cosinesine
Fig. 23 The sine and cosine functions
-270 -225 -180 -135 -90 -45 0 45 90 135 180 225 270-5
-4
-3
-2
-1
0
1
2
3
4
5
Value of tangent
Degrees
Fig. 24 The tangent function
There are values of the trigonometric functions which can be expressed simply as fractions orwith square roots: these occur when the angle in radians is a simple multiple or factor of π. Forexample, we have
sin 0 = 0; cos 0 = 1
sin(π
6
)= 30o =
12; cos 30o =
√3
2
sin(π
4
)= 45o =
√12
= cos 45o
57
sin(π
3
)= 60o =
√3
2; cos 60o =
12
sin(π
2
)= 90o = 1; cos 90o = 0.
For the tangent:
tan 0 = 0; tan 30o =1√3; tan 45o = 1; tan 60o =
√3; tan 90o = infinity. (12.3)
To find the derivatives of the trigonometric functions, we first need to establish the followingidentities:
sin(θ1 + θ2) = sin θ1 cos θ2 + cos θ1 sin θ2, (12.4)
andcos(θ1 + θ2) = cos θ1 cos θ2 − sin θ1 sin θ2. (12.5)
Fig. 25 is useful for establishing (12.4). It shows a right-angle triangle EOD and another smallerright-angle triangle ECJ where the side EC is part of the side ED of the bigger triangle. The rightangles are as shown, as are the angles θ1 and θ2. The angle θ1 is angle COD and is also equal tothe angle CEJ in the smaller triangle. (This can be seen from the fact that angle OCD is equalto angle ECJ, being the angles on opposite sides of the intersecting straight lines ED and OJ, andthe angle OCD must be 90o − θ1, because the three angles of a triangle always add up to 180o).It follows that the triangle ECJ is an exact scaled replica of the triangle OCD (they are ‘similar’triangles). Then we can express the ratio of two sides of one triangle, ECJ, as equal to the ratioof the corresponding sides of the other triangle OCD:
CDOC
=CJEC
. (12.6)
This simply says that sin θ1 in the bigger triangle is equal to sin θ1 in the smaller triangle: obvious,since the angles themselves are both θ1.
Fig. 25 Diagram for expanding sin(θ1 + θ2). Angle COD = θ1 = angle CEJ.
58
ThenCD× EC = CJ×OC. (12.7)
Now by Pythagoras’ theorem applied to the right-angle triangle ECJ, EC2 = CJ2 + EJ2, and weadd this to both sides of (12.7):
EC2 + CD× EC = CJ×OC + CJ2 + EJ2. (12.8)
SoEC (EC + CD) = CJ (OC + CJ) + EJ2, (12.9)
But EC + CD = ED, and OC + CJ = OJ. so
EC× ED = CJ×OJ + EJ2. (12.10)
Then
ED =CJ×OJ + EJ2
EC(12.11)
so, dividing both sides of (12.11) by OE:
EDOE
=CJ×OJ + EJ2
EC×OE. (12.12)
Then finally,EDOE
=CJEC
× OJOE
+EJEC
× EJOE
. (12.13)
But remembering the definitions of sin and cos, we can see that on the left-hand side of (12.13),EDOE = sin(θ1+θ2), whereas on the right-hand side of (12.13), CJ
EC = sin θ1, OJOE = cos θ2, EJ
EC = cos θ1
and EJOE = sin θ2. This establishes (12.4).
To establish (12.5) by algebraic manipulation, we take the square of cos(θ1 + θ2) and use theidentity in (12.2):
cos2(θ1 + θ2) = 1− sin2(θ1 + θ2)
and using the just-established identity for sin(θ1 + θ2):
cos2(θ1 + θ2) = 1− (sin θ1 cos θ2 + cos θ1 sin θ2)2 =
= 1− sin2 θ1 cos2 θ2 − cos2 θ1 sin2 θ2 − 2 sin θ1 sin θ2 cos θ1 cos θ2 = (12.14)
= 1− (1− cos2 θ1) cos2 θ2 − (1− sin2 θ1) sin2 θ2 − 2 sin θ1 sin θ2 cos θ1 cos θ2 =
= 1− cos2 θ2 + cos2 θ1 cos2 θ2 − sin2 θ2 + sin2 θ1 sin2 θ2 − 2 sin θ1 sin θ2 cos θ1 cos θ2. (12.15)
But since, as is true for any angle, cos2 θ2 + sin2 θ2 = 1, the right-hand side of (12.15) simplifies to
cos2 θ1 cos2 θ2 + sin2 θ1 sin2 θ2 − 2 sin θ1 sin θ2 cos θ1 cos θ2
and this is the square of (cos θ1 cos θ2 − sin θ1 sin θ2), so remembering that the left-hand side of(12.14) or (12.15) is cos2(θ1 + θ2), taking square roots establishes (12.5).
Returning to Fig. 22(a), we now look at the case where θ is very small; call it δθ. Fig. 22(a),which shows θ as defined by an arc of a circle divided by the radius, can now be drawn as Fig.26. Because the angle is very small, the length of the arc from A to B, measured along the arc,becomes very close to the straight-line distance AB. We can take the angle OBA to be a rightangle (or the angle OAB; it hardly matters, because the angle is so small). Then the sine of δθ isstraight line AB
OA and this is very close to arc length ABOA , but this last expression is just the definition
of the angle θ in the first place. So, when θ is very small, namely δθ, we have (and this is theso-called ‘first-order approximation’ for sin δθ):
sin δθ ≈ δθ. (12.16)
59
Fig. 26 Diagram for approximating sin δθ and cos δθ
How about cos θ when θ is very small? In Fig. 26, taking OBA to be a right angle as above, cos δθis OB
OA , but these straight lines are very nearly equal in length, so that their ratio is ≈ 1:
cos δθ ≈ 1. (12.17)
Again, this is the ‘first-order approximation’ for cos δθ.
Then tan δθ, which by the definition of the tangent is sin δθcos δθ , is δθ
1 = δθ, so tan δθ ≈ δθ as afirst-order approximation.
We can now find the derivatives of sin θ and cos θ with respect to θ. Suppose the function y isgiven by
y = sin θ, (12.18)
so as usual we increase θ to θ + δθ, whereupon y increases to y + δy:
y + δy = sin(θ + δθ), (12.19)
and the right-hand side of (12.19) can be expanded as a particular case of sin(θ1 + θ2):
y + δy = sin θ cos δθ + cos θ sin δθ (12.20)
and for very small θ (12.20) becomes:
y + δy = sin θ × 1 + cos θ × δθ, (12.21)
so subtracting y = sin θ from both sides of (12.21) gives:
δy = δθ cos θ. (12.22)
Dividing both sides of (12.22) by δθ and going to the limit of zero δθ gives the required derivative:
dy
dθ= cos θ. (12.23)
Very similarly, to find the derivative of cos θ with respect to θ: we start with
y = cos θ (12.24)
so thaty + δy = cos(θ + δθ) = cos θ cos δθ − sin θ sin δθ =
= cos θ × 1− sin θ × δθ (12.25)
so subtracting y = cos θ from both sides of (12.25) gives:
δy = −δθ sin θ (12.26)
60
and dividing both sides of (12.26) by δθ and going to the limit of zero δθ gives:
dy
dθ= − sin θ. (12.27)
When we remember that a derivative measures a rate of change, it is intuitively plausible thatd sin θ
dθ = cos θ and that d cos θdθ = − sin θ. In Fig. 23 that shows the variation of sin θ as a function
of θ, we see that when θ = 0, sin θ = 0 also, but Fig. 23 shows that when θ = 0, by contrast cos θhas its maximum value 1. In Fig. 23 when θ = 0 the rate of change of sin θ, namely the slope ofthe curve at θ = 0, is a maximum. This maximum value is 1, which is the value of cos θ at thatpoint θ = 0. No surprises here, since the rate of change, or the derivative of sin θ, is cos θ!
Conversely, when θ = 0, cos θ = 1 and as we have seen the derivative of cos θ with respect to θis − sin θ. At θ = 0, the derivative − sin θ is zero. So the rate of change of cos θ at θ = 0 is zero.Fig. 23 shows that this is plausible, because the curve of cos θ has a plateau at θ = 0, meaning thatat that point the rate of change is zero. The negative sign in − sin θ is also plausible, because as θchanges from zero to a slightly positive value, the rate of change goes from zero to a small negativevalue and hence a negative slope: cos θ goes ‘downhill’ as θ leaves zero in a positive direction (andalso, of course, in the negative direction).
The derivative of y = tan θ is found by using its definition as
tan θ =sin θ
cos θ. (12.28)
This is a particular case of y = u(x)v(x)
as described in section 9. We can write y = sin θcos θ , so that
using (11.5):dy
dθ=
cos2 θ − (−) sin2 θ
cos2 θ(12.29)
and since the top of the fraction in (12.29) is cos2 θ + sin2 θ = 1, we have
d tan θ
dθ=
1cos2 θ
= sec2 θ. (12.30)
Using the chain rule, we can easily find the derivative of a function such as sin 3x with respect tox. This procedure is carried out when, as is often the case, x is a time, t, and the coefficient 3 (forexample) in front represents a frequency of oscillation (strictly, an angular frequency). The chainrule (putting w = 3x) immediately gives
d sin 3x
dx= 3 cos 3x. (12.31)
61
13 Derivatives of the hyperbolic functions
These are not necessarily functions of angles, so we denote the argument of these functions by x.They are the hyperbolic sine, sinh, the hyperbolic cosine, cosh and the hyperbolic tangent, tanh.They are defined by:
sinh x =12
(ex − e−x)
, cosh x =12
(ex + e−x)
, and tanh x =sinhx
coshx. (13.1)
Fig. 27 shows the variation of sinh x, cosh x and tanh x with x. Unlike their trigonometricalcounterparts, there is no oscillatory behaviour.
-5 -4 -3 -2 -1 0 1 2 3 4 5
-4
-2
0
2
4
Value
x
sinh x
cosh x
tanh x
sinh x
Fig. 27 The hyperbolic functions sinh, cosh and tanh
The equation linking cosh x and sinh x and corresponding to (12.2) is:
cosh2 x− sinh2 x = 1. (13.2)
To see this, we note that from their definitions in (13.1), we have cosh2 x = 14
(e2x + e−2x + 2
),
and sinh2 x = 14
(e2x + e−2x − 2
). The lone 2’s come from the product 2× ex× e−x = 2× ex−x =
2× e0 = 2× 1 = 2. So in evaluating cosh2 x− sinh2 x the e2x and e−2x cancel and we are left with14 (2 + 2) = 1.
The derivatives of sinh x and cosh x are found simply by noting that, as we have seen, dex
dx = ex,
and de−x
dx = −e−x. Then:
d sinh x
dx= cosh x, and
d cosh x
dx= sinh x. (13.3)
The derivative of y = tanh x = sinh xcosh x is found similarly to the derivative of tan θ above. We define
u(x) = sinh x and v(x) = cosh x, whence
dy
dx=
cosh2 x− sinh2 x
cosh2 x=
1cosh2 x
. (13.4)
62
14 From errors to uncertainties and bias, using the deriva-tive and differentials
We saw earlier (equ. (5.2)) that an error δy in the output y can be related to an error δx in theinput x through the notion of a derivative dy
dx (calculated at the particular relevant value of x):
δy ≈ dy
dxδx. (14.1)
Very roughly, and as an intuitively plausible extension of (14.1), we can say that (14.1) impliesthat
spread of output values =dy
dx× spread of input values (14.2)
because the ‘spread’ in the output is a measure of the total contribution of many errors in theinput; each of these input errors contributes to an error in the output, through the relationship in(14.1). The many errors in the input themselves contribute to the ‘spread’ in the input.
The technical term closely related to ‘spread’, when the spread compromises the precision of ameasurement, is uncertainty. At this point it is worth noting that the spread need not be an obvious‘scatter’ in the results of measurements; it may also be an ‘implicit’ spread, as when a single valueis obtained from previous results or from a calibration report, look-up tables or instrument manual,this single value having an implicit reported uncertainty. An implicit uncertainty, ‘fossilised’ soto speak, and which both implies and summarises a statistical analysis carried out in the past, isreferred to as a Type B uncertainty. Where a statistical analysis is carried out in the present, aswhen dealing with scatter of results, the resulting uncertainty is a Type A.
Ignoring that single value will of course impart a bias to the measurand y. As a general rule,any such bias should immediately be corrected for, by adding it to or subtracting it from the valueof the measurand. Thus if a mass has been measured as 10.000 grams but the balance measures toolow by 1 milligram, as stated in the calibration report for the balance, then the bias-corrected massis 10.001 grams. The component of standard uncertainty in this 10.001 grams that is inheritedfrom the report equals the standard uncertainty of the 1 milligram correction, also stated in thereport.
There are, admittedly, cases where any bias is only roughly known and may even signify asimple discrepancy in the results obtained by two or more laboratories when determining the valueof a measurand. With two laboratories, the question may boil down to: does lab A have the biaswith respect to lab B (the ‘better’ lab), or does lab B have the (negative) bias with respect tolab A (the ‘better’ lab from this point of view)? This is a matter for a case-by-case study, butwhen several laboratories are involved and have agreed equal capability, one possible expedientmight be to make no correction for ‘bias’ but to assign a component of standard uncertainty tothe measurand equal to the standard deviation of the discrepant results.
Equ. (14.1) provides the basis for the relationship between input uncertainty and outputuncertainty. This will now be illustrated using a simple artificial example of only four input errorsδx and therefore only four output errors δy.
Suppose that the input x = 1 and the input errors δx have the values: +0.2, −0.1, 0.0 and+0.3. Let us suppose that the derivative dy
dx has the value 2 (at the value x = 1), so that theerrors δy in y must then be: +0.4, −0.2, 0.0 and +0.6. The mean of the δx is +0.1 and the meanof the δy is, of course, +0.2. (It may be noted that although we need to know where to evaluatethe derivative of y with respect to x, in this example at the value x = 1, this value x = 1 playsno part in determining how the errors in x affect y; by contrast, of course, the actual value of thederivative, in this example 2, does play an essential part in this determination).
How to measure the spread in the errors in x and y? One might think that this could beestimated as the mean of the differences between the individual errors and their mean, for bothinput and output; but as can be easily seen, this difference is always zero. In this example, for theinput errors
mean input error = ((+0.2− 0.1) + (−0.1− 0.1) + (0.0− 0.1) + (0.3− 0.1)) /4 = 0.0, (14.3)
63
and the same zero is obtained for the output errors. In fact it is always the case that the meandifference between individual values and their mean must be zero.
Another way of measuring the spread would be to take simply the difference of the maximumand minimum values. But information would then be lost about the distribution of the remainingvalues, intermediate between these. The universally used way of measuring a spread is indirectlythrough a ‘squared spread’, meaning that a kind of ‘mean’ (to be explained in a moment) istaken between the squared differences just listed above. The technical term for a squared spread,calculated as in (14.4) below, is the variance:
variance of δx =((+0.2− 0.1)2 + (−0.1− 0.1)2 + (0.0− 0.1)2 + (0.3− 0.1)2
)/3 (14.4)
= (0.01 + 0.04 + 0.01 + 0.04)/3 = 0.03333,
and taking the square root gives a spread of√
0.03333 = 0.18257 in the input. (The divisor 3 in(14.4) will be explained shortly, and extra decimal places have been kept, to minimise round-offerrors later). As is almost always the case, this measure of the spread is less than the differencebetween the maximum and minimum of the values.
The square root of the variance is the standard deviation or (in the context of the estimationof uncertainties) the standard uncertainty. (There are many contexts where a standard deviationis not a standard uncertainty; these occur when a spread is a natural expression of variabilitywithout any implication of error, as in the case of heights of adult females (or males) in a specifiedpopulation). The variance and the standard deviation, calculated from a sample (in the aboveexample, the sample was a small one of 4) are estimates (in the case of the standard deviation, anapproximate estimate) of the corresponding quantities in the entire population. In this statisticalcontext, it should be noted, ‘population’ does not necessarily refer to people, but to any largenumber of measurements from which a representative sample can be drawn.
In general, if the sample size is n measurements, with a mean x of values x1, x2, x3,...xn (sothe sum of the x’s divided by n is x), the variance is calculated as
variance =
((x1 − x)2 + (x2 − x)2 + (x3 − x)2 + .... + (xn − x)2
)
n− 1. (14.5)
Why is the divisor in the general equation (14.5) n − 1 rather than n, and accordingly 3 ratherthan 4 in the sample of 4 just discussed in the particular example above? This can be explainedin a rigorous way, but for an intuitive feel for the situation, let us look at the terms being squaredin (14.4): they are +0.1, −0.2, −0.1, +0.2. These sum to zero, and must always do as stated justafter (14.3). If they must sum to zero, then we in effect have only three independent measurementsof the variance. Which three we do not know, but this does not matter: we divide by the effectivenumber of independent measurements. This effective number has the technical term degrees offreedom. In the general case of a sample of size n, the calculation of the variance involves nsquared terms, and the terms themselves sum to zero (being differences from the mean). So thereare only n− 1 degrees of freedom. If n is large, then of course the difference between n and n− 1may be negligible.
We note that the variance and standard deviation are ‘origin-independent’, meaning that thevariance and standard deviation of a set of measurements remain the same if a constant value isadded to all the measurements. This is of course an essential property if the ‘spread’ of results isto be quantified in a meaningful way. Thus the variance and standard deviation of 1, 4, 3, 4 arethe same as the variance and standard deviation of 6, 9, 8, 9, since a constant (5) has been addedto every measurement.
Here we interrupt the discussion about the route from errors to uncertainties, in order tointroduce a rather obvious topic that leads to the GUM formula for the propagation of uncertainties.
64
15 Functions of more than one variable, and partial differ-entiation
Up till now all functions have been functions of only one variable. There is of course no reason whythey should not be functions of more than one variable, and in fact this is by far the more commonsituation when uncertainties are to be estimated. The uncertainty of any output or measurand isafter all very likely to be determined by the uncertainties of several inputs. In what follows weshall continue to denote a function by the symbol y, but the input variables will be denoted x, w,t ....; thus for two input variables, y = f(x,w). (The symbol w here is simply a symbol for anotherinput variable; it is not of course the w of the chain rule introduced in section 5.2).
As established earlier (in (14.1)), the error δy in y resulting from the error δx in a variable xcan be written to a good approximation
δy =dy
dxδx, (15.1)
where the derivative dydx must be evaluated at the particular relevant value of the input x (unless,
as explained earlier (after (5.3)) the relationship between input and output is linear).
When there is more than one input variable, the symbol for the derivative, dydx is replaced by
the symbol for the partial derivative:∂y
∂x. (15.2)
This symbol ∂ can be pronounced ‘partial d’ or simply ‘d’, and in the case of (15.2) the partialderivative is taken with respect to the input x. In so doing, all other input variables are temporarilytreated as constant.
For example, suppose that
y = x sin 2w + (1− x) cos 2w. (15.3)
Then:∂y
∂x= sin 2w − cos 2w, (15.4)
and∂y
∂w= 2x cos 2w − 2(1− x) sin 2w. (15.5)
Previously, plotting functions of only one variable gave a line or a curve on a two-dimensionalplane. Plotting a function of two variables gives a surface in three-dimensional space. For interest,the surface corresponding to (15.3), y = x sin 2w + (1− x) cos 2w, is shown in Fig. 28. Along thew-axis, there is the expected oscillation created by the sine and cosine functions of w, whereasalong the x-axis a more ‘linear’ dependence is seen.
A simple form of the chain rule has been used to get from (15.3) to (15.4) and (15.5). If t isany variable, then d(sin 2t)
dt = d(sin 2t)d(2t)
× d(2t)dt = cos(2t) × 2 = 2 cos 2t. A near-identical approach
establishes that d(cos 2t)dt = −2 sin 2t.
If, for the case of a single input, its influence on the output is described by (15.1), it is intuitivelyclear (and moreover it is correct!) that with two inputs x and w, their combined influence on theoutput y is given by
δy =∂y
∂xδx +
∂y
∂wδw, (15.6)
and for more than two inputs, a longer expression is required but one that has the same ratherobvious structure.
In (15.6), δx is an error in input x, and δw is an error in input w. Through those partialderivatives, which are evaluated at the particular relevant values of x and w, and which are ineffect a measure of the sensitivity of y to x and w respectively, the errors δx and δw create anerror δy in the output y.
65
Fig. 28 The surface y = x sin 2w + (1− x) cos 2w
We now continue with the same numerical example as in section 14, where x = 1 and its errorsare +0.2, −0.1, 0.0 and +0.3, and (because there will now be a second input, we use the partialderivative notation) where ∂y
∂x = 2 (evaluated at x = 1). We now introduce a second input w,having the value (say) w = 5, and having a partial derivative ∂y
∂w = 3 at that value w = 5. (Asin the case of x = 1, this value w = 5, although needed for deciding where to evaluate the partialderivative of y with respect to w, plays no part in the following discussion).
Suppose there are four errors in w, namely δw with the values −0.1, −0.2, −0.3, −0.3. Thenet effect on δy can be seen from Table 2.
Table 2
δx ∂y/∂x = 2 δy from δx δw ∂y/∂w = 3 δy from δw net δy+0.2 +0.4 −0.1 −0.3 +0.1−0.1 −0.2 −0.2 −0.6 −0.80.0 0.0 −0.3 −0.9 −0.9
+0.3 +0.6 −0.3 −0.9 −0.3
We now find the relationship between the variance of the errors δx, δw and δy. The varianceof δx has already been calculated (see (14.4)) as 0.03333. The mean of the δw, from the table, is−0.22500. So the variance of the four values of δw is (and we note that since the mean is negative,−0.22500, taking differences from it gives positive signs in front of 0.22500):
variance of δw =
((−0.1 + 0.22500)2 + (−0.2 + 0.22500)2 + (−0.3 + 0.22500)2 + (−0.3 + 0.22500)2
)
3(15.7)
=0.015625 + 0.000625 + 0.005625 + 0.005625
3= 0.009167.
The table gives −0.475 as the mean of the errors δy propagated from the errors δx and δw, so the
66
variance of δy is:
variance of δy =
((+0.1 + 0.475)2 + (−0.8 + 0.475)2 + (−0.9 + 0.475)2 + (−0.3 + 0.475)2
)
3(15.8)
=0.330625 + 0.105625 + 0.180625 + 0.030625
3= 0.2158.
To summarise: we have
variance of δx = 0.03333
variance of δw = 0.009167
variance of δy = 0.2158.
Is there a simple relationship between these? Equ. (15.6) is restated here:
δy =∂y
∂xδx +
∂y
∂wδw
The values that were assigned to the partial derivatives were:∂y∂x = 2; ∂y
∂w = 3.
It may be checked numerically that:
variance of δy =(
∂y
∂x
)2
× variance of δx +(
∂y
∂w
)2
× variance of δw, (15.9)
because: 22 × 0.03333 + 32 × 0.009167 = 4× 0.03333 + 9× 0.009167 = 0.2158.
A useful, abbreviated notation for the variance of errors such as δx, δw and δy is the squaredstandard uncertainty denoted respectively by u2(x), u2(w) and u2(y). So the square root, u(x), isthe standard uncertainty of x, and similarly for other inputs or output. We note that the argumentof u2 is conventionally not the errors, but the actual inputs or output. So the following relationshiphas just been established:
u2(y) =(
∂y
∂x
)2
u2(x) +(
∂y
∂w
)2
u2(w). (15.10)
This relationship, and a more general one that includes it as a special (but frequent) case, will bediscussed further below. But we might well ask at this point: does the relationship (15.10), holdfor any two input variables? The answer is no. To illustrate this, suppose that the four errors δx
were as previously, and also ∂y∂x = 2 and ∂y
∂w = 3 as previously, but that now the four errors δw arefollows: +0.6, +0.3, +0.4, +0.7. The net effect on δy can be seen in Table 3.
Table 3
δx ∂y/∂x = 2 δy from δx δw ∂y/∂w = 3 δy from δw net δy+0.2 +0.4 +0.6 +1.8 +2.2−0.1 −0.2 +0.3 +0.9 +0.70.0 0.0 +0.4 +1.2 +1.2
+0.3 +0.6 +0.7 +2.1 +2.7
We now find the relationship between the variance of the errors δx, δw and δy. The varianceof δx has already been calculated (see (14.4)) as 0.03333. The mean of the δw, from the table, is+0.5. So the variance of the four values of δw is:
variance of δw = u2(w) =
((+0.6− 0.5)2 + (+0.3− 0.5)2 + (+0.4− 0.5)2 + (+0.7− 0.5)2
)
3(15.11)
=0.01 + 0.04 + 0.01 + 0.04
3= 0.03333.
67
This is the same as the variance of δx, and inspection of the values of δx and δw shows that thereason they have the same variance is that they differ by a constant value of 0.4; as mentionedearlier, the variance is origin-independent.
The table gives +1.7 as the mean of the errors δy propagated from the errors δx and δw, sothe variance of δy is:
variance of δy = u2(y) =
((+2.2− 1.7)2 + (+0.7− 1.7)2 + (+1.2− 1.7)2 + (+2.7− 1.7)2
)
3(15.12)
=0.25 + 1.00 + 0.25 + 1.00
3= 0.8333. (15.13)
To summarise: we have
variance of δx = 0.03333
variance of δw = 0.03333
variance of δy = 0.8333.
and as previously, ∂y∂x = 2; ∂y
∂w = 3.
Suppose we assume that, as previously,
u2(y) =(
∂y
∂x
)2
u2(x) +(
∂y
∂w
)2
u2(y) (15.14)
So u2(y) = 0.8333. Inserting numerical values into the right-hand side of (15.14):
right side = 22 × 0.03333 + 32 × 0.03333 = 4× 0.0333 + 9× 0.03333 = 13× 0.03333 = 0.4333
and this is not equal to 0.8333, the left-hand side of (15.14)!
What is missing? The following term, as will be explained shortly. We note first that, if thevariance of δx is 0.03333, denoted as u2(x), then the standard uncertainty u(x) of x is the squareroot of this, or 0.18257. Evidently we also have u(w) = 0.18257. The missing term is
2r(x,w)(
∂y
∂x
∂y
∂w
)u(x)u(w) (15.15)
where r(x, w) is the correlation coefficient between x and w and in this particular case has thevalue r(x,w) = +1. In general, two variables (here x and w) are partially mutually positivelycorrelated if they tend to follow each other: if one increases, the other is also likely to increase,and if one decreases, the other is also likely to decrease. As might be expected, they are partiallynegatively correlated when an increase in one is likely to accompany a decrease in the other, andso on. Correlation may or may not imply causation (which is why statements such as: ‘diet Apredisposes you to condition X’ are so often contentious). Correlation coefficients will be discussedin a little more detail shortly. So, numerically, this term in (15.15) is:
2× (+1)× 2× 3× 0.18257× 0.18257 = 0.4000. (15.16)
Adding this to 0.4333 gives 0.8333, which is indeed u2(y).
So it looks as if this case is described not by (15.10), but by this same equation with anadditional term. The correct complete equation is:
u2(y) =(
∂y
∂x
)2
u2(x) +(
∂y
∂w
)2
u2(y) + 2r(x,w)(
∂y
∂x
∂y
∂w
)u(x)u(w). (15.17)
The case tabulated in Table 2 is the case where r(x,w) = 0; this is why (15.10) held for that case.
The general equation (15.17) above can, in fact, be written more simply by noting that bothleft and right sides are perfect squares if r = +1, so taking the square root:
u(y) =∂y
∂xu(x) +
∂y
∂wu(w). (15.18)
68
Numerically in (15.18): on the left-hand side u(y) =√
0.8333 = 0.9129, and on the right-hand sideu(x) = u(w) = 0.18257, so the right-hand side is 2× 0.18257+3× 0.18257 = 5× 0.18257 = 0.9129,equal to the left-hand side of (15.18).
As a final simplified numerical example of the two-input case x and w, suppose that the fourerrors δx were still as previously, and also ∂y
∂x = 2 and ∂y∂w = 3 as previously, but that now the
four errors δw are the negative of their values in Table 3. The net effect on δy can now be seen inTable 4.
Table 4
δx ∂y/∂x = 2 δy from δx δw ∂y/∂w = 3 δy from δw net δy+0.2 +0.4 −0.6 −1.8 −1.4−0.1 −0.2 −0.3 −0.9 −1.10.0 0.0 −0.4 −1.2 −1.2
+0.3 +0.6 −0.7 −2.1 −1.5
We now find the relationship between the variance of the errors δx, δw and δy. The varianceof δx has already been calculated (see (14.4)) as 0.03333. The mean of the δw, from the table, is−0.5. So the variance of the four values of δw is:
variance of δw =
((−0.6 + 0.5)2 + (−0.3 + 0.5)2 + (−0.4 + 0.5)2 + (−0.7 + 0.5)2
)
3(15.19)
=0.01 + 0.04 + 0.01 + 0.04
3= 0.03333.
This is the same as the variance of δw obtained from Table 3. We expect this to be so, since the δwin Table 4 are simply the negatives of the δw in Table 3, and the variance (and standard deviation)of a set of values must remain unaltered if all the values are changed in sign. (The squares thatappear in the expression for the variance ensure this). Moreover, of course, the variance of the δxis still the same at 0.03333.
Table 4 gives −1.3 as the mean of the errors δy propagated from the errors δx and δw, so thevariance of δy is:
variance of δy =
((−1.4 + 1.3)2 + (−1.1 + 1.3)2 + (−1.2 + 1.3)2 + (−1.5 + 1.3)2
)
3(15.20)
=0.01 + 0.04 + 0.01 + 0.04
3= 0.03333. (15.21)
To summarise: we have
variance of δx = 0.03333
variance of δw = 0.03333
variance of δy = 0.03333.
and as previously, ∂y∂x = 2; ∂y
∂w = 3.
So u2(x), u2(w) and u2(y) are all equal. Equ. (15.10) cannot hold in this case either. It heldonly for the first case, tabulated above in Table 2. However, as in the case tabulated in Table 3,there is a missing term. The general equation (15.17) is repeated here:
u2(y) =(
∂y
∂x
)2
u2(x) +(
∂y
∂w
)2
u2(y) + 2r(x,w)(
∂y
∂x
∂y
∂w
)u(x)u(w).
The missing term, again, is 2r(x,w)(
∂y∂x
∂y∂w
)u(x)u(w) and the value of r(x,w) here is r(x, w) =
−1. Let us check this by evaluating numerically the right-hand side of (15.17), noting first thatu(x) =
√0.03333 = 0.18257 = u(y):
22×0.03333+32×0.03333+2×(−1)×2×3×0.18257×0.18257 = 4×0.03333+9×0.03333−12×0.03333
69
0 1 2 3 4 5-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Value
w
x
r = 0
Reading no.
Fig. 29 δx and δw for Table 2
= 1× 0.03333 = 0.03333
which equals the value of u2(y).
Again, as in the previous case for r(x,w) = +1, the present case r(x,w) = −1 permits asimplified equation (because (15.17) is now again a perfect square, and therefore we can takesquare roots). The counterpart to (15.18) is:
u(y) =∂y
∂xu(x)− ∂y
∂wu(w) (15.22)
oru(y) = −∂y
∂xu(x) +
∂y
∂wu(w). (15.23)
The reason for these two choices is that a square root can always have either sign, positive ornegative (unless it is zero). Since a standard uncertainty cannot be negative, we are free to chooseeither (15.22) or (15.23). In this case, (15.23) is the one to choose: the right side is
−2× 0.18257 + 3× 0.18257 = 0.18257,
which equals the square root of u2(y) = 0.03333.
It should be noted that having u(x), u(w) and u(y) all equal is by no means a general findingwhen r = −1; it is particular to the data chosen here.
The correlation coefficient r(x,w) between inputs x and w, can vary from r(x,w) = −1(complete negative correlation) through r(x,w) = 0 (no correlation), to r(x,w) = +1 (completepositive correlation). A correlation coefficient can never lie outside the range −1 through zero to+1. With the correlation coefficient r(x,w) between x and w having the values 0 in Table 2, +1in Table 3 and −1 in Table 4, it is instructive to compare Figs. 29, 30 and 31 for the three casesrespectively, with one another. Fig. 29 shows no correlation (as far as one can tell with a sample ofonly 4), whereas Fig. 30 shows a choreography of exactly duplicated steps by δx and δw, hence thevalue r(x,w) = +1. By contrast, Fig. 31 shows exactly duplicated anti-steps (to coin a phrase),hence the value r(x,w) = −1.
70
0 1 2 3 4 5-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Value
Reading no.
x
w
r = +1
Fig. 30 δx and δw for Table 3
0 1 2 3 4 5-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Value
Reading no.
x
w
r = -1
Fig. 31 δx and δw for Table 4
71
16 Propagation of uncertainties from inputs to output
As a preliminary to obtaining the basic propagation formula, we look again at the definition ofthe variance of n quantities, equ. (14.5). This was (using the abbreviated notation u2 introducedabove):
u2(x) =(
(x1 − x)2 + (x2 − x)2 + ... + (xn − x)2
n− 1
)(16.1)
where x is the mean of the x’s.
It will be useful to write (16.1) another way. Since (x1− x)2 = x21 + (x)2− 2x1x, (16.1) can be
written
u2(x) =
(x2
1 + (x)2 − 2x1x + x22 + (x)2 − 2x2x + ... +x2
n + (x)2 − 2xnx)
n− 1(16.2)
In this expression, there are n (x)2 in the top line, and we also have −2x1x− 2x2x− ...− 2xnx
= −2(x1 + x2 + ... + xn)x = −2n(x)2, since the sum of all the xs is nx. So the top line can bewritten
u2(x) =
(x2
1 + x22 + ... + x2
n − n(x)2)
n− 1. (16.3)
If n is large, n− 1 can be replaced by n and then (16.3) says that a variance can be described asthe ‘mean square minus squared mean’. (In passing, it should be noted that when computing avariance or standard deviation of quantities which vary only a little, for example only in the fifthor sixth decimal place, (16.3) can lead to serious round-off errors, and (16.1), involving squareddifferences from the mean, is to be preferred).
The correlation coefficient r(x,w) between two sets of quantities x1, x2,...,xn and theircorresponding w-values w1, w2,...,wn, with respective means x and w, can be defined as
r(x,w) =(x1 − x)(w1 − w) + (x2 − x)(w2 − w) + ... + (xn − x)(wn − w)
(n− 1)u(x)u(w). (16.4)
This is a general formula, but for our purposes we can consider the x’s and w’s to be errors, thus(δx)1,(δx)2...(δx)n and similarly (δw)1, (δw)2,...(δw)n. The mean of the δx is denoted δx, and themean of the δw is denoted δw.
So (16.4) can be written:
r(x,w) =((δx)1 − δx)((δw)1 − δw) + ((δx)2 − δx)((δw)2 − δw) + ... + ((δx)n − δx)((δw)n − δw)
(n− 1)u(x)u(w).
Similarly, (16.1) can be written:
u2(x) =(
((δx)1 − δx)2 + ((δx)2 − δx)2 + ... + ((δx)n − δx)2
n− 1
).
Equ. (16.4) can be written in another way, analogously to (16.1) and (16.3). In the top line of(16.4) we have
(δx)1(δw)1 + (δx)2(δw)2 + ... + (δx)n(δw)n
+ [−(δx)1 − (δx)2 − ...− (δx)n] δw + [−(δw)1 − (δw)2 − ...− (δw)n] δx
+nδx δw
= (δx)1(δw)1 + (δx)2(δw)2 + ... + (δx)n(δw)n − nδx δw − nδw δx + nδx δw =
(δx)1(δw)1 + (δx)2(δw)2 + ... + (δx)n(δw)n − δx δw. (16.5)
So r(x,w) may be written
r(x,w) =(δx)1(δw)1 + (δx)2(δw)2 + ... + (δx)n(δw)n − δx δw
(n− 1)u(x)u(w). (16.6)
72
In (16.5), we have used the obvious definitions for the mean x-error δx as δx =(δx)1+(δx)2+...+(δx)n
n , and similarly for the mean w error δw = (δw)1+(δw)2+...+(δw)nn .
For obtaining the propagation formula with a general n values of two inputs x and w (in theexamples in section 15, n = 4), as stated above we denote the errors in x as (δx)1, (δx)2,...,(δx)n,and similarly (δw)1, (δw)2,...,(δw)n. For example, in Table 4, we had (δx)1 = +0.2 and (δw)3 =−0.4. We now have:
(δy)1 =∂y
∂x(δx)1 +
∂y
∂w(δw)1
(δy)2 =∂y
∂x(δx)2 +
∂y
∂w(δw)2
...
(δy)n =∂y
∂x(δx)n +
∂y
∂w(δw)n (16.7)
The mean y-error δy can be written
δy =(δy)1 + (δy)2 + ... + (δy)n
n
=∂y
∂x
((δx)1 + (δx)2 + ... + (δx)n)n
+∂y
∂w
((δw)1 + (δw)2 + ... + (δw)n)n
=∂y
∂xδx +
∂y
∂wδw. (16.8)
Bearing in mind the general definition for variance or squared standard uncertainty in (16.1),we see that the variance u2(y) of y may be written
(n− 1)u2(y) =((δy)1 − δy
)2 +((δy)2 − δy
)2 + ... +((δy)n − δy
)2
=(
∂y
∂x(δx)1+
∂y
∂w(δw)1 − δy
)2
+(
∂y
∂x(δx)2+
∂y
∂w(δw)2 − δy
)2
+... +(
∂y
∂x(δx)n+
∂y
∂w(δw)n − δy
)2
. (16.9)
To simplify the mathematics, we can now assume that the errors δx in x are measured relative totheir mean δx, and similarly the errors δw in w are measured relative to their mean δw. In effect,this now says that δx = 0 and that δy = 0. In consequence, we will now have δy = 0, following(16.8). Thus in the case of (say) Table 2, the x-errors, which have a mean of +0.1, will now be(δx)1 = +0.2 − 0.1 = +0.1, (δx)2 = −0.1 − 0.1 = −0.2, (δx)3 = 0.0 − 0.1 = −0.1 and (δx)4 =+0.3−0.1 = +0.2. These now add up to zero. Similarly, the w-errors in Table 2 will now be (sincetheir existing mean is −0.225) (δw)1 = −0.1 + 0.225 = +0.125, (δw)2 = −0.2 + 0.225 = +0.025,(δw)3 = −0.3 + 0.225 = −0.075 and (δw)4 = −0.3 + 0.225 = −0.075, also adding up to zero.
It should be emphasised that assuming zero means for the errors (that is, measuring errorsrelative to their means) makes no difference to the final result in (16.14) or (16.15) below. Ifthe errors were taken ‘as given’, without taking differences from means, the following analysiswould have a more complicated form, but the same final result would be obtained. In this morecomplicated analysis, all terms involving means such as δx, δw and δy would finally cancel out.
Accordingly, (16.9) can be written
(n− 1)u2(y) =((δy)21+ (δy)22+ ... + (δy)2n
)
=(
∂y
∂x(δx)1+
∂y
∂w(δw)1
)2
+(
∂y
∂x(δx)2+
∂y
∂w(δw)2
)2
+ ... +(
∂y
∂x(δx)n+
∂y
∂w(δw)n
)2
,
and expanding the brackets, remembering that ‘(a + b)2 = a2 + b2 + 2ab’:
(n− 1)u2(y) =(
∂y
∂x
)2 ((δx)21 + (δx)22 + ... + (δx)2n
)
73
+(
∂y
∂w
)2 ((δw)21 + (δw)22 + ... + (δw)2n
)
+2[
∂y
∂x
∂y
∂w
]((δx)1(δw)1 + (δx)2(δw)2 + ... (δx)n(δw)n) (16.10)
=(
∂y
∂x
)2 (u2(x)(n− 1)
)+
(∂y
∂w
)2 (u2(w)(n− 1)
)
+2[
∂y
∂x
∂y
∂w
]((δx)1(δw)1 + (δx)2(δw)2...+ (δx)n(δw)n) . (16.11)
The u2(x) and u2(w) were introduced in (16.11) because, with δx = 0, we can now put (n −1)u2(x) = (δx)21 + (δx)22 + ... + (δx)2n, and similarly for the δw.
But the correlation coefficient r(x,w) as in (16.6) may be written (since δx = δw = 0)
r(x, w) =(δx)1(δw)1 + (δx)2(δw)2 + ... + (δx)n(δw)n)
(n− 1)u(x)u(w). (16.12)
So substituting (16.12) into part of (16.11), we obtain
(n− 1)u2(y) =(
∂y
∂x
)2 (u2(x)(n− 1)
)+
(∂y
∂w
)2 (u2(w)(n− 1)
)
+2(n− 1)r(x,w)∂y
∂x
∂y
∂wu(x)u(w), (16.13)
and cancelling out the (n− 1) on both sides of (16.13):
u2(y) =(
∂y
∂x
)2 (u2(x)
)+
(∂y
∂w
)2 (u2(w)
)+ 2r(x,w)
∂y
∂x
∂y
∂wu(x)u(w). (16.14)
As mentioned above, exactly the same equation (16.14) is obtained if the original errors δx and δyare treated without subtracting their means, so that they no longer need sum to zero. This impliesthat if a constant is added to one set of errors, say the δx, and another constant (or possibly thesame constant) to the δw, (16.14) remains valid for determining how the uncertainties propagateinto the output y. A constant in this context is often called a bias, and the bias (provided itremains constant) plays no part in the propagation of uncertainties from inputs to output. Thepropagation of the bias itself is, of course, determined by an equation such as (16.8), which relatesδy to δx and δw through the usual sensitivity coefficients (the partial derivatives).
The step from (16.13) to (16.14), when the n− 1 is canceled out from the left- and right-handsides, implies that there are n errors δx, embodied in the u2(x), and also n errors δw, embodiedin the u2(w). But what if the number of errors differs for the two inputs? Equ. (16.14) remainsvalid, because if m is the number of errors δw with m less than n, we merely need to take thedeficit n−m as consisting of errors δw that are all zero.
With more than two inputs, say three, namely x, w and t, (16.14) is generalised as follows:
u2(y) =(
∂y
∂x
)2 (u2(x)
)+
(∂y
∂w
)2 (u2(w)
)+
(∂y
∂t
)2 (u2(t)
)
+2r(x,w)∂y
∂x
∂y
∂wu(x)u(w) + 2r(w, t)
∂y
∂w
∂y
∂tu(w)u(t) + 2r(t, x)
∂y
∂t
∂y
∂xu(t)u(x). (16.15)
So for three inputs there are three correlation coefficients to consider. At this point it is worthnoting what is intuitively obvious, namely that (for example) r(x,w) = r(w, x): if one variable iscorrelated with a second, then the second is correlated to exactly the same extent with the first.
With n inputs, the number of correlation coefficients is (1/2)×n× (n−1). For n = 4, there aretherefore six correlation coefficients: if we call the inputs a, b, c and d, the correlations are r(a, b),r(a, c), r(a, d), r(b, c), r(b, d) and r(c, d).
74
For this general number of inputs, (16.15) is generalised in an obvious way, keeping the samemathematical structure. We obtain the propagation equation in section 5.1.2 of the GUM [1] foruncorrelated inputs, and the propagation equation in section 5.2.2 for correlated inputs.
Correlations might appear to impose a significant extra experimental and computational burdenwhen evaluating propagated uncertainties. In practice, it is often found that acceptable results areobtained if such correlations are assumed negligible. The key word here is ‘often’, not ‘always’ !The possible presence of correlations should be kept in mind.
It should be noted, in passing, that ‘mutual independence’ of two variables may appear to beanother way of saying that they have zero correlation, but in fact zero correlation is a weakercondition than independence. Independence implies zero correlation, but zero correlation does notimply independence. As an example, the four points P, Q, R and S on the border of the circlein Fig. 14 have zero correlation between their respective x and y co-ordinates, but these x and yvalues are not independent, since they are connected by the equation of a circle: y =
√a2 − x2.
In one important case correlations must be taken into account: this case arises when a straightline or curve is ‘fitted’ to data. For example, a variable y may depend on a variable x in a ‘linear’fashion, that is: y = K+ax, and the constants K (the intercept on the y-axis) and a (the slope) arefitted using any of several techniques, commonly by ‘ordinary least-squares’ (OLS) (this techniqueassumes that the x variables are much more precisely known than are the y-variables). OLSautomatically yields values of K, its standard uncertainty u(K), a and its standard uncertaintyu(a), and also the correlation coefficient r(K, a) between K and a. (The correlation will be zeroif the x-values have zero mean). Then if a value y0 is to be determined for a given x0, so thaty0 = K + ax0 with K, a and r(K, a) known, the standard uncertainty u(y0) in y0 is given by astraightforward application of the propagation equation (16.14). Since ∂y0
∂K = 1, ∂y0∂a = x0, (16.14)
gives:u2(y0) = u2(K) + x2
0u2(a) + 2x0r(K, a)u(K)u(a).
Similar considerations apply to the estimation of u(y0) when there is a quadratic or higher-powerdependence as well, for example y = K + ax + bx2. There are many practical applications of suchcases, linear and quadratic: for example, if x is a temperature and y is a temperature-dependentquantity such as electrical resistance or the speed of a chemical reaction. The case of a resistancevarying linearly with temperature is discussed in section 16.3.
Another case where correlations may need to be considered arises if two working instrumentsare used both of which have been calibrated against the same standard instrument.
An additional cautionary note is that for estimating correlations, it is quite useless to have onlyfour pairs of values of the two inputs; the values in Tables 2, 3 and 4 were only four in numberpurely for simplicity of illustration. In practice, at least 10 pairs of values are recommended inthe case of fitting of constants as just described, and ideally at least 30 will be needed if there is alarge scatter around the fitted line or curve. Given ample data, correlations roughly between +0.7and +1.0 (or, less frequently, between −0.7 and −1.0) are usually significant, whereas between,say, −0.3 and +0.3 they may be no more than random sampling fluctuations with no significantcorrelation.
It is worth noting that in establishing the equation for the propagation of uncertainties (suchas (16.14) for two inputs and (16.15) for three inputs), there has been no need to assume anyparticular statistical distribution for the inputs. By ‘statistical distribution’, more accuratelytermed a probability density distribution, PDF, is meant a functional form that describes how theprobability that a variable is located within a small range varies over the entire permitted range.The total probability of the variable being somewhere within its entire permitted range is of course1, namely certainty. The prime exhibit of a PDF, because it occurs so commonly, is the Gaussianor ‘normal’ distribution (‘normal’ because it is normal. One of the few examples where a technicalterm has an everyday meaning!). The functional form of the Gaussian PDF is:
f(x)dx = (1/σ)√
1/(2π) exp(−1
2(x− µ)2/σ2
)dx, (16.16)
and this is depicted in Fig. 32.
75
-4 -3 -2 -1 0 1 2 3 40.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Probability density of normal variable
x
68% point (1 )
95% point (1.96 )
99% point (2.58 )99% point (2.58 )
Fig. 32 The Gaussian or normal distribution
The reason for the frequent occurrence of the normal distribution is a consequence of theso-called Central Limit Theorem [16, chapter 6], which states that the combination of variousprobability density distributions all of roughly comparable variance often results in a goodapproximation to the normal distribution.
The ‘small range’ mentioned in the paragraph preceding (16.16) is dx. The Gaussian PDFis symmetric, with a bulky middle and rapidly thinning tails. The probability that a Gaussianvariable lies somewhere in the tails is therefore of course low. The mean of the distribution is µ(‘mu’), and its standard deviation is σ (‘sigma’). In Fig. 32, µ = 0 and σ = 1. The 68%, 95% and99% points are shown. The 68% points are located one standard deviation from the mean, so 68%of the total area under the curve lies between the ±1σ points. For the output or measurand, themore commonly used 95% points delimit the range such that the probability that the variable inquestion lies outside the range is only 5% (and because of the symmetry, 2.5% in the right-handtail and 2.5% in the left-hand tail). Similarly for the 99% points. When the standard uncertaintyof an output as obtained from (16.14) or (16.15) (or their generalisation to many inputs) is to beconverted to such an expanded uncertainty (given the upper-case symbol U), then an assumptionmust be made about the most likely statistical distribution of the measurand; the usual assumptionis of a Gaussian distribution.
The 95% points are situated approximately 1.96 standard deviations from the mean (which isalso the peak, in the case of symmetric distributions such as the Gaussian). Further approximating1.96 as 2 explains the common use of ‘95% expanded uncertainty’ as ±2× the standard uncertainty(we recall that in the context of uncertainties, ‘standard deviation’ and ‘standard uncertainty’ meanthe same thing). Thus if a temperature is measured as 25.0oC with a standard uncertainty of 0.1oC,this implies an expanded uncertainty U = 2× 0.1o = 0.2oC, and hence asserts that the chance ofthe temperature being greater than 25.2oC or less than 24.8oC is only 5%. The temperature canbe quoted as 25.0oC ± 0.2oC at a level of confidence of 95%. The ± should be reserved for suchexpanded (not standard) uncertainties.
Some commentators on the GUM have described the notion of an uncertainty-causing ‘dis-persion’ of the measurand as an inconsistency in the GUM. A further criticism is that in somecases the assumption of a Gaussian distribution is untenable, and this and some further perceiveddifficulties make for intensive discussions that the interested reader could follow in [6].
16.1 The standard uncertainty of an arithmetic mean
Among the simplest applications of (16.14) or (16.15) is the expression for the standarduncertainty of an arithmetic mean of n quantities x1, x2,...xn. These quantities are assumedto make up a sample drawn from the same population, which usually implies that each of thex’s can be assumed to have the same standard uncertainty u(x). Intuitively it may be thoughtthat the standard uncertainty of the mean of the x’s must be somewhat less than u(x), otherwise
76
what is the point of taking a mean value? We expect that the mean, after all, should be better‘buffered’ against measurement errors than any of the individual component measurements are.If we examine this intuition more closely, we see that we have implicitly assumed that the errorscontaminating each of the individual measurements are mutually uncorrelated, so that to someextent, at least, they ‘cancel out’. This assumption is indeed critical to the correct use of thestandard expression (16.21) for the standard uncertainty of the mean, which is derived as follows.
The arithmetic mean x is defined as
x =1n
(x1 + x2 + ... + xn) . (16.17)
Each of the x’s is now an ‘input’, and all the x’s have the same standard uncertainty u(x), andthe mean x is the ‘output’. We now use the obvious generalisation of (16.15) to n inputs (insteadof just three in (16.15)). Taking all correlations as zero in (16.15), as just discussed, and puttingy = x for the output gives:
u2(x) =(
∂x
∂x1
)2
u2(x) +(
∂x
∂x2
)2
u2(x) + ... +(
∂x
∂xn
)2
u2(x). (16.18)
The partial derivatives, from (16.17), are:
∂x
∂x1=
1n
;
∂x
∂x2=
1n
;
and so on until∂x
∂xn=
1n
;
All the partial derivatives are equal, and inserting them into (16.18) gives:
u2(x) =(
1n
)2
u2(x) +(
1n
)2
u2(x) + ...
(1n
)2
u2(x)(n terms added up) (16.19)
=(
1n
)2
× nu2(x) =u2(x)
n. (16.20)
This implies that
u(x) =u(x)√
n. (16.21)
Equ. (16.21) is the standard expression for the standard uncertainty of a mean value, andindicates that the mean is ‘more accurate’ by roughly the square root of the number of componentmeasurements. The implicit assumption, however, is that the component measurements aremutually uncorrelated. As a general rule, if there is some significant mutual correlation, it isoften possible to predict roughly where a particular measurement is likely to be located relativeto the two immediately adjacent measurements. As an extreme case, if the measurements followa linear relationship in time (that is, with constant drift), then obviously such a prediction canbe made with very high confidence, and this is a case of perfect correlation (r = +1 or r = −1).When there is significant mutual correlation, the standard uncertainty of the mean is reduced,relative to the standard uncertainty of the individual measurements, by less than
√n, and a more
complicated analysis than the one just given is necessary [7,8].
16.2 The standard uncertainty of a geometric mean
The geometric mean enters into the definition of the INR, which was discussed as example 4 insection 1. There, the quantity N was the geometric mean of the clotting time of normal subjects.It is of interest to obtain a general expression for the standard uncertainty of a geometric mean,and to compare this expression with that for the standard uncertainty of an arithmetic mean asin section 16.1. The geometric mean mg of n readings x1, x2,...xn is defined as
mg = (x1x2...xn)1n . (16.22)
77
Thus, for example, suppose x1 = 1, x2 = 2, x3 = 3 and x4 = 10. The arithmetic mean is x = 4.000
and the geometric mean is mg = (1 × 2 × 3 × 10)14 = 60
14 ≈ 2.783. The geometric mean is less
sensitive than is the arithmetic mean to the presence of any outliers, in this case the reading 10.It can also be quickly checked that if the quantities are all identical, the two means will coincide.
In practice, the geometric mean is calculated not by multiplying together individual readings,but by first taking the natural logarithm (namely, to base e) of (16.22). (The logarithm to base10 could be used equally well; the relationship between the two logarithms was given in (7.4) and(7.5), but whichever base is used, the same result, (16.33) below, will be obtained).
log mg =(
1n
)(log x1 + log x2 + ... + log xn), (16.23)
using, for example, rule (7) in section 7. We define L = log mg, so that
L =(
1n
)(log x1 + log x2 + ... + log xn), (16.24)
and the required partial derivatives, using (8.1), are then:
∂L
∂x1=
1nx1
;∂L
∂x2=
1nx2
; ...∂L
∂xn=
1nxn
. (16.25)
When all n inputs x1, x2,...xn are assumed mutually uncorrelated, the GUM propagation equationprovides the output u2(L) as
u2(L) =(
∂L
∂x1
)2
u2(x1) +(
∂L
∂x2
)2
u2(x2) + ... +(
∂L
∂xn
)2
u2(xn) (16.26)
=1n2
(u2(x1)
x21
+u2(x2)
x22
+ ... +u2(xn)
x2n
). (16.27)
As in the case of the arithmetic mean, we assume equal standard uncertainties u(x) for the x’s,and so (16.27) becomes:
u2(L) =u2(x)n2
(1x2
1
+1x2
2
+ ... +1x2
n
). (16.28)
But if L = log mg, then ∂L∂mg
= 1mg
. So now taking mg as the single input and L as output givesthe propagation relation:
u2(L) =(
∂L
∂mg
)2
u2(mg) =u2(mg)
m2g
. (16.29)
Then using (16.27):u2(mg)
m2g
=u2(x)n2
(1x2
1
+1x2
2
+ ... +1x2
n
), (16.30)
giving
u2(mg) =(
mgu(x)n
)2 (1x2
1
+1x2
2
+ ... +1x2
n
), (16.31)
or
u(mg) =mgu(x)
n
√(1x2
1
+1x2
2
+ ... +1x2
n
). (16.32)
so that a proportional standard uncertainty u(mg)mg
arises naturally in the case of a geometric mean:
u(mg)mg
=u(x)n
√(1x2
1
+1x2
2
+ ... +1x2
n
). (16.33)
In the INR discussion as example 4 in section 1, the geometric mean mg was denoted N , andu(mg) is u(N).
78
x
y
larger slope, smaller intercept
smaller slope, larger intercept
on y-axis
on y-axis
Fig. 33 Two best-fit lines to data points (not shown), illustrating negative intercept-slope correlation
For comparison with an arithmetic mean, u(x) = u(x)√n
as in (16.21). The standard uncertainty ofan arithmetic mean decreases as the square root of the number of observations, but it is interestingto note that the same relationship applies, roughly, to the geometric mean. This is because thereare n summed terms, often of comparable magnitude, under the square-root sign in (16.33), andthere is n outside the square-root sign in the bottom line of (16.33). So overall the dependence onn, for the geometric mean, goes as
√n
n = 1√n, as for the arithmetic mean.
16.3 Standard uncertainty of an interpolated point on a best-fit line or curve.
In this example, taking a correlation into account is essential. A best-fit line or curve to a set of(x, y) data points is obtained (usually by the technique known as ordinary least-squares (OLS) [15,chapters 6 and 7] or [16, chapter 3]. Each point has an x- and a y-value, with x usually measuredalong the horizontal axis and y along the vertical axis. The assumption is that the y-values dependon the x values; the experimenter sets the x-values (with an assumed negligible uncertainty), andmeasures the resulting y-values that each of them produces.
A case in point is the measurement of the temperature coefficient of a standard resistor. Thestandard resistor is made of a special alloy that confers on it a low temperature-coefficient, hencethe resistor’s role as a standard, but nevertheless the temperature-coefficient must be measuredover a small range of temperatures so that at a given temperature within this range, the value ofthe resistor can be accurately known. For simplicity we assume a linear change of resistance withtemperature:
y = K + ax (16.34)
where y is the resistance at temperature x, K is the intercept on the y-axis (that is, when x = 0),and a is the temperature-coefficient and for this linear case is the slope of the best-fit line. Theleast-squares procedure will yield numerical values for five quantities: the intercept K, its standarduncertainty u(K), the value of the temperature-coefficient a and its standard uncertainty u(a), andthe correlation coefficient r(K, a) between K and a. At a given temperature x0, the correspondingresistance y0 is given by y0 = K + ax0. The problem is to determine u2(y0). The required partialderivatives are:
∂y0
∂K= 1, (16.35)
and∂y0
∂a= x0. (16.36)
The propagation equation must now include the correlation term:
u2(y0) = u2(K) + x20u
2(a) + 2r(K, a)x0u(K)u(a). (16.37)
If the values of x are roughly equally spaced and are mainly positive, so that most of the datapoints are to the right of the y-axis, then r(K, a) will be negative. (A more accurate conditionfor the negative correlation is that the mean of the x-values is positive). Fig. 33 shows why, in
79
exaggerated form. If because of small errors the slope a (the temperature-coefficient) increases,then the intercept of the line on the y-axis, namely K, decreases, and vice versa. The correlationcoefficient can easily exceed 0.8 (in absolute value), and so the third term on the right-hand sideof (16.37) will affect u(y0) significantly.
In OLS, which is also known as Model I regression analysis, the assumption is made that thex-values have negligible uncertainty. If this is not so, a more complicated analysis is needed, oftenknown as total least-squares (TLS) but also as Model II regression analysis, and is the methodpertinent to many applications in clinical biochemistry [17].
Acknowledgments
As a physical metrologist with scant knowledge of biochemistry, I am greatly indebted to IanFarrance of RMIT, with whom I have co-authored two papers on statistics as applied to problemsin biochemistry and from whom I have learnt what little biochemistry I know. Ian made somevery useful comments on the present document. I am also very grateful to Daniel Burke for hiscomments and suggestions that have indisputably improved its educational value. Matthew Footalso made some pertinent comments on an early version of the manuscript. However, any remainingerrors are completely my responsibility. Last but certainly not least, I wish to express my gratitudeto my wife, Liliane, for her patience and support during the writing of this document.
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