experimental error error analysis - michael burns analysis.pdf · antilog (-2.224) =...

1

Error AnalysisExperimental Error

• Experimental Error– The uncertainty obtained in a measurement of

an experiment– Results can from systematic and/or random

errors• Blunders• Human Error• Instrument Limitations

– Relates to the degree of confidence in ananswer

– Propagation of uncertainties must becalculated and taken into account

Experimental Error

It is impossible to make an exactmeasurement. Therefore, all experimentalresults are wrong. Just how wrong they aredepends on the kinds of errors that were madein the experiment.

As a science student you must be careful tolearn how good your results are, and to reportthem in a way that indicates your confidence inyour answers.

Types of Errors

• Systematic Errors– These are errors caused by the way in which

the experiment was conducted. In otherwords, they are caused by the design of thesystem or arise from flaws in equipment orexperimental design or observer

– Sometimes referred to as determinate errors– Reproducible with precision– Can be discovered and corrected

2

Systematic Error

The cloth tape measure that you use to measurethe length of an object had been stretched outfrom years of use. (As a result, all of your lengthmeasurements were too small)

The electronic scale you use reads 0.05 g toohigh for all your mass measurements (because itwas improperly zeroed at the beginning of yourexperiment).

Examples:

Detection of Systematic Errors

• Analyze samples of known composition– Use standard Reference material– Develop a calibration curve

• Analyze “blank” samples– Verify that the instrument will give a zero

result• Obtain results for a sample using multiple

instruments– Verifies the accuracy of the instrument

How to Eliminate Systematic Errors

How would you measure the distance between twoparallel vertical lines

-most would pull out a ruler, align one end withone bar, read of the distance.

-You should put ruler down randomly (asperpendicular as you can). Note where each markhits the ruler, then subtract the two readings.Repeat a number of times and average theresult.

-Minimize the number of human operations youcan

Elimination of systematic error can best beaccomplished by a well planned and well executedexperimental procedure

Types of Errors• Random Errors

– Sometimes referred to as indeterminateerrors or noise errors

– Arises from things that cannot be controlled• Variations in how an individual or individuals read the

measurements• Instrumentation noise

– Always present and cannot always becorrected for, but can be treatedstatistically

– The important property of random error isthat it adds variability to the data but doesnot affect average performance for the data

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Random Errors

Examples:

You measure the mass of a ring three timesusing the same balance and get slightly differentvalues: 12.74 g, 12.72 g, 12.75 g

The meter stick that is used for measuring,slips a little when measuring the object

Accuracy or Precision

• PrecisionReproducibility of resultsSeveral measurements afford the same

resultsIs a measure of exactness

• AccuracyHow close a result is to the “true” value“True” values contain errors since they too

were measuredIs a measure of rightness

Accuracy vs Precision

YESYES3.1415926

NOYES3.14

YESNO7.18281828

NONO3

PrecisionAccuracy=

Calculating Errors

TerminologySignificant Figures – minimum number of digits

required to express a value in scientific notation withoutloss of accuracy

Absolute Uncertainty – margin of uncertaintyassociated with “a” measurement

Relative Uncertainty – compares the size of theabsolute uncertainty with the size of its associatedmeasurement (a percent)

Propagation of Uncertainty – The calculationto determine the uncertainty that results from multiplemeasurements

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Significant Figures

How to determine which digits are SignificantWrite the number as a power of 10Zero’s are significant and must be included when they

occur• In the middle of a number• At the end of a number on the right hand side of the

decimal point– This implies that you know the value of a measurement

accurately to a specific decimal point

The significant figures (digits) in a measurementinclude all digits that can be known precisely, plus alast digit that is an estimate.

Significant Figures

Let’s look at 123.45

1.2345x102

Scientific Notation

We have 5 significant digits

Let’s look at 0.000123

1.23x10-4

We have 3 significant digits

Significant Figures

Determine the number of significantdigits in the following numbers

– 142.7– 142.70– 0.000006302– 0.003050– 10.003 x 104

– 9.250 x 104

– 9000– 9000.

Significant Figures

142.7 1.427x102

4 significant digits

142.70 1.4270x102


5

Significant Figures

0.000006302 6.302x10-6


0.003050 3.050x10-3


Significant Figures

9.250x104 9.250x104


10.003x104 1.0003x105


Significant Figures

9000 9x103

1 significant digit

9000. 9.000x103


Significant Figures

The last significant digit in ameasured quantity is the first digitof uncertainty

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Significant Figures

58.3 ± 0.158.358% Transmittance

0.234 ± 0.0010.2340.23Absorbance

True expression1 degree ofuncertainty

Certain values

Determine the significant figures from thediagram below

Significant Figures

When adding or subtracting, the last digitretained is set by the first doubtfulnumber.

When multiplying or dividing, the numberof significant digits you use is simply thenumber of significant figures as is in theterm with the fewest significant digits.

Adding Significant Digits

4503+34.90+550= 5090

3 is the first doubtful number



The 87.9 are the doubtful numbers


Via Calculator: 5087.9

Adding Significant Digits

2456.2345+23.21=

23400.00+111.49=

23400+111.49=

234000-2340=

2479.44

23511.49

23500

232000

2479.4445

23511.49

23511.49

231660

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Multiplying Significant Digits

2.7812x1.7= 4.72804

Rounded to 4.7 because 1.7 only has 2 significant digits

4.7

Multiplying Significant Digits

14.200x3.2400=

1.00x150.03=

1200x1.234=

45.35.2345=

48.008

150.03

1480.8

8.654121…

48.008

150

1500

8.65

Significant Figures in Logarithms andAntilogarithms

• Logarithm of n– n = 10a or log n = a

• 2 parts to a logarithm– Characteristic – integer part– Mantissa – decimal part

• Logarithm – the number of significant digits found in n = thenumber of significant digits in the mantissa

• Antilogarithm – the number of significant digits in the mantissa =the number of significant digits expressed in the answer

1.23451.2345

Express the answer of each of the following withthe correct # of Significant Figures

• Logarithms and Antilogarithms– log 339 = log 1237 =– log (3.39 x 10-5) = log 3.2 =– antilog (-3.42) = antilog 4.37 =– Log 0.001237 = 104.37 =– 10-2.600 = log (2.2 x 10-18) =– antilog (-2.224) = 10-4.555 =

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Logarithms

log 339 = 2.530 log 1237 =3.0924

log (3.39 x 10-5) =-4.470 log 3.2 =0.51

antilog (-3.42) =3.8x10-4 antilog 4.37 =23000

Log 0.001237 =-2.9076 104.37 =23000

10 -2.600 =2.51 log (2.2 x 10-18) =-17.66

antilog (-2.224) = 5.97x10-310 -4.555 =2.79x10-5

Rounding

Rounding is the process of reducing the numberof significant digits in a number. The result ofrounding is a "shorter" number having fewernon-zero digits yet similar in magnitude. Theresult is less precise but easier to use. Thereare several slightly different rules forrounding.

Rounding

Common method• This method is commonly used, for example in

accounting.• Decide which is the last digit to keep.• Increase it by 1 if the next digit is 5 or more

(this is called rounding up)• Leave it the same if the next digit is 4 or less

(this is called rounding down)• Example: 7.146 rounded to hundredths is 7.15

(because the next digit [6] is 5 or more).

RoundingThis method is also known as statistician's rounding . It is identicalto the common method of rounding except when the digit(s)following to rounding digit start with a five and have no non-zerodigits after it. The new algorithm is:

Decide which is the last digit to keep. Increase it by 1 if the next digit is 6 or more, or a 5 followed by one or more

non-zero digits. Leave it the same if the next digit is 4 or less Round up or down to the nearest even digit if the next digit is a five followed

(if followed at all) only by zeroes. That is, increase the rounded digit if it iscurrently odd; leave it if it is already even.

Examples:

7.016 rounded to hundredths is 7.02 (because the next digit (6) is 6 or more) 7.013 rounded to hundredths is 7.01 (because the next digit (3) is 4 or less) 7.015 rounded to hundredths is 7.02 (because the next digit is 5, and the

hundredths digit (1) is odd) 7.045 rounded to hundredths is 7.04 (because the next digit is 5, and the

hundredths digit (4) is even) 7.04501 rounded to hundredths is 7.05 (because the next digit is 5, but it is

followed by non-zero digits)

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Increasing Precision withMultiple Measurements

One way to increase your confidence in experimentaldata is to repeat the same experiment many times.

When dealing with repeated measurements, there arethree important statistical quantities

Mean (or average)

Standard Deviation

Standard Error

Mean

What is it:

An estimate of the true value of the measurement

Statistical Interpretation:

The central value

Symbol:

x

Standard Deviation

What is it:

A measure of the spread in the data


You can be reasonably sure (about 70% sure)that if you repeat the same experiment onemore time, that the next measurement will beless than one standard deviation away fromthe average

Symbol:

Standard Error

What is it:

An estimate in the uncertainty in the averageof the measurements


You can be reasonably sure (about 70% sure)that if you repeat the entire experiment againwith the same number of repetitions, theaverage value from the new experiment will beless than one standard deviation away fromthe average value of this experiment

Symbol:M N

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Example

Measurements: 0.32, 0.54, 0.44, 0.29, 0.48

Calculate the Mean: 0.41

Calculate the Standard Deviation: 0.11

Calculate the Standard Error: 0.05

M N

Therefore: 0.41±0.05

Relative Compared to AbsoluteUncertainty

Absolute uncertainty illustrates theuncertainty in a measurement

6.3302± 0.001

Relative uncertainty illustrates themagnitude of uncertainty with regardto the measurement.

Relative Compared to AbsoluteUncertainty

Relative uncertainty – compares theabsolute uncertainty with the size ofthe associated measurement

Relative uncertainty = absolute uncertainty / measurement

Percentage Relative Uncertainty

% relative uncertainty = relative uncertainty x 100

Propagation of Uncertainty

• Since measurements commonly willcontain random errors that lead to adegree of uncertainty, arithmeticoperations that are performed usingmultiple measurements must takeinto account this propagation oferrors when reporting uncertaintyvalues

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Systematic Errors

Errors calculated from data are Random Errors

Errors from the instrument are called System Errors(usually labeled on instrument or told by instructor asa percent)

15.23 0.05 0.17random systemk k k

22

15.23 0.1815.2 0.2

ran systemk k k

or

Error Propagation

There are 3 different ways of calculating orestimating the uncertainty in calculated results

Significant digits (The easy way out)Useful when a more extensive uncertainty analysis is notneeded.

Error Propagation (Not as bad as it looks)Useful for limited number or single measurements

Statistical Methods (When you have lots ofnumbersUseful for many measurements

Dependent Error Propagation

Adding and Subtracting

...e x y z

Multiplying and Dividing

...x y z

e Ex y z

Average

v v

Dependent (approx)

(121)+(52)-(73) =(12+5-7) (1+2+3)=106

(121)*(52)*(73)

1 2 312 5 7 12 5 7

12 5 7420 383

Average is 25, then 25 5

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Propagation of ErrorsBasic Rule

If x and y have independent random errorsand , then error in z=x+y is

2 2z x y

x y

3 0.14 0.2

xy

3 47

z

2 2

0.1 0.2

0.223

z

7 0.2Therefore we have



1.76 (0.03) + 1.89 (0.02) – 0.59 (0.02) =

Z=1.76+1.89-0.59=3.06

2 2 2

3

0.03 0.02 0.02

1.7 100.041231056

z

Therefore Z=3.06 0.04

Propagation of ErrorsBasic Rule

If x and y have independent randomerrors and , then error in z=xy is

22x yz z

x y

x y

3 0.1

4 0.2

x

y

3 412

z

2 20.1 0.212

3 40.72211102555

z

12 0.7Therefore we have



[1.76(0.03) x 1.89(0.02)] / 0.59(0.02) =

Z=

1.76 1.89

5.6379661020.59

2 2 21.76 1.89 0.03 0.02 0.020.59 1.76 1.89 0.59

0.222083034

z

Therefore z=5.6 0.2

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Putting it Together

x=200 2Y=50 2z=40 2

xq

y z

x, y, z are independent, find q

Let d=y-z 2 250 40 2 2

10 2 2

10 3

d

20010

20

xq

d

2 22 320

200 10

20 0.901

6

q

Therefore q=20 6

What about Functions of 1 Variable

Find error for with s=20.023V s

We cannot use because

s, s, s are not independent

2 2 2s s s

z zs s s

What to the rescue???

Calculus

V=s3

Let’s take the derivative of V withrespect to s 23

dVs

ds

2

2

3

3 2 0.02

0.24

dV s ds

Therefore the value for V is V=80.2

32

8

V

Think of dV and ds as a small change(error) in V and s

x=100 6 then find V when V x

A function of one variable… CALCULUS

12

121

2

26

2 1000.3

V x

x

dV xdx

dxdV

x

Therefore V=10.0 0.3

10010

V x

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What about a Function witha Constant?

You measure the diameter of a circle to be 20.02

Determine the area of the circle2

2

2

214

A r

d

d

Calculus

12121

2 0.0220.06

dAd

dd

dA d dd

2

21

A r

The area is 3.14 0.06

If q=f(x1, x2, x3, …xn)

22 2

1 21 2

... nn

q q qq x x xx x x

then

Let q=x1+x2

2 2

1 21 2

2 21 2

2 2

1 1

1 1

q qq x xx x

x x

x x

2 21 1q x x

Previous rule

PROOF

If q=f(x1, x2, x3, …xn)

Let q=x1*x2

2 2

1 1

1 2

x xq q

x x

Previous rule

PROOF

2 2

1 21 2

2 22 1 1 2

2 22 22 1 1 2

2 22 2

1 22 21 2

2 2

1 2

1 2

q qq x xx x

x x x x

x x x x

q qx x

x x

x xq

x x

21

qxx

The Atwood Machine consists of two masses M and mattached to the ends of a light, frictionless pulley. Whenthe masses are released, the mass M is show to acceleratedown with an acceleration:

M ma g

M m

Suppose the M and m are measured as M=100 1g and m=50 1 g.Find the uncertainty in a

2

2

1 1

2

M m M mag

M M m

mgM m

2

2

1 1

2

M m M ma gm M m

MgM m

The Partial Derivatives are:

15

2 2

2 2

2 2

2 22 22

2 2 2 2

2

2 2

2

2 9.850 1 100 1

100 50

0.1

a aq M m

M m

mg MgM mM m M m

gm M M m

M m

100 509.8

100 503.3

M ma g

M m

Therefore a=3.3 0.1 m/s2

Uncertainty Focal Length

pqf

p q

Determine the focal length plus uncertainty when p=100±2cm and q=30±1 cm

2

2

2

1q p q pqfp p q

qp q

2

2

2

1p p q pqfq p q

pp q

Focal Length

2 2

2 24 4

2

2 24 4

2

30 2 100 1

100 30

0.6112

f ff p q

p q

q p p q

p q

100 30100 30

23.07623.1

f

The focal length is 23.1±0.6 cm or 23±1 cm

Ugly Trig Problem

2

cos 4xq

x y

Determine q and error is x=10±2, y=7±1, Ø=400±30

2

cos 4 2

cos 4

yqx x y

2

2 cos 4

cos 4

xqy x y

2

4 2 sin 4

cos 4

x yq

x y

=-0.732

=0.963

=9.813

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Still the Ugly Trig Problem

2

2 20.732 2 0.963 1 9.813 3

180

3.31.8165

q

Therefore q=3.5±2

Trig shouldbe in

radians

experimental error error analysis - michael burns analysis.pdf · antilog (-2.224) =...

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