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Uncertainty 1

Treatment of Uncertainties

PHYS 244, 246

© 2003



Uncertainty 2

Types of Uncertainties

Random Uncertainties: result from the randomness of

measuring instruments. They can be dealt with by making

repeated measurements and averaging. One can calculate thestandard deviation of the data to estimate the uncertainty.

Systematic Uncertainties: result from a flaw or limitation in the

instrument or measurement technique. Systematic uncertainties

will always have the same sign. For example, if a meter stick is

too short, it will always produce results that are too long.



Uncertainty 3

Accuracy vs. Precision

Accurate: means correct. An accurate measurement

correctly reflects the size of the thing being

measured.

Precise: repeatable, reliable, getting the same

measurement each time. A measurement can be

precise but not accurate.



Uncertainty 4

Standard Deviation

§!

! N

i

i x N

x1

1

§!

! N

i

i x x N

1

2)(1

1W

The average or mean of a set of data is

The formula for the standard deviation given below is the one

used by Microsoft Excel. It is best when there is a small set of measurements. The version in the book divides by N instead of

N-1.

Unless you are told to use the above function, you may use the

Excel function µ=stdev(B2:B10)¶



Uncertainty 5

Absolute and Percent

Uncertainties

If x = 99 m ± 5 m then the 5 m is referred to as an absolute

uncertainty and the symbol x (sigma) is used to refer to it. You

may also need to calculate a percent uncertainty ( % x):

%5%100m99

m5% !v¹¹

º

¸©©ª

¨! xW

Please do not write a percent uncertainty as a decimal ( 0.05) because the reader will not be able to distinguish it from an

absolute uncertainty.



Uncertainty 6

Standard Deviation



Uncertainty 7

Standard Deviation



Uncertainty 8

Expressing Results in terms of the

number of

In this course we will use to represent the uncertainty in ameasurement no matter how that uncertainty is determined

You are expected to express agreement or disagreement

between experiment and the accepted value in terms of a

multiple of .

For example if a laboratory measurement the acceleration

due to gravity resulted in g = 9.2 ± 0.2 m / s2 you

would say that the results differed by 3 from the accepted

value and this is a major disagreement

To calculate N

32.0

2.98.9exp!

!

!

W W

er imental accepted N



Uncertainty 9

Propagation of Uncertainties with

Addition or Subtraction

22

y x z W W W !

If z = x + y or z = x ± y then the absolute uncertainty in z is

given by

Example:



Uncertainty 10

Propagation of Uncertainties with

Multiplication or Division

22

y x z W W W !

If z = x y or z = x / y then the percent uncertainty in z is given

by

Example:



Uncertainty 11

Propagation of Uncertainties in mixed

calculations

If a calculation is a mixture of operations, you propagateuncertainties in the same order that you perform the calculations.



Uncertainty 12

Uncertainty resulting from

averaging N measurements

N

xav g

W W !

If the uncertainty in a single measurement of x is statistical, then

you can reduce this uncertainty by making N measurements and

averaging.

Example: A single measurement of x yields

x = 12.0 ± 1.0, so you decide to make 10 measurements and

average. In this case N = 10 and x = 1.0, so the uncertainty

in the average is

3.0100.1 !!! N

xav g

W

W

This is not true for systematic uncertainties- if your meter stick

is too short, you don¶t gain anything by repeated

measurements.



Uncertainty 13

Special Rule:

Uncertainty when a number is multiplied by a

constant

This is actually a special case of the rule for multiplication anddivision. You can simply assume that the uncertainty in the

constant is just zero and get the result given above.

Example: If x = 12 ± 1.0 = 12.0 ± 8.3 % and z = 2 x, then z =24.0 ± 8.3 % or z = 24 ± 2. It should be noted that you would

get the same result by multiplying 2 (12 ± 1.0)= 24 ± 2.

%3.8%)3.8(%)0(% 22!! zW



Uncertainty 14

Uncertainty when a number is raised to a

power

%251728%)3.8(31728%)3.812(33

s!s!s! z

400170043017283s!s! z

%1.446.3%)3.8(

2

146.3%)3.812( 2

12

1

s!s!s! z

Example: If z = 12 ± 1.0 = 12.0 ± 8.3 % then

If z = xn then % z = n ( % x )

14.46.321

s! z



Uncertainty 15

Uncertainty when calculation involves a

special function

Example: If = 120 ± 2.00

sin(140) = 0.242

sin(120) = 0.208

sin(100) = 0.174

For a special function, you add and subtract the uncertaintiesfrom the value and calculate the function for each case.

Then plug these numbers into the function.

And thus sin(120 ± 20 ) = 0.208 ± 0.034

0.034

0.034



Uncertainty16

Percent Difference

100valueaccepted

valuealexperiment-valueaccepteddiff v¹¹

º ¸

©©ª¨!

Calculating the percent difference is a useful way to compare

experimental results with the accepted value, but it is not a

substitute for a real uncertainty estimate.

4100

sm8.9

sm4.9

sm9.8

diff 2

22

!v

¹¹¹

º

¸

©©©

ª

¨ !

Example: Calculate the percent difference if a measurement

of g resulted in 9.4 m / s2 .

uncertainty notes 1

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