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 the standard 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

iix

Nx

1

1

N

ii xx

N 1

2)(1

1

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 ‘=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 σx (sigma) is used to refer to it. You may also need to calculate a percent uncertainty ( %σx):

%5%100m99

m5%

x

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 σ

•In this course we will use σ to represent the uncertainty in a measurement 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 σ.

•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 3σ from the accepted value and this is a major disagreement

•To calculate Nσ 3

2.02.98.9exp

erimentalacceptedN

Uncertainty 9

Propagation of Uncertainties withAddition or Subtraction

22yxz

If z = x + y or z = x – y then the absolute uncertainty in z is given by

Example:

Uncertainty 10

Propagation of Uncertainties withMultiplication or Division

22 %%% yxz

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 propagate uncertainties in the same order that you perform the calculations.

Uncertainty 12

Uncertainty resulting from averaging N measurements

Nx

avg

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 σx = 1.0, so the uncertainty in the average is

3.0100.1

Nx

avg

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 and division. 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 z

Uncertainty 14

Uncertainty when a number is raised to a power

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

400170043017283 z

%1.446.3%)3.8(2146.3%)3.812( 2

12

1z

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

If z = xn then %σz = n ( % σx )

14.46.321

z

Uncertainty 15

Uncertainty when calculation involves a special function

Example: If θ = 120 ± 2.00

sin(140) = 0.242

sin(120) = 0.208

sin(100) = 0.174

For a special function, you add and subtract the uncertainties from 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

Uncertainty 16

Percent Difference

%100 valueaccepted

valuealexperiment- valueaccepteddiff%

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.

%4%100s

m8.9s

m4.9sm9.8

diff%2

22

Example: Calculate the percent difference if a measurement of g resulted in 9.4 m / s2 .