NOTES ON UNCERTAINTY ANALYSIS FOR MEL LABS by Matt Young For more detail, see also http://www.mines.edu/Academic/courses/physics/phgn471/uncertainty.pdf Copyright © 2002 by Matt Young. All rights reserved. Matt Young’s Home Page

1. Definitions Error = deviation from True Value Uncertainty = estimate of probable error

2. Types of uncertainty •

Measured or statistical uncertainty •

•

for fluctuating or random variables

All other uncertainties •

for variables whose statistics are not known

•

calculated from estimates

•

include but not limited to systematic errors

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3. Uncertainty due to measured or statistical errors •

Data are noisy, as due to voltage fluctuations

•

Temporarily ignore other sources of uncertainty

•

Measurand = quantity to be measured •

Individual measured values are mi

•

Mean = µ, calculated from mean of data set

•

N = number of data points

•

True value = M = ??? •

µ is called an estimator of M

Calculate the standard deviation of the mean, or SDOM: N

σm = •

∑ i =1

(mi − µ ) 2

N ⋅ ( N − 1)

)m is the standard uncertainty due to random errors alone

•

)m decreases in proportion to (N)

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4. Significance of standard uncertainty ) •

M falls between µ – ) and µ + ) with 68 % probability •

•

68 % confidence interval

M falls between µ – 2) and µ + 2) with 95 % probability •

•

95 % confidence interval

M falls between µ – 3) and µ + 3) with 99.7 % probability •

99.7 % confidence interval

Half the battle is learning the vocabulary!

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Cheap and dirty way to estimate ) •

Calculate the maximum value of the dataset minus the minimum value

•

Divide by 6·(N)

•

Result is a fair estimate of )

•

Use to check your calculation

¡Important note!

) is the 68 % confidence interval; the ’s

below are your best estimates of 99.7 % confidence intervals. They are not directly comparable.

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5. Estimated errors (all other errors) •

Usually errors whose statistics are not known

P artial  U n certain ty      =  + ( G u essin g )  A n alysis   D ifferen tiatio n  Suppose m = m(a,b,c) & we calculate m by measuring

a, b, & c Let a = extreme value of error of a •

½ scale division, for example, or

•

mechanical tolerance in a part, for example

a is •

an estimate or a guess

•

of the 99.7 % confidence interval

Estimate error m of m due to error a of a:

∂m ∆m a = ⋅ ∆a ∂a • •

Assumes a