Uncertainty Analysis

by Matt Young

Based on a talk given to the Laser Measurements Short Course, Boulder, Colorado, August, 2000

No measurement is exact

A measured value is therefore meaningless without some statement of its accuracy


Postmodernism: No fact can be known with absolute certainty

GUM: True value does not exist; only measurements exist

I take the approach deprecated in Annex E.5 of GUM (Guide to the Expression of Uncertainty in Measurement; see References)

The result is the same as the GUM's but, uh, intelligible and more nearly consistent with common parlance


Mean value

    µ = (1/N) S1N (xi)

is an estimator of X

Measurements may or may not cluster about the true value

(1) The values xi cluster about the true value X.

    (a) The spread is narrow, so µ is a good approximation to X.

    (b) The spread is wide, so µ is only a fair approximation to X.

(2) The values xi do not cluster about the mean, so µ is not a very good approximation to X

Statistical errors

Error ei is the difference

    ei = xi - X

We do not (and cannot) know X

Approximate X by the mean µ of the experimental values:

    ei* = xi - µ

s = sqrt[(1/(N-1)) S1N (ei*2)] Gaussian or normal distribution of errors
  Standard deviation of the mean

Peak of the Gaussian curve estimates the value for which ei = 0 (not X = 0)

       s = s/sqrt(N)

is the standard deviation of the mean
as opposed to the sample standard deviation s

Type A uncertainty

Uncertainty that is measured by statistical means

Mean µ of N measurements has a 68 % probability of being between µ  - s and µ + s , or         ur = s

due to random errors

Standard uncertainty is always positive

mean ± k times standard uncertainty Type B uncertainty

Uncertainty that is not measured statistically

I think of Type B errors as belonging to several subsets Estimated errors (my terminology)


Measure long distance with steel tape

Rectangular distribution         u = em/sqrt(3)


Digital voltmeter, no electronic noise

Note no division by sqrt(N) since repeated measurements will be identical

Imported errors (my terminology)

Manufacturer specifies an uncertainty


More generally,

        u = 1.5 uf/sqrt(3)

(f for manufacturer)

Systematic errors (deprecated term)

Errors that generally have only one sign

Example: thermal expansion in a steel tape Example adapted from micrometry: Example continued
  Uncertainty of correction

Assume maximum error emof the correction = one-half the correction itself

This approach allows us to specify a result ± a single uncertainty To sum it up:
Table 1. Standard uncertainties.
Type of uncertainty Distribution of errors Standard uncertainty Value
Random or statistical (Type A) Gaussian Standard deviation of mean
Estimated (including uncertainty of systematic error or bias) (Type B) Rectangular Standard deviation of rectangular distribution em/sqrt(3)
Imported (Type B) Rectangular Manufacturer's specification 1.5 um/sqrt(3)
Systematic (Type B) Bias --  b


Combined standard uncertainty

    uc = sqrt(ur2 + u12 + u22 + u32 + ...)

Express experimental results in form

        µ ± 2 uc


¡Uncertainty analysis is approximate and subjective!

Dirty secret:

A statement of uncertainty tells what we think about our measurement more than it tells about the measurement itself (thanks to Ron Wittmann)

Appendix *

Older sources add systematic errors arithmetically. In our notation,

    uc = sqrt(ur2) + u1 + u2 + u3 + ...

where ui here represents a systematic error

* A vestigial part of a book for which no one has yet discovered a use

Copyright 2000 by M. Young.  All rights reserved.


Anonymous, Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization, Geneva, 1993.

Barry N. Taylor and Chris E. Kuyatt, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results, Natl. Inst. Stand. Technol. Tech. Note 1297, Washington, 1994. Available on the Web at http://physics.nist.gov/Pubs/guidelines/outline.html.  This is sort of a guide to the GUM.

John Taylor An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, University Science Books, Mill Valley, California, 1997. (This excellent book does not teach or conform to the methodology of the GUM.)