Errors

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…from CheLabWiki, an online resource for chemical-engineering laboratories located at www.chelabwiki.org; Site Revision #456; 6 January 2009.

Let y represent the (unknown) exact value of a measurable quantity and let y_i represent one value measured for y. Then the error e_i in y_i is the difference

We divide any error into two parts: systematic error and statistical error.

Systematic Error: Accuracy

Systematic error is the difference between the (unknown) exact value y and the mean y_m of an infinite number of repeated measurements,

Systematic errors are caused by some consistent bias in the measurement, such as a faulty instrument calibration, impurities in samples, consistent misreading of a meter. Such errors cannot be found by mathematical manipulations of measurements repeated at a single x-value. Instead, estimates may be obtained

(a) by thoughtful consideration of how a measurement might be biased and

(b) by comparing measured values to other, independent experiments or theories.

When systematic errors can be estimated, we may compensate for them by applying correction factors to measured values.

Design and execution of experiments focus on controlling or eliminating systematic error; that is, good experiments push systematic error into the background, so that the phenomena of interest are made apparent and accessible.^[1] This becomes more challenging as we move to higher accuracy. The difficulties are often problems of scale—not only must we control large effects more tightly, but additional subtle effects may become important when high accuracy is sought. To paraphrase Galison,^[1] some systematic errors leave light footprints. The total systematic error determines the accuracy of an experiment.

Statistical Error: Precision

Statistical error (also called random error) is the difference between one measured value and the mean of an infinite number of repeated measurements,

Statistical errors are caused by momentary bias or erratic fluctuations, such as a voltage surge, a pressure fluctuation, or a momentary lapse in reading a meter.

Figure 1. Accuracy differs from and is independent of precision. Let the cross in each of the above panels represent the exact value of a quantity. We measure the quantity ten times; each panel above shows one of four possible outcomes from the ten measurements: (a) low in both accuracy and precision, (b) low accuracy but high precision, (c) high accuracy but low precision, or (d) high in both accuracy and precision.

(a) The sum of (2) and (3) gives the total error (1).

(b) It is not the source of an error, but its regularity that dictates whether the effect is systematic or statistical.

(c) We do not know the exact value y nor can we perform an infinite number of measurements to obtain the mean y_m; hence, systematic and statistical errors are unknown and unknowable.^[2]

The total statistical error determines the precision of an experiment. Since systematic errors are independent of statistical errors, accuracy and precision are separate, independent concepts, although they are commonly confused with one another in everyday discourse. Since they are independent, we can identify four possibilities, as in Figure 1:^[2]

1. low accuracy and low precision
2. low accuracy and high precision
3. high accuracy and low precision
4. high accuracy and high precision

Since errors are both unknown and unknowable, we do not try to evaluate them; instead, we evaluate estimates for uncertainties.

References

1. ↑ ^{1.0 1.1} P. J. Galison, How Experiments End, University of Chicago Press, Chicago, 1987, ISBN 0-226-27915-4.
2. ↑ ^{2.0 2.1} J. M. Haile, Analysis of Data, Macatea Productions, Central, SC, 2003. ISBN 0-9728602-0-7.