Error Anaysis 2

Error has to do with uncertainty in measurements that nothing canbe done about. A metre rule might measure to the nearest mm. If ameasurement is repeated, the length obtained may differ and none ofthe measurements can be preferred. Although it is not possible to doanything about such errors, they can be characterized using standardmethods.

Classification ofError

Generally, errors can be divided into two broad and rough butuseful classes: systematic and random.

Systematic errors are errors which tend to shift all measurementsin a systematic way so their mean value is displaced. This may be dueto such things as incorrect calibration of equipment, consistentlyimproper use of equipment or failure to properly account for someeffect. In a sense, a systematic error is rather like a blunder andlarge systematic errors can and must be eliminated in a goodexperiment. On the other hand random errors will always be presentdue to inaccurate calibration or measurement.

Random errors are errors which fluctuate from one measurement tothe next. They yield results distributed about some mean value. Theycan occur for a variety of reasons.

• They may occur due to lack of sensitivity. For a sufficientlya small change an instrument may not be able to respond to it or toindicate it or the observer may not be able to discern it.

• They may occur due to noise. There may be extraneousdisturbances which cannot be taken into account.

• They may be due to imprecise definition.

• They may also occur due to statistical processes such as theroll of dice.

Propagation of Errors

Frequently, the result of an experiment will not be measureddirectly. Rather, it will be calculated from several measuredphysical quantities (each of which has a mean value and an error).What is the resulting error in the final result of such anexperiment?

For instance, what is the error inwhereandaretwo measured quantities with errorsandrespectively?

A first thought might be that the error inwouldbe just the sum of the errors inandThisassumes that, when combined, the errors inandhavethe same sign and maximum magnitude; that is that they always combinein the worst possible way. This could only happen if the errors inthe two variables were perfectly correlated, (i.e.. if the twovariables were not really independent).

If the variables are independent then sometimes the error in onevariable will happen to cancel out some of the error in the other andso, on the average, the error inwillbe less than the sum of the errors in its parts. A reasonable way totry to take this into account is to treat the perturbations in Z produced by perturbations in its parts as if they were"perpendicular" and added according to the Pythagoreantheorem,

That is, ifandthensince

This idea can be used to derive a general rule. Suppose there aretwo measurements,andandthe final result isforsome functionIfisperturbed by thenwillbe perturbed by

whereisthe derivative ofwithrespect towithheldconstant. Similarly the perturbation indueto a perturbation inis,

Combining these by the Pythagorean theorem yields

,

so this gives the same result as before. Similarly ifthen,

which also gives the same result. Errors combine in the same wayfor both addition and subtraction. However, ifthen,

or the fractional error inisthe square root of the sum of the squares of the fractional errors inits parts. (You should be able to verify that the result is the samefor division as it is for multiplication.) For example,

It should be noted that since the above applies onlywhen the two measured quantities are independent of each other itdoes not apply when, for example, one physical quantity is measuredand what is required is its square. Ifthenthe perturbation indueto a perturbation inis