There are specific rules for how to consistently express the uncertainty associated with a number. In general, the last significant figure in any result should be of the same order of magnitude (i.e.. in the same decimal position) as the uncertainty. Also, the uncertainty should be rounded to one or two significant figures. Always work out the uncertainty after finding the number of significant figures for the actual measurement.
For example,
The following numbers are all incorrect.
is wrong butis fine
is wrong butis fine
is wrong butis fine
In practice, when doing mathematical calculations, it is a good idea to keep one more digit than is significant to reduce rounding errors. But in the end, the answer must be expressed with only the proper number of significant figures. After addition or subtraction, the result is significant only to the place determined by the largest last significant place in the original numbers. For example,
should be rounded to get 90.4 (the tenths place is the last significant place in 1.1). After multiplication or division, the number of significant figures in the result is determined by the original number with the smallest number of significant figures. For example,
(2.80) (4.5039) = 12.61092
should be rounded off to 12.6 (three significant figures like 2.80).
]]>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
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