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Errors and Uncertainty

Error has to do with uncertainty in measurements that nothing can be done about. If a measurement is repeated, the values obtained will differ and none of the results can be preferred over the others. Although it is not possible to do anything about such error, it can be characterized. For instance, the repeated measurements may cluster tightly together or they may spread widely. This pattern can be analysed systematically.

When we measure something the measurement is meaningless without knowing the uncertainty in the measurement. This leads us to the idea of errors in measurement. Other factors such as the conditions under which the measurements are taken may also affect the uncertainty of the measurements. Thus when we report a measurement we must include the maximum and minimum errors in the measurement.

As an example, take measuring the height of a person, the measure may be accurate may have a scale of 1 mm. But depending on how the person being measured holds themself during the measurement we might be accurate in measuring to the nearest cm.

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

Systematic errors are errors which tend to shift all measurements in a systematic way so their mean value is displaced. This may be due to such things as incorrect calibration of equipment, consistently improper use of equipment or failure to properly account for some effect. In a sense, a systematic error is rather like a blunder and large systematic errors can and must be eliminated in a good experiment. But small systematic errors will always be present. For instance, no instrument can ever be calibrated perfectly.

Other sources of systematic errors are external effects which can change the results of the experiment, but for which the corrections are not well known. In science, the reasons why several independent confirmations of experimental results are often required (especially using different techniques) is because different apparatus at different places may be affected by different systematic effects. Aside from making mistakes (such as thinking one is using the x10 scale, and actually using the x100 scale), the reason why experiments sometimes yield results which may be far outside the quoted errors is because of systematic effects which were not accounted for.

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

  • They may occur due to lack of sensitivity. For a sufficiently a small change an instrument may not be able to respond to it or to indicate it or the observer may not be able to discern it.
  • They may occur due to noise. There may be extraneous disturbances which cannot be taken into account.
  • They may be due to imprecise definition.
  • They may also occur due to statistical processes such as the roll of dice.
  • Random errors displace measurements in an arbitrary direction whereas systematic errors displace measurements in a single direction. Some systematic error can be substantially eliminated (or properly taken into account). Random errors are unavoidable and must be lived with.