#### Quantitative and Physical History

# Significance, Uncertainty and Error

#### By Mark Ciotola

First published on February 27, 2019

## Significance

There is a limit in the significance of quantities. Here, we are only referring to mathematical significance. For example, a quality is only significant up to half of the smallest unit being measured. For example, if you measure a distance with a meter stick, and the stick is only ruled in 1 centimeter units, then you can express the distance in terms of the smallest subdivision of the rulings: in this case, that would be 1/2 of a centimeter. So the significance would be 0.5 cm, which is three digits, e.g. 91.5 cm.

## Uncertainty

Measurements involve a degree of uncertainly. Once again, here, we are only referring to mathematical uncertainly. Using the previous example, the distance is likely not exactly at any specific centimeter ruling. It is somewhere between centimeter marks, and it can sometimes be a bit of a judgment call to determine which is the closest mark. So the uncertainly here would be plus or minus 0.5 cm. So if the distance was measured as 91.5 cm, then the measurement would be expressed as 91.5 cm +/- 0.5 cm. This sort of error cannot typically be eliminated.

Determining uncertainty for historical data will often be much more challenging. At this state of the field, it is reasonably to determine a rational basis for the quantity of uncertainty, include an explanation of the rationale.

## Error

Measurements can be subject to systematic error. This type of error occurs due to a consistent flaw in the measurement system. For example, suppose the end of the meter stick was once cut off at the 1 cm mark, so that it always understates the distance by 1 cm. Such sources can sometimes be identified through examination of the measuring apparatus, and eliminated if identified.

in history, there are some patterns that may comprise the equivalent of systematic error. First is the tendency to overestimate battle casualties, especially for the enemy. Second, it the likelihood to understate production when reported for taxes and to overestimate it when told in a good story.

## Statistical Approaches to Improving Quality of Data

There are several statistical approaches to improving the quality of the data. If measurements can be repeated for the same quantity, such as of a distance, length or volume, then it is often possible to use a combination of individual measurements to achieve a more accurate one. Unfortunately, some historical quantities are not subject to repeat measurement. It may be possible to combine measurements of different, but similar phenomena, to improve the quality, but this approach must be handled with care and an extra degree of critical analysis and review.

A rough measure of uncertainty is variation between several neighboring data points. While that error could be due to actual variation, some of it is likely due to error.

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