9.1.10, 1.2.5, 2.2.11, D.0.1, D.0.2, D.0.3, D.1.1, D.1.2, D.1.6, D.1.9 - Describing and checking data

Mean, median and mode

A measure of central tendency summarises the centre or typical value of a set of observations, but different measures answer different questions.

Consider these invented recall scores, measured as words remembered out of 20: 4, 6, 6, 8, 11. Add all five scores to obtain 35; divide by five to obtain a mean of 7 words. The ordered middle score is 6, so the median is 6 words. The value 6 appears twice while each other value appears once, so the mode is also 6 words. A mean need not be one of the observed values.

Always order the scores before finding the median. With an even number of numerical scores, take the mean of the two middle scores. For 3, 5, 7, 9, the middle positions are the second and third, so the median is $(5+7)/2=6$. This is not the same as choosing the second score. A dataset can have two equally frequent modes; if no value is more frequent than the others, report no useful unique mode rather than inventing one.

The calculations describe this sample. They do not by themselves show that a difference between two samples is statistically significant or that an experimental manipulation caused it.

Choosing and comparing summaries

The most useful summary depends on what the scores mean and what the distribution contains.

The mean uses every numerical score, making it useful for approximately symmetric data measured in meaningful equal units. Its sensitivity is also a weakness: one extreme score can pull it away from most participants. The median depends on position, so it often describes a typical participant better in skewed data or data containing an extreme value. For ranked or ordered categories, the median locates the middle response; do not assume the gaps between category numbers are equal. The mode works for nominal categories such as a preferred revision method, where calculating an arithmetic mean would be meaningless.

For example, invented completion times 12, 13, 13, 14, 48 seconds have mean 20 seconds but median 13 seconds. Four people finished within 12–14 seconds; the mean conceals that cluster because of the 48-second value. A researcher should check whether 48 is a recording error or a genuine observation. A genuine inconvenient result must not be deleted simply to improve the mean. Reporting the median and the spread can be a sensible response.

To compare two conditions, use the same operational definition, units and summary. Invented recall scores 4, 6, 6, 8, 11 have mean 7 words; 6, 7, 7, 9, 11 have mean 8 words. The second condition remembered one more word on average. This statement describes direction and size; it does not claim every person in the second condition performed better or that the difference generalises to the population. The groups overlap, and spread and research design matter too.

Unequal group sizes matter when combining means. If ten people average 6 words and thirty average 10 words, the combined total is $10\times6+30\times10=360$ words across forty people. The overall mean is 9 words, not the unweighted average of the two group means, 8. The larger group contributes more observations.