Lessons · Teaching · mean, median, mode and range, and which one answers the question

Four numbers that summarise a set, and when each one lies

The mean is the total shared out equally, the median is the middle value in order, the mode is the most common value, and the range is the spread from lowest to highest. The mean is pulled by an extreme value; the median is not.

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What it is for

A teacher reports a class average and a parent hears a typical child. If one pupil scored 5 and the rest scored near 80, the mean is not typical of anybody and the median is. Knowing which measure answers the question asked is the difference between reporting and misleading.

How to think about it

Sort the data first, every time. Mean is total divided by count. Median is the middle after sorting, or the mean of the two middles if the count is even. Mode is the most frequent, and there may be none or several. Range is highest minus lowest, and it is a spread, not a centre.

Worked example

Set 2, 4, 4, 5, 10: the mean is 25 / 5 = 5

Total divided by how many there are.

Sorted, the middle value is 4, so the median is 4

Sorting first is the step people skip and then get wrong.

The mode is 4, as the only repeated value; the range is 10 - 2 = 8

Range measures spread and says nothing about the centre.

With an even count, the median is the mean of the two middle values

So a median need not be a value that appears in the set at all.

Your turn

Name the measure of centre that an extreme value does not pull.

The  is not dragged by one very high or very low value

The trap

Finding the median without sorting. The middle of the list as written is not the median unless the list happens already to be in order, and it usually is not.

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