Mean against median

After the summer vacation a teacher asks his class how many short stories each has read. Most answers are small numbers, but one student has read 40. Before calculating, can you guess whether the mean will be less than or greater than the median?

It will be GREATER. That single very high value adds a great deal to the sum, so it pulls the mean upwards, while the median — which only counts positions — barely moves. The median here is 6, meaning half the class read 6 or more stories.

That is the outlier at the HIGH end. Poovizhi's family had one at the LOW end, where the mean was dragged BELOW the median.

Now a third case with no outlier at all: the number of pages in a newspaper from Monday to Sunday — 16, 18, 20, 22, 26, 16, 10. The values are spread fairly evenly, and here the mean and median come out close together.

So the three cases give a simple reading rule. When the data is balanced or uniformly spread out, the mean and median are CLOSE. When there is an outlier at the lower end, the mean shifts down, so mean < median. When there is an outlier at the higher end, the mean shifts up, so mean > median.

That makes the gap between mean and median a useful signal in itself. Before you even look for the outlier, the gap tells you which direction to look in. (What happens if there are outliers on BOTH sides? Try some example data and see — they can pull against each other.)

Mean and median are together called measures of central tendency: they measure the tendency of the values to pile up around a particular value, that is, they represent the 'centre' of the data. Marking the data, the mean and the median on a dot plot shows all three at once and makes the pull of an outlier visible.