Mean, median and mode.

Step 1 of 6

The balance point

The board holds $20$⁠ readings of the same quantity, say the time a ball takes to roll down a track. The $x$⁠-axis is only the order in which they were taken; what matters is each bead’s height, its $y$⁠ value. Every reading is a little different from the others, and still you want one number that says “about how much”.

The mean is the total of all the values divided by how many there are:

$\bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}$$\bar{x} =$$\frac{x_1 + x_2 + \dots + x_n}{n}$⁠

Think of it as a balance point: if each value is a weight on a ruler, the ruler balances exactly at the mean. That is why the distances from the mean, those above minus those below, always add to $0$⁠. On the board the amber line is $\bar{y}$⁠, and it runs exactly at $5$⁠.

Now a short list, without the board: work out the mean.

Step 2 of 6

The value in the middle

The median is the value in the middle once the data are in order of size. Half the values are smaller and half are larger. When the number of values $n$⁠ is odd, the median is the value in place $\frac{n + 1}{2}$⁠ of the sorted list.

The first step is always to sort. Try it.

Step 3 of 6

An even count

When the number of values is even, there is no single value in the middle. There are two, in places $\frac{n}{2}$⁠ and $\frac{n}{2} + 1$⁠, and the median is their mean. That is how the median of the $20$⁠ readings on the board works too: the mean of the tenth and the eleventh in order.

Step 4 of 6

The most common value

The mode is the value that appears most often. It is the only one of the three that also works for data that are not numbers, such as a class’s favourite colour, and it is always a value that really occurred. But when every value is different, as with the readings on the board, it means nothing: each value appears once.

For a shoe shop the mode is the number that matters: that is the size to order more of.

Step 5 of 6

A bad reading

The board shows Mean and spread: $\bar{y} = 5$⁠. The median of the twenty readings is close to it, $5.10$⁠.

Now someone wrote one reading down wrong, $0$⁠ instead of about $5$⁠. Bad reading will add it to the board as a pink bead. Predict what happens to the mean and to the median, then look.

Step 6 of 6

The new mean

The bad reading is on the board, and the board hides the mean again. You do not need all $21$⁠ values to work it out: go back from the old mean to the total, $\bar{x} \cdot n$⁠, add the new value, and divide by the new count.

A value like this, far from all the rest, is called an outlier. Before you decide which average to report, ask whether it is a mistake. If it is, correct it or leave it out. If not, like one huge salary in a small company, the median describes the typical value better than the mean.