Mean, median and mode

You do not have to work with numbers for long before you feel the need to describe the broad picture. Mean, median and mode are three such measures.

The definitions

Imagine 9 people going to the cinema. Each brings their own pocket money for sweets: 50, 80, 150, 150, 220, 300, 350, 450 and 500 kroner. Together, they have 2,250 kroner.

The mean is the sum of all the values divided by the number of values. Here, that is 2,250 / 9 = 250 kroner.

The median is the middle value when the numbers are sorted from smallest to largest. Here, the median is 220 kroner.

The mode is the value that occurs most often. Here, it is 150 kroner.

Try it

Here is the cinema group again. Give some of them more money and see what happens to the mean, the median and the mode. Who ends up in the middle?

When is the mode useful?

The mode is most useful for categories, such as the most common shoe size or the best-selling sweet. If you run a shoe shop that only sells sizes 36, 37, 38, 46 and 48, it makes no sense to stock only the median size (38) or the mean size (around 41). Here, the mode is what matters.

When are the median and mean useful?

In other situations, the choice is usually between the mean and the median. Let's go back to the cinema group. Imagine the person with 500 kroner receives another 4,500 kroner by phone transfer. The mean jumps from 250 to 750 kroner! The new mean says very little about the other 8.

In official statistics, for example on income, the median is often the more interesting figure. Changes in the median tend to reflect the majority better than the mean, especially when there are large differences within a group. At the same time, the mean has strengths the median lacks.

The mean is still useful:

• It uses all the information. Every value counts. The median only cares about who is in the middle, and does not change whether the richest person gets another 10 kroner or 10 billion.
• It is tied to the total. The mean times the count is the total. If you know the mean is 250 kroner and 9 people are going to the cinema, you know they have 2,250 kroner between them. This is why GDP per person and tax revenue use the mean: it is the total that pays the bill, not (just) the people in the middle.
• It can be combined. If two cinema groups each have their own mean, you can work out the mean for everyone, weighted by the size of each group. You cannot do that with medians.

So the question is not which measure is best, but what you want to know: how the typical person is doing (median), or how much there is in total (mean).

Different kinds of mean

The mean we have used so far is called the arithmetic mean: add everything up and divide by the count.

When numbers are multiplied together rather than added, this gives the wrong answer. Imagine you put 1,000 kroner into shares. In the first year they rise by 50 per cent, and in the second year they fall by 50 per cent.

• The arithmetic mean of +50% and −50% is 0%. It sounds as if you are standing still.
• But 1,000 × 1.5 = 1,500, and 1,500 × 0.5 = 750. You have lost 250 kroner.

For growth over time, we therefore use the geometric mean. You multiply the growth factors together and take the nth root:

So the average annual growth was about −13.4%. And 1,000 × 0.866 × 0.866 ≈ 750, exactly what you ended up with.

The geometric mean is never larger than the arithmetic mean, and the more the numbers fluctuate, the bigger the gap. This is why quoting the arithmetic average return for shares, funds or house prices over many years paints too rosy a picture.

Rule of thumb: Use the arithmetic mean when the numbers are added, such as kroner, people or hours. Use the geometric mean when they are multiplied, such as growth, interest, returns and price rises over several years.