Mean and Average: Are They the Same Thing?

Average is the family; the mean is one member

Three averages appear on every school syllabus, and they answer the same question in three different ways: which single number is typical of this set? The mean shares the total out equally. The median finds the value in the middle once the data are in order. The mode names the value that occurs most often.

So mean and average are not synonyms, though they are not rivals either. Average is the wider word, and the mean is what most people intend by it. The habit worth building is naming which one you used, because a figure labelled only as an average is a figure nobody can check.

• Mean: add every value, divide by how many values there are
• Median: put the values in order, take the middle one - or the mean of the middle two if there is an even number
• Mode: the value occurring most often; there can be more than one, or none at all
• Range is not an average. It measures spread, and quoting it as an average is a common and costly slip

One set of marks, three defensible answers

Seven students take a test marked out of 20 and score 3, 4, 4, 5, 6, 7 and 20.

The mean is 49 ÷ 7 = 7. The median is the fourth value in order, which is 5. The mode is 4. Three different numbers, all correctly calculated, all describing the same seven results.

Look at what the mean has done. It is 7, and five of the seven students scored below it. That is not a mistake in the arithmetic - it is what a mean does when one value sits a long way from the rest. The single score of 20 pulls the total up by more than the other six can pull it down, and the typical student ends up below average.

Which figure is honest depends on the use it is being put to. Describing the class to a parent, the median of 5 represents the typical student far better. Working out how many marks the class earned altogether, only the mean will do, because the mean is the one that can be multiplied back: 7 × 7 = 49, and no other average has that property.

Choosing between them, and saying why

Exam questions test this directly, usually by asking which average is most appropriate and why. The mark is for the reason rather than the choice, and the reason is nearly always about an extreme value or about the data not being numerical.

Outside an exam, the choice is where the argument lives. Anyone quoting an average without saying which one has left themselves room to pick the flattering figure, and knowing that all three exist is what lets you ask for the other two.

• Use the mean when the values sit reasonably close together and you need a figure that can be scaled back up to a total.
• Use the median when one or two values are far from the rest - incomes, house prices, waiting times, a test one student aced.
• Use the mode when the data are not numbers at all: most common shoe size stocked, most popular subject, most frequent grade awarded.
• Say which one you used and on what data. An average with no method attached cannot be argued with, which is often exactly why it was chosen.

Why grouped data gives an estimate rather than a mean

Once data are grouped - twenty students scored between 11 and 15 marks, and so on - the individual values are gone. You cannot add up numbers you no longer have. The method is to assume every value in a group sits at that group's midpoint, multiply each midpoint by its frequency, add the products, and divide by the total frequency.

What comes out is an estimate, and questions say estimate for that reason. The assumption is that values are spread evenly inside each group, which is rarely exactly true, and the error grows as the groups get wider. It is worth stating on the page that the answer is an estimate; it costs nothing and it shows you know why.

Watch for an open-ended top group - over 60, with no upper limit. It has no midpoint, so either the question supplies one or you must choose and say what you chose. This is the one place in the topic where two students can produce different correct answers.