Mean, Median, Mode and Range

The mean is the average most people picture, and the idea behind it is sharing out fairly. Imagine five stacks of counters, three, five, six, eight and eight tall. Push them all into one pile and you have thirty counters, then deal them back into five equal stacks and each one is six tall. That is the mean. The working is always the same two steps: add every value to get thirty, then divide by how many values there are, which is five, giving six. The mistake to avoid is dividing by the largest number instead of by how many values you have. And do not be worried if the mean is not one of your numbers — that happens constantly and is perfectly correct.

The median is the value in the middle, but there is one step you must do first: put the numbers in order. Take seven, three, nine, four and five. Sorted they read three, four, five, seven, nine, and the middle one is five, with two numbers on each side. Now try an even number of values. Ten, two, seven and five sorts to two, five, seven, ten, and there is no single middle, so you take the two in the middle and average them: five plus seven is twelve, halved gives six. The number one mistake in this whole topic is forgetting to sort. Cross off the smallest and largest in pairs and you will land on the middle every time.

The mode is simply the value that appears most often, and it behaves a little differently from the other two. In four, four, six, seven, nine only the four repeats, so the mode is four. In two, two, five, seven, seven there are two values repeating equally, so both are modes and you write two and seven. In one, three, six, eight nothing repeats at all, and then the honest answer is that there is no mode — never invent one just because the question asks. And here is the part that makes the mode special: it is the only one of the three averages that works on words. You cannot average eye colours, but you can certainly say which colour was most common.

The range is different from the other three, because it does not try to describe a typical value at all. It tells you how spread out the data is, and you find it by taking the smallest value away from the largest. Look at these two teams. Team A scored seven, eight, eight, nine and eight, so its range is nine take away seven, which is two. Team B scored two, five, eight, twelve and thirteen, giving a range of thirteen take away two, which is eleven. Both teams total forty and both have a mean of eight, yet they are nothing alike. Team A is reliable every game; Team B is wildly up and down. That is exactly what the range captures and the mean cannot.

Here is everything on one set of seven test scores, already written in order: six, seven, seven, seven, eight, ten, eleven. The mean is the total, fifty-six, divided by seven, giving eight. The median is the fourth score, because that leaves three scores on each side, and the fourth is seven. The mode is also seven, since it appears three times. The range is eleven take away six, which is five. Notice what those four numbers do. Mean, median and mode are all trying to describe a typical score, while the range describes how spread out the scores are. And since a question will always name the one it wants, read it twice before you start calculating.