Why one number is never enough to describe a data set

0:00How tall is your class? It sounds like a question with an answer. Here is one. Twenty eight children, every one of them measured. The shortest is a hundred and twenty eight centimetres. The tallest is a hundred and fifty eight. Add all twenty eight heights together and you get four thousand and forty six. Divide by twenty eight, and the average is a hundred and forty four point five.

0:28Sort them and the middle value is a hundred and forty five. Four numbers, and not one of them answers the question on its own. Together they do. Split the class in two and look again. Seventeen boys. Eleven girls. The boys average a hundred and forty two point nine. The girls, a hundred and forty six point nine. The boys' middle value is a hundred and forty four. The girls', a hundred and forty eight.

1:01So on both counts the girls are the taller group. That much is true, and it is about as far as any single number will carry you. Now put the ends back in. The girls run from a hundred and thirty six up to a hundred and fifty six. Twenty centimetres from end to end. The boys run from a hundred and twenty eight to a hundred and fifty eight. Thirty.

1:29So the shortest child in this class is a boy. And so is the tallest. The boys hold both ends, and they are spread half again as wide as the girls. And their average is still the lower of the two. None of that is a contradiction. The ends and the centre are answering different questions, and here you can watch them disagree. Now the sentence to be careful with.

1:57The girls average about four centimetres more than the boys. That is a fact about two groups. It says nothing whatsoever about any particular boy and any particular girl. Take every boy and pair him with every girl. Seventeen times eleven is a hundred and eighty seven pairs. In sixty three of them, the boy is the taller one. A third of all the pairs. Twenty two of these twenty eight children stand inside the stretch where the two groups overlap.

2:31An average is a statement about a group. Push it down onto two people and it simply stops being true. Now set each average against its own middle value. The boys: average a hundred and forty two point nine, middle value a hundred and forty four. The girls: average a hundred and forty six point nine, middle value a hundred and forty eight. In both groups the average sits below the middle value, by just over a centimetre each time.

3:06That is the low end pulling. In each group the shorter children reach further from the pack than the taller ones do. Two numbers, side by side, and you can hear the shape of the data behind them. So what do the ends carry that no average can? They carry how far apart the values are. Nothing else in the description does. But the ends also say something about the centre, and it is worth keeping.

3:35An average can never be larger than the largest value. Every height going into that total is at most a hundred and fifty eight, so twenty eight of them total at most twenty eight lots of a hundred and fifty eight, and dividing by twenty eight leaves the average at or below it. The same argument runs at the other end. And the middle value is one of the values, so it is inside as well.

4:04Both centres always live between the two ends. Every data set, always. If yours does not, the arithmetic is wrong. Two groups were asked to close their eyes and say when they thought a minute was up. These are drawn guesses, not measured ones. Fourteen in each group. Both averages fall short of a minute. Fifty seven point six seconds for the first group, fifty nine point four for the second.

4:36Not quite two seconds between them. Close enough to call them the same. Now stop reading the numbers and look at the two pictures. The first group spans twenty one seconds. The second spans ten. Similar averages, and one group is more than twice as scattered as the other. No average can tell you that. You have to look, or you have to ask for the ends. A cricket innings closes at four hundred and seven runs for ten wickets, so eleven batters batted.

5:11Nineteen of those runs were extras. Nobody batted them. Three hundred and eighty eight came off the bat, shared between eleven players. Divide, and the average is thirty five point two seven. Now sort those eleven scores and take the middle one. It is zero. Six of the eleven were out without scoring, and six out of eleven is more than half, so the middle score has to be nothing.

5:41Two players made a hundred and fifty eight and a hundred and eighty four. Between them, most of the runs in the innings. An average of thirty five and a middle value of zero, describing the same eleven people. Here is a smaller trap, and it costs marks. A player's last seven matches: fifty seven, thirteen, nought, eighty four, then a match missed, then fifty one and twenty seven. The nought and the blank are not the same thing.

6:15A nought is a match played and a match failed. It counts in how many. A blank is not a match at all. It counts in neither. Two hundred and thirty two runs, over the six matches played, is thirty eight point six seven. Divide by seven instead and you get thirty three point one four. Throw the real nought away as well and you get forty six point four.

6:42The same runs, three different answers. Only the number underneath ever moved. And sometimes the question is which values belong in the set at all. A mango tree, counted every month for a year. Nothing, nothing, eight, twenty four, forty one, sixteen, five, and then nothing at all for the rest of the year. Ninety four mangoes across twelve months is seven point eight three a month. But the tree does not fruit for seven of those months.

7:16Across the five months it does, it is eighteen point eight. Nearly two and a half times as much. And the middle value over the whole year is zero, on a tree that gave ninety four mangoes. Deciding what goes into the set is part of the work, not something that happens before it starts. One more, from five hundred years ago, before anybody had the word average. To fix the length of a foot, they took sixteen men, stood them heel to toe in one line, and called that line a rod.

7:53Then they cut the rod into sixteen equal pieces. One piece was one foot. Add sixteen lengths and divide by sixteen. That is an average, taken three centuries before it had a name. And it is a good one. One unusually short man moves the answer by a sixteenth of his own difference, and no further. Last one. Five sumo wrestlers and six ballet dancers, weighed. Roughly how many times heavier is a wrestler than a dancer?

8:28You cannot start without choosing a summary, and the choice changes the answer. By the averages, five point six seven times. By the middle values, five point five five. Near enough the same answer. But add each group up and divide one total by the other, and you get four point seven two. That one is wrong, and not by a little. There are five wrestlers and six dancers, so the two totals are counting different numbers of people.

9:00Which is where we came in. One number is never the description of a set. It is the one thing you chose to keep.