Which of the three averages a given question actually wants

0:00Thirty scripts, marks out of a hundred, sorted into six classes. Two students in the lowest class, three in the next, then seven, six, six and six. One table. Now put all three averages on it. The mean comes out at sixty-two exactly. The median comes out at sixty-two point five. And the mode comes out at fifty-two. Fifty-two, sixty-two, sixty-two point five. Three numbers, one table, and nobody has made an arithmetic slip.

0:34So which one is the average of this data? That question has no answer, and this video is about why. The three are not three attempts at one quantity. They are exact answers to three different questions, and which question you are asking is decided by what the number is for. Everything that follows comes from one difference between them: how much of the table each one actually reads. Start with the mean, because it is the greediest of the three.

1:09The mean reads every observation, without exception. Every count in the table enters the total, and every class mark is multiplied by its own count. There is a way to test that rather than assert it. Take a table, add one observation to a class, and see whether the answer moves. Do that for every class in turn. On a table like this one, every single bump moves the answer, because every count is standing in the arithmetic.

1:42Nothing in the table is ignored, and nothing in it is rounded away. That is the mean's whole character, and it is the reason for both of its properties. It compares two distributions well, because it has looked at all of both of them. And it can be dragged, because one far-out group is part of all of them too. Here is the drag, on a real table. Thirty-five regions, counted by students per teacher, in classes five wide.

2:15Three regions, then eight, nine, ten, three - and then two empty classes, and two regions right out at the end. With all thirty-five regions the mean is twenty-nine point two one. Now set those two far-out regions aside and work with the remaining thirty-three. The mean falls to twenty-seven point eight. One and four tenths, moved by two regions out of thirty-five. That is not a mistake in the arithmetic. It is the arithmetic working exactly as designed.

2:51The mean was asked to represent every region, and it did. Whether that is a virtue depends entirely on what you wanted the number for, which is the whole point. The median reads something else. It reads positions, and never sizes. Once an observation is past the middle, the median knows it is there and does not know how big it is. You can test that too, and the test is sharper than the last one.

3:22Take the same table and push the far-out group further out - three whole class widths further out - without changing a single count. Nobody has arrived and nobody has left. The data has the same shape and the same total. Only the size of those far-out observations has changed. Run all five of our tables through that, and the mean moves every time. The median does not move at all. Not on one of the five.

3:53That is not resistance to being dragged a little. That is the median being structurally unable to see the change. Positions moved nowhere, so the middle stayed exactly where it was. Now, is that steadiness good? It depends, and the honest answer has two halves. When your far-out values are errors, or one-offs, or the sort of thing that will not happen again, ignoring their size is exactly what you want.

4:25A typical wage, a typical house price, a typical time taken - these are all median questions, because a handful of enormous values would otherwise decide the answer for everybody. But the cost is real. Those far-out values did not go away. They are still in the data, and sometimes they are the most important part of it. Choosing the median is a decision to set them aside. It is a perfectly good decision. It should just be a conscious one, rather than something that happened because a formula was convenient.

5:05The median ignoring extreme values does not mean extreme values do not matter. The mode is the narrowest reader of the three, and by a long way. It looks at the busiest class and at its two immediate neighbours. Three adjacent counts. Everything else in the table is invisible to it. Test that the same way. Change a count anywhere outside those three bars, by as much as you like, and as long as the busiest bar is still the busiest, the mode does not move.

5:42Which explains something on the regions table. We set two far-out regions aside and the mean shifted by one and four tenths. The mode was thirty point six two five before, and thirty point six two five after. Not approximately. Exactly, to the last digit. The busiest class was thirty to thirty-five, its neighbours held nine and three, and none of those three numbers was touched. The mode had nothing to notice.

6:15So here is the map, and it is a map from purposes, not from data. If the number will be added, or shared out, or compared with another distribution, you want the mean, because it is the only one built out of the whole of the data. Total pay divided among the staff. Average rainfall across regions. Anything you intend to multiply back up by the count. If the number has to survive a few extreme values, you want the median.

6:47A typical income, a typical waiting time, a typical price. And if the question is which single group is busiest - which shoe size to stock, which class to timetable a room for, which band most customers fall into - you want the mode. Notice that none of those three sentences mentions the data at all. They are all about the job the number has to do. That is the answer to which average is right: name the job first, and the measure follows.

7:21Sometimes the choice barely matters, and it is worth knowing when. Sixty-eight households, counted by electricity used, in classes twenty wide. Four, five, thirteen, twenty, fourteen, eight, four. That shape rises to a peak and comes down again at about the same rate. It is close to balanced about its busiest class. Its mode is a hundred and thirty-five point seven seven. Its median is a hundred and thirty-seven, exactly.

7:55Its mean is a hundred and thirty-seven point zero six. The whole spread of the three is one point two nine. On numbers of that size, that is a rounding error. For a balanced distribution the three questions nearly coincide, so choosing between them is nearly free. Which is a good habit to have: look at the shape before you agonise over the choice. Now back to the thirty scripts, where they do not coincide at all.

8:28Fifty-two, sixty-two, sixty-two point five. The mean and the median are half a mark apart. The mode is ten marks below the mean, and ten and a half below the median. Why is the mode so far out on its own? Because of what it reads. The busiest class here is forty to fifty-five, holding seven. And the busiest class is simply not where the weight of this distribution sits.

8:56Count the students above fifty-five - above the top of the busiest class. Eighteen of them. Against twelve below. The mode is answering the question it was asked, faithfully: which single class is busiest. Forty to fifty-five is. It was never asked where the bulk of the data is, and it does not know. That is not lopsidedness dragging the mean. It is a peak in one place and a mass of data in another.

9:28There is a rule students memorise, and it is worth destroying properly. The rule says mode below median below mean. Here is a table where that is exactly right. A hundred family names, counted by how many letters they have, in classes three wide. Its mode is seven point eight eight, its median eight point zero five, its mean eight point three two. Mode, median, mean. Just as promised. And here is a table where the order is precisely reversed. Eighty patients by age, in classes ten wide.

10:07Mean thirty-five point three seven five, median thirty-five point eight seven, mode thirty-six point eight two. Mean, median, mode. The exact opposite order. And the thirty scripts we started with run a third way again: mode, then mean, then median. Three tables, three orderings, and none of them a special case invented to embarrass you. There is no ordering to memorise. There is only what each measure reads. There is also a thumb rule that ties the three together. Three times the median is roughly the mode plus twice the mean.

10:50Roughly is doing a great deal of work in that sentence, so let us find out how much. Run it on all four tables and measure the miss. On the patients, three times the median is a hundred and seven point six one, against a hundred and seven point five seven. Out by four hundredths. That is nought point nought four per cent. On the households, out by one point one one. Nought point three per cent.

11:21On the family names, out by nought point three seven. One point five per cent. And on the thirty scripts, three times the median is a hundred and eighty-seven point five, against a hundred and seventy-six. Out by eleven and a half. Six per cent. Look at where the rule works and where it fails. It is excellent on the three tables whose measures were already close together, and poor on the one where they were far apart.

11:54Which is to say it is least reliable exactly where you would most want to lean on it. One caution, because it would be easy to overclaim. Across those first three, the size of the miss does not follow the spread in order. The safe statement is that the rule falls apart once the three separate, not that you can predict the error from how far apart they are. Use it to check an answer. Never use it to compute a missing one.

12:26Last, the conditions these formulas come with, and they are not arbitrary. The mode formula and the median formula both want classes of equal width, and both want classes that meet. Meeting matters because both of them place an answer inside a class by assuming the observations are spread evenly across it. A gap between classes is a stretch of the number line the assumption says nothing about. The mean is free of all three restrictions.

13:00Give it classes of ten, ten and thirty and it answers perfectly happily, because it never places anything inside a class. It only ever uses the class marks. The step-deviation shortcut does need equal widths, but that is the shortcut's requirement and not the mean's. And one honest refusal. If two classes tie for busiest, the mode construction has no answer, and neither does the formula. The construction joins corners of the busiest bar to its neighbours and crosses them. With a flat top, those two lines are parallel and never meet.

13:43A refusal is the right answer there. A number would be worse. One table, three numbers, and the whole difference between them is what each one reads. The mean reads every observation, which is why it compares distributions and why it can be dragged. The median reads positions and never sizes, which is why a far-out value can be pushed three class widths further out without moving it a hair.

14:14The mode reads three adjacent counts, which is why it did not flinch when two regions left the table. None of the three is the real average with the other two as approximations. So do not ask which average is right. Ask what the number is for. Then the measure is not a choice at all. It is a consequence.