Signed deviations cancel to nothing, so the sign has to be discarded

0:00A centre tells you where a set of numbers sits. It does not tell you how far apart they are, and no amount of care in choosing the centre will make it. So a second number is needed, and we already know roughly what it has to be made of. Distances. Pick a centre; ask how far each observation is from it; combine those distances into one figure. That is the recipe, and this video is about the first and most natural way of carrying it out.

0:33It fails completely. It is worth watching closely anyway, because the reason it fails is worth more than the construction it destroys. Here is what anybody would write down first. One: find the centre. Take the mean, which is the total of the observations divided by how many of them there are. Two: subtract that centre from every observation in turn. Each of those differences has a name; it is called a deviation.

1:07Three: average the deviations. Look at what that construction has going for it. It uses every observation, not just the two ends. It is built out of differences from a centre, which is exactly what was asked for. And it hands back a single number. It looks like the answer. So run it. Eight readings: six, seven, ten, twelve, thirteen, four, eight and twelve. Step one, the centre. They add to seventy-two, and there are eight of them, so the mean is nine.

1:43Step two, subtract nine from each. Six is three below, so minus three. Seven is minus two. Ten is plus one. Twelve is plus three, thirteen is plus four, four is minus five, eight is minus one, and the last twelve is plus three again. Step three. Add those eight deviations up. Minus three, minus five, minus four, minus one, plus three, minus two, minus three, and zero. Zero. Divide zero by eight and the answer is still zero.

2:22The construction has reported that these eight numbers are not strewn about at all. The instinct now is to try a different set of numbers and see whether it happens again. Resist that instinct. It is the single most misleading thing you could do here, because a second example that also gives zero teaches exactly the wrong lesson. It teaches you that this is a coincidence which happens rather often.

2:53It is not a coincidence. It is not even a fact about these eight numbers. One line of algebra shows that this construction returns zero for every set of numbers that has ever been written down, and it is that line, and not a second example, that is worth your attention. Adding up the deviations means adding up every observation, and then subtracting the centre once for each observation. So the total of the deviations is the total of the observations, less the count times the centre.

3:32That is not a new fact; it is the same sum with the brackets rearranged. Now look at those two quantities on the right. The total of the eight observations is seventy-two. The count times the centre is eight nines, which is seventy-two as well. Seventy-two less seventy-two is nothing, which is where the zero came from. And here is the part that matters. The second of those two quantities was never going to be anything else.

4:04The mean is the total divided by the count. Multiply it back by the count and you have the total again. That is not arithmetic about six and seven and ten. That is the definition of the mean, read backwards. Go back through that argument and look for the place where the eight numbers were used. There is no such place. The observations were added; they were never inspected. Which means the conclusion cannot be about them.

4:37About any set of numbers at all, the deviations from its own mean add to nothing, and the average of those deviations is nothing. Four hundred records nobody chose, generated rather than picked: the number whose deviations fail to add to nothing is zero, and the number with an average deviation other than nothing is zero as well. Not nearly zero. Zero. A quantity that reads the same for every set of numbers in existence cannot report anything about any of them.

5:15Before going further, it is worth checking that this machinery is capable of saying anything other than nothing. A comparison that answers zero to everything would produce these same results while measuring nothing at all. So take the same eight deviations about fifty instead of about nine, and they add to minus three hundred and twenty-eight. Across those four hundred records, taken about the middle value instead of the mean, three hundred and eighty-eight fail to cancel.

5:49Taken about a fixed twenty, three hundred and ninety-five fail. So the cancelling is not an artefact of the arithmetic. It is a property of one particular centre, and the next question is whether choosing a different one is the way out. Here are eleven readings: three, nine, five, three, twelve, ten, eighteen, four, seven, nineteen and twenty-one. Sorted, they run three, three, four, five, seven, nine, ten, twelve, eighteen, nineteen, twenty-one.

6:24Eleven observations, so the middle value is the sixth one along, and that is nine. Take the deviations about that nine. Sorted, they are minus six, minus six, minus five, minus four, minus two, nought, one, three, nine, ten and twelve. Add them up and you get twelve. Not zero. So the cancelling really is about the mean and not about centres in general, and anyone who states it about the centre has overstated it.

6:56But watch what happens if we take the same eleven readings about their own mean, which is a hundred and eleven over eleven. They cancel again. That is the derivation working in both directions, and it is worth seeing it as three separate quantities rather than as one conclusion. For the eight numbers about their mean: the deviations total nothing, the observations total seventy-two, and the count times the centre is seventy-two.

7:30The first is nothing precisely because the other two agree. Now the eleven about the middle value of nine: the deviations total twelve, the observations total a hundred and eleven, and the count times the centre is ninety-nine. A hundred and eleven less ninety-nine is twelve. The deviations did not cancel, and the amount by which they failed to cancel is exactly the amount by which those two quantities disagree.

8:03The identity never broke. It simply stopped being interesting, because only at the mean are those two quantities forced to be the same number. There is an informal way to see why the mean must have observations on both sides of it. The mean sits strictly inside the record. The eight readings run from four up to thirteen and their mean of nine is between those ends. The eleven run from three up to twenty-one and their mean is between those ends too.

8:37Across the four hundred generated records, the number of exceptions is zero. So something has to be below it and something has to be above it, and the signs have to fight. But do not carry that reasoning across to any other centre. On four readings of one, one, one and five the middle value is one, which is also the smallest of them, and not a single deviation about it comes out negative.

9:07That record had to be built on purpose. Across four hundred generated records of odd length, the number whose middle value equals their smallest reading is zero, while the number whose deviations about their middle value fail to cancel is three hundred and ninety-four. Which is the honest picture: about the middle value the total is usually not zero, the signs still mix, and the strewing is still being under-reported.

9:39So changing the centre is not the repair. About the mean the total is nailed to zero by an identity. About some other centre it is not nailed to anything, but the positive and negative deviations still work against each other and still cancel part of the strewing away. The trouble was never which centre was chosen. The trouble is what is being averaged. Look at what the cancelling is actually made of.

10:11Of the eight deviations, four came out positive: one, three, three and four, and those add to eleven. Four came out negative: minus one, minus two, minus three and minus five, whose sizes are one, two, three and five, adding to eleven as well. Eleven against eleven. That is not a happy accident of these eight numbers either. The two halves must balance, because their difference is the total, and the total is nothing.

10:44Across the four hundred records, the number where the positive total differs from the total of the negative sizes is zero. Which rules out the next thing somebody usually suggests: keeping only the deviations that came out positive. Twice their total, over the full count of eight, is the same number the distances give you. Across four hundred records it fails zero times. One half of a perfectly balanced pair carries nothing the whole pair did not.

11:20There is a trap in that last statement and it is worth stepping into deliberately. Averaging the positive deviations gives half the answer only if you divide by the full count of observations. Divide instead by how many of them came out positive, which is the natural reading of the word averaging, and the halving vanishes. On these eight, eleven over four is two point seven five, which is the whole answer rather than half of it.

11:52The reason is a coincidence of this particular record: exactly four of its eight deviations came out positive. And it really is a coincidence. Across the four hundred records, a hundred and ninety-seven split exactly in half, a hundred and ninety-seven show that coincidence, and the number belonging to one of those groups but not the other is zero. The two go together, and neither is general. So the divisor has to be named or the claim is simply wrong.

12:29Now the repair, and it is not a trick to make the answer come out nicer. It is a change of question. Draw the eight readings as points on a line, with the mean marked at nine. A signed deviation is an arrow. It records two things at once: which side of nine the observation is on, and how far away it is. That first piece of information is what does the damage, because for every arrow pointing left there is enough arrow pointing right to swallow it.

13:03Now rub out the arrowheads and keep only the lengths. The length of the segment between an observation and the centre does not point anywhere. It is a distance, and distances have no signs to cancel with. Written down, the distance between an observation and a fixed value is the size of their difference: subtract them either way round and take the sign off the answer. That is what the two upright bars mean.

13:36Apply that to the eight deviations and the minus signs simply fall away. Three, two, one, three, four, five, one, three. Those add to twenty-two, not to zero. Divide by the eight observations and you get two point seven five. A number that finally moves when the readings move. That is the mean deviation: the total of the distances of the observations from a chosen central value, divided by how many observations there are.

14:09It is written em dee of a, where a is the central value you chose and the notation carries it deliberately, because the definition does not care which centre you pick. In practice two get used: the mean and the middle value. One last check, on a record with nothing strewn about it at all. Five readings of seven. The mean is seven, the deviations add to nothing, none of them is above the mean and none is below, and the mean deviation is nothing.

14:45Both constructions agree here, and they should: there genuinely is no strewing to report. That is the only kind of record for which a reading of nothing is the truth. Which leaves the lesson worth keeping. The failed construction was not a bad idea badly executed. It was undone by an identity, and identities cannot be worked around by choosing better inputs. The deviations about the mean add to nothing for every record that exists, so the average of those deviations is a number that never changes, and a number that never changes measures nothing.

15:27Discarding the sign is what turns a displacement into a distance, and a distance is what the question about strewing was asking for in the first place.