WEBVTT 00:00:00.035 --> 00:00:03.588 Every field that measures anything keeps meeting the same curve. 00:00:03.785 --> 00:00:09.671 Biologists find it in heights, physicists in noise, pollsters in polls: a symmetric 00:00:09.671 --> 00:00:14.745 bell, over and over, in data that has no business agreeing about anything. 00:00:15.058 --> 00:00:16.149 Today we find out 00:00:16.149 --> 00:00:20.701 where that bell comes from, and the answer is one theorem about averages. 00:00:22.214 --> 00:00:25.395 So let me build a world with no bell in it anywhere. 00:00:25.592 --> 00:00:27.230 Here is a population: 00:00:27.334 --> 00:00:30.446 the number of sales a small shop makes in a day. 00:00:30.758 --> 00:00:34.413 Most days are quiet, and one sale is the commonest day of all, 00:00:34.496 --> 00:00:36.216 at thirty-two percent. 00:00:37.012 --> 00:00:41.448 Two sales happen on twenty-two percent of days, and three on sixteen. 00:00:41.644 --> 00:00:43.317 Then the tail begins: 00:00:43.514 --> 00:00:46.229 four, five, six, 00:00:46.312 --> 00:00:49.249 and every so often a wild day of seven sales. 00:00:50.046 --> 00:00:52.601 Two numbers will follow us all lecture. 00:00:52.705 --> 00:00:57.698 The mean, mu, is two point eight, the balance point of the whole shape. 00:00:57.801 --> 00:01:00.064 And the standard deviation, sigma, 00:01:00.147 --> 00:01:05.012 is one point eight, the typical distance a single day lands from that mean. 00:01:05.715 --> 00:01:07.759 And look at the shape itself. 00:01:07.863 --> 00:01:09.034 It is lopsided: 00:01:09.117 --> 00:01:13.692 the tail runs off to the right, and a seven sits far from everything else. 00:01:14.005 --> 00:01:18.313 Nothing about this picture is symmetric, and nothing about it is a bell. 00:01:19.110 --> 00:01:20.933 Now the destination. 00:01:21.130 --> 00:01:24.601 This curve on the right is the normal curve: the bell that every one 00:01:24.601 --> 00:01:29.687 of those fields keeps finding, symmetric about its middle and thin in both tails. 00:01:29.883 --> 00:01:31.995 Keep the two pictures side by side, 00:01:32.078 --> 00:01:34.609 because together they are the whole lecture. 00:01:35.522 --> 00:01:39.109 The claim is that the lopsided shop on the left will manufacture the bell 00:01:39.109 --> 00:01:42.721 on the right, and the machine that does it is averaging. 00:01:43.033 --> 00:01:47.295 Take four days at random, average them, and keep the average. 00:01:47.492 --> 00:01:50.999 Do it again and again, and watch what the averages do. 00:01:52.065 --> 00:01:56.940 Here is the machine we ended on, and its one rule: draw four days at random, 00:01:57.023 --> 00:02:00.042 average them, and keep nothing but the average. 00:02:00.239 --> 00:02:03.328 On the left, the population we draw from. 00:02:03.524 --> 00:02:07.426 On the right, an empty frame where the averages will live. 00:02:08.478 --> 00:02:09.791 First sample. 00:02:09.987 --> 00:02:11.764 The shop hands us a two, 00:02:11.764 --> 00:02:16.896 a one, a five, and a three, and there they are, sitting on the population. 00:02:17.693 --> 00:02:19.992 Now the arithmetic, in full. 00:02:20.189 --> 00:02:23.415 Two plus one plus five plus three is eleven, 00:02:23.498 --> 00:02:26.493 and eleven over four is two point seven five. 00:02:26.690 --> 00:02:31.950 That is the sample mean, written x bar, and it is the only thing we keep. 00:02:32.747 --> 00:02:34.106 Over it goes. 00:02:34.302 --> 00:02:37.308 The tick marks two point seven five on the new axis, 00:02:37.391 --> 00:02:41.954 and then the whole sample becomes one gray brick, standing in the bin it landed in. 00:02:42.151 --> 00:02:44.415 The four days that made it are gone. 00:02:45.212 --> 00:02:46.234 Again. 00:02:46.431 --> 00:02:49.334 A one, a four, and a pair of twos: 00:02:49.531 --> 00:02:53.082 the sum is nine, nine over four is two point two five, 00:02:53.165 --> 00:02:56.254 and a second brick lands just to the left of the first. 00:02:57.306 --> 00:03:01.823 Once more: a six, a one, a three, and a two. 00:03:02.275 --> 00:03:05.944 Twelve over four is three exactly, and a third brick lands. 00:03:06.741 --> 00:03:08.332 Now a lucky draw. 00:03:08.436 --> 00:03:11.885 A seven comes up, with a one, a one, and a two. 00:03:12.081 --> 00:03:16.958 Add them: eleven again, so the average is two point seven five, again. 00:03:17.271 --> 00:03:18.686 Even a wild day, 00:03:18.769 --> 00:03:23.007 averaged with three quiet ones, gets dragged back to the middle of the pile. 00:03:23.804 --> 00:03:25.744 From here, let it run. 00:03:25.940 --> 00:03:28.170 Sample, average, brick. 00:03:28.367 --> 00:03:30.550 Sample, average, brick. 00:03:30.863 --> 00:03:34.065 Five more spins of that loop, five more bricks, 00:03:34.148 --> 00:03:38.236 and already the pile stands tallest in the middle and thin at both ends. 00:03:39.032 --> 00:03:41.796 Nine averages hint at a shape without showing it. 00:03:41.993 --> 00:03:45.871 To see it properly we need not nine but dozens, so let the machine take 00:03:45.871 --> 00:03:47.822 over and run while we watch. 00:03:48.905 --> 00:03:50.796 Sixty-four averages now: 00:03:50.879 --> 00:03:55.488 the machine kept sampling while we talked, and every average left its brick. 00:03:55.801 --> 00:03:57.227 Look at the wall they built. 00:03:57.310 --> 00:04:01.932 It stands tallest just short of three, and it falls away on both sides. 00:04:02.415 --> 00:04:04.703 Set the wall against where it came from. 00:04:04.899 --> 00:04:08.476 The population leans left and drags its tail out to seven. 00:04:08.672 --> 00:04:12.142 The averages have no tail worth the name: past four and a half, 00:04:12.225 --> 00:04:14.107 the bins are simply empty. 00:04:14.303 --> 00:04:18.182 Averaging erased the skew, and the question is how. 00:04:18.978 --> 00:04:22.111 So watch a lucky sample try to build a tail. 00:04:22.194 --> 00:04:26.351 A seven comes up, and with it a one, a two, and a three. 00:04:27.148 --> 00:04:30.481 Their average: thirteen over four, which is three and a quarter. 00:04:30.584 --> 00:04:34.997 The seven hauled it to the right, and the quiet days hauled it straight back. 00:04:35.193 --> 00:04:38.515 Averaging is a tug of war, and the middle wins. 00:04:39.567 --> 00:04:43.315 An average out at seven would need all four draws to be sevens at once, 00:04:43.398 --> 00:04:46.719 and each has probability zero point zero five. 00:04:47.032 --> 00:04:50.980 Multiply the four together: about six chances in a million. 00:04:51.084 --> 00:04:54.103 That is why the wall is dead long before seven. 00:04:55.155 --> 00:04:59.208 Now bring back the curve from the opening minute and lay it over the wall. 00:04:59.660 --> 00:05:03.028 This is the normal curve, and it hugs the bricks. 00:05:03.340 --> 00:05:06.833 A lopsided population, pushed through nothing but averaging, 00:05:06.916 --> 00:05:08.797 has manufactured the bell. 00:05:10.449 --> 00:05:12.877 What we built deserves its name. 00:05:13.073 --> 00:05:15.045 The sample mean is itself random: 00:05:15.128 --> 00:05:17.799 a different sample gives a different average. 00:05:18.112 --> 00:05:22.303 Its distribution, across every sample you could possibly draw, is called 00:05:22.303 --> 00:05:26.878 the sampling distribution of the mean, and our histogram is a portrait of it. 00:05:27.420 --> 00:05:29.104 And measure the portrait. 00:05:29.416 --> 00:05:33.945 Its centre sits at two point eight, the population mean, exactly. 00:05:34.258 --> 00:05:38.379 Its width is zero point nine, and zero point nine is one point eight divided 00:05:38.379 --> 00:05:42.281 by two: sigma, divided by the square root of four. 00:05:42.594 --> 00:05:46.751 The sample size has crept into the width, and that is the theorem knocking. 00:05:47.866 --> 00:05:50.119 Here it is then, in full: 00:05:50.223 --> 00:05:51.919 the Central Limit Theorem. 00:05:52.231 --> 00:05:56.083 Draw samples of size n from any population at all, 00:05:56.167 --> 00:06:00.800 provided it has a mean mu and a finite standard deviation sigma. 00:06:00.997 --> 00:06:05.943 As n grows, the distribution of the sample mean approaches the normal curve, centred 00:06:05.943 --> 00:06:09.879 on mu, with width sigma over the square root of n. 00:06:10.792 --> 00:06:13.591 That one line is the quantitative heart. 00:06:13.694 --> 00:06:17.372 The width of the bell of averages is sigma over root n. 00:06:17.456 --> 00:06:21.671 The population's shape is not in it, and the skew is not in it. 00:06:21.868 --> 00:06:25.723 Only mu, sigma, and n survive the averaging. 00:06:27.236 --> 00:06:30.116 Let us watch that square root earn its keep. 00:06:30.568 --> 00:06:32.945 Here is the bell for n equals four, 00:06:33.029 --> 00:06:35.875 centred on the dashed line at two point eight. 00:06:36.071 --> 00:06:38.556 The bracket underneath is one width of it: 00:06:38.660 --> 00:06:42.921 sigma of x bar, the quantity our formula computes. 00:06:43.683 --> 00:06:45.960 The computation sits beside it. 00:06:46.133 --> 00:06:50.429 Sigma is one point eight, the spread we measured back at the start. 00:06:50.603 --> 00:06:52.519 Root four is two. 00:06:52.611 --> 00:06:56.176 And one point eight over two is zero point nine. 00:06:56.950 --> 00:07:02.183 Now buy more data. Quadruple the sample to sixteen: root sixteen is four, 00:07:02.267 --> 00:07:06.572 and the width drops to one point eight over four, zero point four five. 00:07:06.656 --> 00:07:09.559 Watch the bracket shrink to match the new bell. 00:07:09.903 --> 00:07:11.375 Mark the price of that. 00:07:11.459 --> 00:07:15.848 Four times the data bought exactly half the width, not a quarter of it. 00:07:16.648 --> 00:07:20.051 The square root is the reason, and it never relents. 00:07:20.754 --> 00:07:23.240 Quadruple again, to sixty-four. 00:07:23.436 --> 00:07:27.776 Root sixty-four is eight, and the width is zero point two two five: 00:07:27.860 --> 00:07:30.051 the bell has sharpened into a spike. 00:07:30.135 --> 00:07:32.898 It grows taller as it narrows because the area 00:07:32.898 --> 00:07:38.054 under every one of these curves is one, so the probability has nowhere to go but up. 00:07:38.850 --> 00:07:42.226 So this is what averaging buys, and what it costs. 00:07:42.310 --> 00:07:46.199 It forgets the population's shape entirely, and it shrinks the noise 00:07:46.199 --> 00:07:48.034 by exactly root n. 00:07:48.231 --> 00:07:53.015 What remains is the fine print, because a theorem this strong has conditions. 00:07:54.124 --> 00:07:58.475 A theorem this useful earns its fine print, and there are three clauses. 00:07:58.559 --> 00:08:02.403 Beside them, the shop again, to test each clause against. 00:08:03.315 --> 00:08:07.972 First: the draws must be independent, and all from one population. 00:08:08.168 --> 00:08:10.792 Our machine had that by construction. 00:08:10.989 --> 00:08:15.495 Real data, measured over time or across neighbours, has to earn it. 00:08:16.407 --> 00:08:18.927 Second: sigma must be finite. 00:08:19.031 --> 00:08:21.656 Here is a population that breaks the rule: 00:08:21.852 --> 00:08:23.919 this red tail dies away so slowly 00:08:23.919 --> 00:08:28.735 that the spread is infinite, and averages drawn from it never settle into any bell. 00:08:28.819 --> 00:08:30.921 No sigma, no theorem. 00:08:31.717 --> 00:08:36.165 Third: how large n must be depends on the shape you start from. 00:08:36.361 --> 00:08:40.678 Our shop's mild tail was nearly gone by n equals four; 00:08:40.762 --> 00:08:43.418 a harsher skew takes more averaging. 00:08:43.502 --> 00:08:45.763 The theorem promises the bell in the limit, 00:08:45.847 --> 00:08:48.518 and the shape decides the price of getting there. 00:08:49.659 --> 00:08:53.952 Now spend the theorem on the thing you meet everywhere: the error bar. 00:08:54.036 --> 00:08:56.263 A lab measures something noisy, 00:08:56.347 --> 00:09:01.072 averages n readings, and reports the average: three point one. 00:09:01.385 --> 00:09:05.426 An honest report also says how far off that number is likely to be. 00:09:06.223 --> 00:09:08.127 The theorem is the answer. 00:09:08.324 --> 00:09:13.256 That reported average was one draw from a bell whose width is sigma over root n, 00:09:13.340 --> 00:09:15.976 and here that width is zero point two. 00:09:16.079 --> 00:09:19.559 So draw the bar: from two point nine to three point three, 00:09:19.643 --> 00:09:21.746 one width each side of the reading. 00:09:21.942 --> 00:09:24.439 That is the error bar, drawn. 00:09:25.236 --> 00:09:30.101 So a quoted plus or minus is not decoration, and it is not a guess. 00:09:30.298 --> 00:09:32.442 It is this theorem, applied: 00:09:32.526 --> 00:09:35.372 the width of the normal curve the average came from. 00:09:35.475 --> 00:09:39.296 Wherever an error bar stands, a bell stands behind it. 00:09:40.600 --> 00:09:42.934 One curve carries the whole lecture. 00:09:43.131 --> 00:09:44.951 Here is the noise in the average, 00:09:45.035 --> 00:09:49.180 sigma over root n, plotted against the sample size. 00:09:49.493 --> 00:09:53.093 At n equals four it stood at zero point nine. 00:09:53.197 --> 00:09:56.042 At sixteen, zero point four five. 00:09:56.238 --> 00:09:59.536 At sixty-four, zero point two two five. 00:10:00.449 --> 00:10:02.954 The curve falls like one over root n, 00:10:03.038 --> 00:10:06.835 so each halving of the noise costs a quadrupling of the data. 00:10:07.032 --> 00:10:09.525 That is why large experiments are large, 00:10:09.609 --> 00:10:13.267 and why the last decimal place is always the expensive one. 00:10:13.808 --> 00:10:16.328 And that is the Central Limit Theorem. 00:10:16.432 --> 00:10:20.203 Averages forget the shape they were drawn from, remember the mean, 00:10:20.287 --> 00:10:22.482 and close in on it like root n. 00:10:22.585 --> 00:10:24.928 The bell was never hiding in the population. 00:10:25.012 --> 00:10:27.219 Averaging builds it, every time.