{"version":1,"lectureId":"01M14V04E76DMNTAVA3A7H0RXH","attempt":0,"publication":{"slug":"central-limit-theorem-by-sampling","title":"The Central Limit Theorem, Shown by Sampling","subject":"statistics","summary":"A skewed population, samples of four days at a time, and the wall of gray bricks their averages build: this lecture watches the Central Limit Theorem happen before stating it. The tail dies because averaging is a tug of war; the bell's width comes out as sigma over root n; growing the sample from four to sixteen to sixty-four halves the width twice; and the closing chapters read the fine print and cash the theorem in as the error bar on every reported measurement.","metaDescription":"Watch a skewed population turn into the normal curve: samples of four, a wall of averages, and the sigma over root n law behind every error bar.","transcript":"Every field that measures anything keeps meeting the same curve. Biologists find it in heights, physicists in noise, pollsters in polls: a symmetric bell, over and over, in data that has no business agreeing about anything. Today we find out where that bell comes from, and the answer is one theorem about averages. So let me build a world with no bell in it anywhere. Here is a population: the number of sales a small shop makes in a day. Most days are quiet, and one sale is the commonest day of all, at thirty-two percent. Two sales happen on twenty-two percent of days, and three on sixteen. Then the tail begins: four, five, six, and every so often a wild day of seven sales. Two numbers will follow us all lecture. The mean, mu, is two point eight, the balance point of the whole shape. And the standard deviation, sigma, is one point eight, the typical distance a single day lands from that mean. And look at the shape itself. It is lopsided: the tail runs off to the right, and a seven sits far from everything else. Nothing about this picture is symmetric, and nothing about it is a bell. Now the destination. This curve on the right is the normal curve: the bell that every one of those fields keeps finding, symmetric about its middle and thin in both tails. Keep the two pictures side by side, because together they are the whole lecture. The claim is that the lopsided shop on the left will manufacture the bell on the right, and the machine that does it is averaging. Take four days at random, average them, and keep the average. Do it again and again, and watch what the averages do. Here is the machine we ended on, and its one rule: draw four days at random, average them, and keep nothing but the average. On the left, the population we draw from. On the right, an empty frame where the averages will live. First sample. The shop hands us a two, a one, a five, and a three, and there they are, sitting on the population. Now the arithmetic, in full. Two plus one plus five plus three is eleven, and eleven over four is two point seven five. That is the sample mean, written x bar, and it is the only thing we keep. Over it goes. The tick marks two point seven five on the new axis, and then the whole sample becomes one gray brick, standing in the bin it landed in. The four days that made it are gone. Again. A one, a four, and a pair of twos: the sum is nine, nine over four is two point two five, and a second brick lands just to the left of the first. Once more: a six, a one, a three, and a two. Twelve over four is three exactly, and a third brick lands. Now a lucky draw. A seven comes up, with a one, a one, and a two. Add them: eleven again, so the average is two point seven five, again. Even a wild day, averaged with three quiet ones, gets dragged back to the middle of the pile. From here, let it run. Sample, average, brick. Sample, average, brick. Five more spins of that loop, five more bricks, and already the pile stands tallest in the middle and thin at both ends. Nine averages hint at a shape without showing it. To see it properly we need not nine but dozens, so let the machine take over and run while we watch. Sixty-four averages now: the machine kept sampling while we talked, and every average left its brick. Look at the wall they built. It stands tallest just short of three, and it falls away on both sides. Set the wall against where it came from. The population leans left and drags its tail out to seven. The averages have no tail worth the name: past four and a half, the bins are simply empty. Averaging erased the skew, and the question is how. So watch a lucky sample try to build a tail. A seven comes up, and with it a one, a two, and a three. Their average: thirteen over four, which is three and a quarter. The seven hauled it to the right, and the quiet days hauled it straight back. Averaging is a tug of war, and the middle wins. An average out at seven would need all four draws to be sevens at once, and each has probability zero point zero five. Multiply the four together: about six chances in a million. That is why the wall is dead long before seven. Now bring back the curve from the opening minute and lay it over the wall. This is the normal curve, and it hugs the bricks. A lopsided population, pushed through nothing but averaging, has manufactured the bell. What we built deserves its name. The sample mean is itself random: a different sample gives a different average. Its distribution, across every sample you could possibly draw, is called the sampling distribution of the mean, and our histogram is a portrait of it. And measure the portrait. Its centre sits at two point eight, the population mean, exactly. Its width is zero point nine, and zero point nine is one point eight divided by two: sigma, divided by the square root of four. The sample size has crept into the width, and that is the theorem knocking. Here it is then, in full: the Central Limit Theorem. Draw samples of size n from any population at all, provided it has a mean mu and a finite standard deviation sigma. As n grows, the distribution of the sample mean approaches the normal curve, centred on mu, with width sigma over the square root of n. That one line is the quantitative heart. The width of the bell of averages is sigma over root n. The population's shape is not in it, and the skew is not in it. Only mu, sigma, and n survive the averaging. Let us watch that square root earn its keep. Here is the bell for n equals four, centred on the dashed line at two point eight. The bracket underneath is one width of it: sigma of x bar, the quantity our formula computes. The computation sits beside it. Sigma is one point eight, the spread we measured back at the start. Root four is two. And one point eight over two is zero point nine. Now buy more data. Quadruple the sample to sixteen: root sixteen is four, and the width drops to one point eight over four, zero point four five. Watch the bracket shrink to match the new bell. Mark the price of that. Four times the data bought exactly half the width, not a quarter of it. The square root is the reason, and it never relents. Quadruple again, to sixty-four. Root sixty-four is eight, and the width is zero point two two five: the bell has sharpened into a spike. It grows taller as it narrows because the area under every one of these curves is one, so the probability has nowhere to go but up. So this is what averaging buys, and what it costs. It forgets the population's shape entirely, and it shrinks the noise by exactly root n. What remains is the fine print, because a theorem this strong has conditions. A theorem this useful earns its fine print, and there are three clauses. Beside them, the shop again, to test each clause against. First: the draws must be independent, and all from one population. Our machine had that by construction. Real data, measured over time or across neighbours, has to earn it. Second: sigma must be finite. Here is a population that breaks the rule: this red tail dies away so slowly that the spread is infinite, and averages drawn from it never settle into any bell. No sigma, no theorem. Third: how large n must be depends on the shape you start from. Our shop's mild tail was nearly gone by n equals four; a harsher skew takes more averaging. The theorem promises the bell in the limit, and the shape decides the price of getting there. Now spend the theorem on the thing you meet everywhere: the error bar. A lab measures something noisy, averages n readings, and reports the average: three point one. An honest report also says how far off that number is likely to be. The theorem is the answer. That reported average was one draw from a bell whose width is sigma over root n, and here that width is zero point two. So draw the bar: from two point nine to three point three, one width each side of the reading. That is the error bar, drawn. So a quoted plus or minus is not decoration, and it is not a guess. It is this theorem, applied: the width of the normal curve the average came from. Wherever an error bar stands, a bell stands behind it. One curve carries the whole lecture. Here is the noise in the average, sigma over root n, plotted against the sample size. At n equals four it stood at zero point nine. At sixteen, zero point four five. At sixty-four, zero point two two five. The curve falls like one over root n, so each halving of the noise costs a quadrupling of the data. That is why large experiments are large, and why the last decimal place is always the expensive one. And that is the Central Limit Theorem. Averages forget the shape they were drawn from, remember the mean, and close in on it like root n. The bell was never hiding in the population. Averaging builds it, every time.","watch":{"version":1,"scenes":[{"title":"A Lopsided Population","start":0,"end":112.51533333333334,"objects":{"bars":"a Polygon [blue] drawn in pop (vertices=((0.6, 0.0), (1.4, 0.0), (1.4, 0.32), (0.6, 0.32)), fill_opacity=0.5)","bars_2":"a Polygon [blue] drawn in pop (vertices=((1.6, 0.0), (2.4, 0.0), (2.4, 0.22), (1.6, 0.22)), fill_opacity=0.5)","bars_3":"a Polygon [blue] drawn in pop (vertices=((2.6, 0.0), (3.4, 0.0), (3.4, 0.16), (2.6, 0.16)), fill_opacity=0.5)","bars_4":"a Polygon [blue] drawn in pop (vertices=((3.6, 0.0), (4.4, 0.0), (4.4, 0.11), (3.6, 0.11)), fill_opacity=0.5)","bars_5":"a Polygon [blue] drawn in pop (vertices=((4.6, 0.0), (5.4, 0.0), (5.4, 0.08), (4.6, 0.08)), fill_opacity=0.5)","bars_6":"a Polygon [blue] drawn in pop (vertices=((5.6, 0.0), (6.4, 0.0), (6.4, 0.06), (5.6, 0.06)), fill_opacity=0.5)","bars_7":"a Polygon [blue] drawn in pop (vertices=((6.6, 0.0), (7.4, 0.0), (7.4, 0.05), (6.6, 0.05)), fill_opacity=0.5)","bell":"an Axes (x_range=(0.0, 8.0), y_range=(0.0, 0.5), x_ticks_every=1.0)","bell_curve":"a FunctionPlot [yellow] drawn in bell (function=<function>)","card":"a Title that says \"Introductory Statistics — The Central Limit Theorem, Shown by Sampling\"","heading":"a Heading that says \"A Lopsided Population\"","lbl_bell":"a Tex [text] that says \"Where we are going\"","lbl_pop":"a Tex [text] that says \"What we have\"","mu_line":"a Line [yellow] labelled \"mu = 2.8\" drawn in pop (start=(2.8, 0.0), end=(2.8, 0.37), dashed=True)","point":"a Point [yellow] drawn in pop (location=(1.0, 0.34))","point_2":"a Point [yellow] drawn in pop (location=(7.0, 0.07))","pop":"an Axes (x_range=(0.0, 8.0), y_range=(0.0, 0.4), x_ticks_every=1.0)","sigma_brace":"a Brace [green] labelled \"sigma = 1.8\" drawn in pop (x_start=2.8, x_end=4.6)"},"beats":[{"start":0,"say":"Every field that measures anything keeps meeting the same curve. Biologists find it in heights, physicists in noise, pollsters in polls: a symmetric bell, over and over, in data that has no business agreeing about anything. Today we find out where that bell comes from, and the answer is one theorem about averages.","live":[],"does":[[0,"card is shown on the screen, written out."],[1.5,"card: enter:write-left-to-right."],[19.864,"card is hidden from the screen — left the board."]]},{"start":21.064,"say":"So let me build a world with no bell in it anywhere. Here is a population: the number of sales a small shop makes in a day. 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This curve on the right is the normal curve: the bell that every one of those fields keeps finding, symmetric about its middle and thin in both tails. Keep the two pictures side by side, because together they are the whole lecture.","live":null,"does":[[79.633,"pop moves to a new place on the board."],[79.633,"lbl_bell is shown on the screen, written out."],[79.633,"bell is shown on the screen, written out."],[83.19699999999999,"bell_curve is shown on the screen, drawn."],[91.347,"lbl_pop is shown on the screen, written out."]]},{"start":95.593,"say":"The claim is that the lopsided shop on the left will manufacture the bell on the right, and the machine that does it is averaging. Take four days at random, average them, and keep the average. 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Twelve over four is three exactly, and a third brick lands.","live":["plan","pop","work","hist","heading","bars","bars_2","bars_3","bars_4","bars_5","bars_6","bars_7","mu_line","polygon","polygon_2"],"does":[[180.1603333333333,"s3_dots is shown on the screen, written out."],[180.91433333333333,"s3_dots_2 is shown on the screen, written out."],[181.50633333333332,"s3_dots_3 is shown on the screen, written out."],[182.38933333333335,"s3_dots_4 is shown on the screen, written out."],[183.44533333333334,"work becomes \"$overline(x) = frac(6 + 1 + 3 + 2, 4) = frac(12, 4) = 3$\"."],[184.98933333333332,"work (the \"3#2\" part) is emphasized."],[186.4873333333333,"polygon_3 is shown on the screen, grown."],[186.73133333333334,"s3_dots is hidden from the screen."],[186.73133333333334,"s3_dots_2 is hidden from the screen."],[186.73133333333334,"s3_dots_3 is hidden from the screen."],[186.73133333333334,"s3_dots_4 is hidden from the screen."],[186.73133333333334,"work (the \"3#2\" part) is no longer emphasized."]]},{"start":188.28333333333333,"say":"Now a lucky draw. A seven comes up, with a one, a one, and a two. Add them: eleven again, so the average is two point seven five, again. Even a wild day, averaged with three quiet ones, gets dragged back to the middle of the pile.","live":["plan","pop","work","hist","heading","bars","bars_2","bars_3","bars_4","bars_5","bars_6","bars_7","mu_line","polygon","polygon_2","polygon_3"],"does":[[190.70933333333332,"s4_dots is shown on the screen, written out."],[192.12633333333332,"s4_dots_2 is shown on the screen, written out."],[192.6723333333333,"s4_dots_3 is shown on the screen, written out."],[193.22933333333333,"s4_dots_4 is shown on the screen, written out."],[194.3673333333333,"work becomes \"$overline(x) = frac(7 + 1 + 1 + 2, 4) = frac(11, 4) = 2.75$\"."],[195.47033333333331,"work (the \"frac(11, 4)\" part) is emphasized."],[197.69933333333333,"work (the \"2.75\" part) is emphasized."],[197.69933333333333,"work (the \"frac(11, 4)\" part) is no longer emphasized."],[203.2493333333333,"work (the \"2.75\" part) is no longer emphasized."],[204.09633333333332,"polygon_4 is shown on the screen, grown."],[204.53733333333332,"s4_dots is hidden from the screen."],[204.53733333333332,"s4_dots_2 is hidden from the screen."],[204.53733333333332,"s4_dots_3 is hidden from the screen."],[204.53733333333332,"s4_dots_4 is hidden from the screen."]]},{"start":205.89233333333334,"say":"From here, let it run. Sample, average, brick. Sample, average, brick. Five more spins of that loop, five more bricks, and already the pile stands tallest in the middle and thin at both ends.","live":["plan","pop","work","hist","heading","bars","bars_2","bars_3","bars_4","bars_5","bars_6","bars_7","mu_line","polygon","polygon_2","polygon_3","polygon_4"],"does":[[209.86333333333332,"polygon_5 is shown on the screen, grown."],[212.18533333333332,"polygon_6 is shown on the screen, grown."],[213.6243333333333,"polygon_7 is shown on the screen, grown."],[213.9743333333333,"polygon_8 is shown on the screen, grown."],[214.3243333333333,"polygon_9 is shown on the screen, grown."]]},{"start":221.71233333333333,"say":"Nine averages hint at a shape without showing it. To see it properly we need not nine but dozens, so let the machine take over and run while we watch.","live":["plan","pop","work","hist","heading","bars","bars_2","bars_3","bars_4","bars_5","bars_6","bars_7","mu_line","polygon","polygon_2","polygon_3","polygon_4","polygon_5","polygon_6","polygon_7","polygon_8","polygon_9"],"does":[[231.69431249999997,"heading is hidden from the screen — left the board."],[231.69431249999997,"hist is hidden from the screen — left the board."],[231.69431249999997,"polygon is hidden from the screen — hist left the board."],[231.69431249999997,"polygon_2 is hidden from the screen — hist left the board."],[231.69431249999997,"polygon_3 is hidden from the screen — hist left the board."],[231.69431249999997,"polygon_4 is hidden from the screen — hist left the board."],[231.69431249999997,"polygon_5 is hidden from the screen — hist left the board."],[231.69431249999997,"polygon_6 is hidden from the screen — hist left the board."],[231.69431249999997,"polygon_7 is hidden from the screen — hist left the board."],[231.69431249999997,"polygon_8 is hidden from the screen — hist left the board."],[231.69431249999997,"polygon_9 is hidden from the screen — hist left the board."],[231.69431249999997,"plan is hidden from the screen — left the board."],[231.69431249999997,"pop is hidden from the screen — left the board."],[231.69431249999997,"bars is hidden from the screen — pop left the board."],[231.69431249999997,"bars_2 is hidden from the screen — pop left the board."],[231.69431249999997,"bars_3 is hidden from the screen — pop left the board."],[231.69431249999997,"bars_4 is hidden from the screen — pop left the board."],[231.69431249999997,"bars_5 is hidden from the screen — pop left the board."],[231.69431249999997,"bars_6 is hidden from the screen — pop left the board."],[231.69431249999997,"bars_7 is hidden from the screen — pop left the board."],[231.69431249999997,"mu_line is hidden from the screen — pop left the board."],[231.69431249999997,"work is hidden from the screen — left the board."]]}]},{"title":"The Shape of the Averages","start":232.73597916666665,"end":351.8072291666666,"objects":{"bars":"a Polygon [blue] drawn in pop (vertices=((0.6, 0.0), (1.4, 0.0), (1.4, 0.32), (0.6, 0.32)), fill_opacity=0.5)","bars_2":"a Polygon [blue] drawn in pop (vertices=((1.6, 0.0), (2.4, 0.0), (2.4, 0.22), (1.6, 0.22)), fill_opacity=0.5)","bars_3":"a Polygon [blue] drawn in pop (vertices=((2.6, 0.0), (3.4, 0.0), (3.4, 0.16), (2.6, 0.16)), fill_opacity=0.5)","bars_4":"a Polygon [blue] drawn in pop (vertices=((3.6, 0.0), (4.4, 0.0), (4.4, 0.11), (3.6, 0.11)), fill_opacity=0.5)","bars_5":"a Polygon [blue] drawn in pop (vertices=((4.6, 0.0), (5.4, 0.0), (5.4, 0.08), (4.6, 0.08)), fill_opacity=0.5)","bars_6":"a Polygon [blue] drawn in pop (vertices=((5.6, 0.0), (6.4, 0.0), (6.4, 0.06), (5.6, 0.06)), fill_opacity=0.5)","bars_7":"a Polygon [blue] drawn in pop (vertices=((6.6, 0.0), (7.4, 0.0), (7.4, 0.05), 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A histogram of many sample means is a portrait of it.\"","fit":"a FunctionPlot [yellow] drawn in hist (function=<function>, x_range=(1.1, 4.9))","heading1":"a Heading that says \"The Shape of the Averages\"","heading2":"a Heading that says \"The Sampling Distribution\"","hist":"an Axes (x_range=(1.0, 5.5), y_range=(0.0, 10.0), x_ticks_every=1.0)","k3_dots":"a Point [yellow] drawn in pop (location=(7.0, 0.02))","k3_dots_2":"a Point [yellow] drawn in pop (location=(1.0, 0.02))","k3_dots_3":"a Point [yellow] drawn in pop (location=(2.0, 0.02))","k3_dots_4":"a Point [yellow] drawn in pop (location=(3.0, 0.02))","mu_line":"a Line [yellow] labelled \"mu\" drawn in pop (start=(2.8, 0.0), end=(2.8, 0.37), dashed=True)","point":"a Point [yellow] drawn in pop (location=(7.0, 0.07))","point_2":"a Point [yellow] drawn in hist (location=(4.9, 0.4))","pop":"an Axes (x_range=(0.0, 8.0), y_range=(0.0, 0.4), x_ticks_every=1.0)","rare":"a Math [text] that says \"$0.05^4 approx 6 times 10^(-6)$\"","sevens":"a Point [yellow] drawn in pop (location=(7.0, 0.02))","sevens_2":"a Point [yellow] drawn in pop (location=(7.0, 0.055))","sevens_3":"a Point [yellow] drawn in pop (location=(7.0, 0.09))","sevens_4":"a Point [yellow] drawn in pop (location=(7.0, 0.125))","stats3":"a Math [text] that says \"$upright(\"centre\") = 2.8, quad upright(\"width\") = 0.9 = frac(1.8, sqrt(4))$\"","tick325":"a Line [yellow] drawn in hist (start=(3.25, 0.0), end=(3.25, 1.2))","work3":"a Math [text] that says \"$overline(x) = frac(7 + 1 + 2 + 3, 4) = frac(13, 4) = 3.25$\""},"beats":[{"start":232.73597916666665,"say":"Sixty-four averages now: the machine kept sampling while we talked, and every average left its brick. Look at the wall they built. It stands tallest just short of three, and it falls away on both sides.","live":[],"does":[[232.73597916666665,"heading1 is shown on the screen, written out."],[232.73597916666665,"pop is shown on the screen, written out."],[232.73597916666665,"bars is shown on the screen, written out."],[232.73597916666665,"hist is shown on the screen, written out."],[232.76597916666665,"bars_2 is shown on the screen, written out."],[232.77097916666665,"bricks is shown on the screen, grown."],[232.79597916666665,"bars_3 is shown on the screen, written out."],[232.81597916666666,"bricks_2 is shown on the screen, grown."],[232.82597916666666,"bars_4 is shown on the screen, written out."],[232.85597916666666,"bars_5 is shown on the screen, written out."],[232.86097916666665,"bricks_3 is shown on the screen, grown."],[232.88597916666666,"bars_6 is shown on the screen, written out."],[232.90597916666664,"bricks_4 is shown on the screen, grown."],[232.91597916666666,"bars_7 is shown on the screen, written out."],[232.91597916666666,"mu_line is shown on the screen, written out."],[232.95097916666666,"bricks_5 is shown on the screen, grown."],[232.99597916666664,"bricks_6 is shown on the screen, grown."],[233.04097916666666,"bricks_7 is shown on the screen, grown."],[233.08597916666665,"bricks_8 is shown on the screen, grown."],[233.13097916666666,"bricks_9 is shown on the screen, grown."],[233.17597916666665,"bricks_10 is shown on the screen, grown."],[233.22097916666667,"bricks_11 is shown on the screen, grown."],[233.26597916666665,"bricks_12 is shown on the screen, grown."],[233.31097916666664,"bricks_13 is shown on the screen, grown."],[233.35597916666666,"bricks_14 is shown on the screen, grown."],[233.40097916666664,"bricks_15 is shown on the screen, grown."],[233.44597916666666,"bricks_16 is shown on the screen, grown."],[233.49097916666665,"bricks_17 is shown on the screen, grown."],[233.53597916666666,"bricks_18 is shown on the screen, grown."],[233.58097916666665,"bricks_19 is shown on the screen, grown."],[233.62597916666664,"bricks_20 is shown on the screen, grown."],[233.67097916666665,"bricks_21 is shown on the screen, grown."],[233.71597916666664,"bricks_22 is shown on the screen, grown."],[233.76097916666666,"bricks_23 is shown on the screen, grown."],[233.80597916666665,"bricks_24 is shown on the screen, grown."],[233.85097916666666,"bricks_25 is shown on the screen, grown."],[233.89597916666665,"bricks_26 is shown on the screen, grown."],[233.94097916666666,"bricks_27 is shown on the screen, grown."],[233.98597916666665,"bricks_28 is shown on the screen, grown."],[234.03097916666664,"bricks_29 is shown on the screen, grown."],[234.07597916666666,"bricks_30 is shown on the screen, grown."],[234.12097916666664,"bricks_31 is shown on the screen, grown."],[234.16597916666666,"bricks_32 is shown on the screen, grown."],[234.21097916666665,"bricks_33 is shown on the screen, grown."],[234.25597916666666,"bricks_34 is shown on the screen, grown."],[234.30097916666665,"bricks_35 is shown on the screen, grown."],[234.34597916666667,"bricks_36 is shown on the screen, grown."],[234.39097916666665,"bricks_37 is shown on the screen, grown."],[234.43597916666664,"bricks_38 is shown on the screen, grown."],[234.48097916666666,"bricks_39 is shown on the screen, grown."],[234.52597916666664,"bricks_40 is shown on the screen, grown."],[234.57097916666666,"bricks_41 is shown on the screen, grown."],[234.61597916666665,"bricks_42 is shown on the screen, grown."],[234.66097916666666,"bricks_43 is shown on the screen, grown."],[234.70597916666665,"bricks_44 is shown on the screen, grown."],[234.75097916666664,"bricks_45 is shown on the screen, grown."],[234.79597916666665,"bricks_46 is shown on the screen, grown."],[234.84097916666664,"bricks_47 is shown on the screen, grown."],[234.88597916666666,"bricks_48 is shown on the screen, grown."],[234.93097916666665,"bricks_49 is shown on the screen, grown."],[234.97597916666666,"bricks_50 is shown on the screen, grown."],[235.02097916666665,"bricks_51 is shown on the screen, grown."],[235.06597916666666,"bricks_52 is shown on the screen, grown."],[235.11097916666665,"bricks_53 is shown on the screen, grown."],[235.15597916666664,"bricks_54 is shown on the screen, grown."],[235.20097916666666,"bricks_55 is shown on the screen, grown."],[235.24597916666664,"bricks_56 is shown on the screen, grown."],[235.29097916666666,"bricks_57 is shown on the screen, grown."],[235.33597916666665,"bricks_58 is shown on the screen, grown."],[235.38097916666666,"bricks_59 is shown on the screen, grown."],[235.42597916666665,"bricks_60 is shown on the screen, grown."],[235.47097916666667,"bricks_61 is shown on the screen, grown."],[235.51597916666665,"bricks_62 is shown on the screen, grown."],[235.56097916666664,"bricks_63 is shown on the screen, grown."],[235.60597916666666,"bricks_64 is shown on the screen, grown."]]},{"start":245.54997916666665,"say":"Set the wall against where it came from. The population leans left and drags its tail out to seven. The averages have no tail worth the name: past four and a half, the bins are simply empty. Averaging erased the skew, and the question is how.","live":["pop","hist","heading1","bars","bars_2","bars_3","bars_4","bars_5","bars_6","bars_7","mu_line","bricks","bricks_2","bricks_3","bricks_4","bricks_5","bricks_6","bricks_7","bricks_8","bricks_9","bricks_10","bricks_11","bricks_12","bricks_13","bricks_14","bricks_15","bricks_16","bricks_17","bricks_18","bricks_19","bricks_20","bricks_21","bricks_22","bricks_23","bricks_24","bricks_25","bricks_26","bricks_27","bricks_28","bricks_29","bricks_30","bricks_31","bricks_32","bricks_33","bricks_34","bricks_35","bricks_36","bricks_37","bricks_38","bricks_39","bricks_40","bricks_41","bricks_42","bricks_43","bricks_44","bricks_45","bricks_46","bricks_47","bricks_48","bricks_49","bricks_50","bricks_51","bricks_52","bricks_53","bricks_54","bricks_55","bricks_56","bricks_57","bricks_58","bricks_59","bricks_60","bricks_61","bricks_62","bricks_63","bricks_64"],"does":[[250.91297916666664,"point is shown on the screen, grown."],[252.91297916666664,"point is hidden from the screen."],[256.57897916666667,"point_2 is shown on the screen, grown."],[258.57897916666667,"point_2 is hidden from the screen."]]},{"start":261.69497916666666,"say":"So watch a lucky sample try to build a tail. A seven comes up, and with it a one, a two, and a three.","live":null,"does":[[265.30597916666665,"k3_dots is shown on the screen, written out."],[267.1049791666667,"k3_dots_2 is shown on the screen, written out."],[267.56997916666666,"k3_dots_3 is shown on the screen, written out."],[268.24297916666666,"k3_dots_4 is shown on the screen, written out."]]},{"start":269.55147916666664,"say":"Their average: thirteen over four, which is three and a quarter. The seven hauled it to the right, and the quiet days hauled it straight back. Averaging is a tug of war, and the middle wins.","live":["pop","hist","heading1","bars","bars_2","bars_3","bars_4","bars_5","bars_6","bars_7","mu_line","bricks","bricks_2","bricks_3","bricks_4","bricks_5","bricks_6","bricks_7","bricks_8","bricks_9","bricks_10","bricks_11","bricks_12","bricks_13","bricks_14","bricks_15","bricks_16","bricks_17","bricks_18","bricks_19","bricks_20","bricks_21","bricks_22","bricks_23","bricks_24","bricks_25","bricks_26","bricks_27","bricks_28","bricks_29","bricks_30","bricks_31","bricks_32","bricks_33","bricks_34","bricks_35","bricks_36","bricks_37","bricks_38","bricks_39","bricks_40","bricks_41","bricks_42","bricks_43","bricks_44","bricks_45","bricks_46","bricks_47","bricks_48","bricks_49","bricks_50","bricks_51","bricks_52","bricks_53","bricks_54","bricks_55","bricks_56","bricks_57","bricks_58","bricks_59","bricks_60","bricks_61","bricks_62","bricks_63","bricks_64","k3_dots","k3_dots_2","k3_dots_3","k3_dots_4"],"does":[[270.11997916666667,"work3 is shown on the screen, written out."],[270.50297916666665,"work3 (the \"frac(13, 4)\" part) is emphasized."],[272.3259791666667,"tick325 is shown on the screen, written out."],[272.3259791666667,"work3 (the \"3.25\" part) is emphasized."],[272.3259791666667,"work3 (the \"frac(13, 4)\" part) is no longer emphasized."],[279.22197916666664,"work3 (the \"3.25\" part) is no longer emphasized."]]},{"start":281.85347916666666,"say":"An average out at seven would need all four draws to be sevens at once, and each has probability zero point zero five. Multiply the four together: about six chances in a million. That is why the wall is dead long before seven.","live":["pop","work3","hist","heading1","bars","bars_2","bars_3","bars_4","bars_5","bars_6","bars_7","mu_line","bricks","bricks_2","bricks_3","bricks_4","bricks_5","bricks_6","bricks_7","bricks_8","bricks_9","bricks_10","bricks_11","bricks_12","bricks_13","bricks_14","bricks_15","bricks_16","bricks_17","bricks_18","bricks_19","bricks_20","bricks_21","bricks_22","bricks_23","bricks_24","bricks_25","bricks_26","bricks_27","bricks_28","bricks_29","bricks_30","bricks_31","bricks_32","bricks_33","bricks_34","bricks_35","bricks_36","bricks_37","bricks_38","bricks_39","bricks_40","bricks_41","bricks_42","bricks_43","bricks_44","bricks_45","bricks_46","bricks_47","bricks_48","bricks_49","bricks_50","bricks_51","bricks_52","bricks_53","bricks_54","bricks_55","bricks_56","bricks_57","bricks_58","bricks_59","bricks_60","bricks_61","bricks_62","bricks_63","bricks_64","k3_dots","k3_dots_2","k3_dots_3","k3_dots_4","tick325"],"does":[[281.85347916666666,"k3_dots is hidden from the screen."],[281.85347916666666,"k3_dots_2 is hidden from the screen."],[281.85347916666666,"k3_dots_3 is hidden from the screen."],[281.85347916666666,"k3_dots_4 is hidden from the screen."],[281.85347916666666,"tick325 is hidden from the screen."],[284.90697916666664,"sevens is shown on the screen, written out."],[285.02697916666665,"sevens_2 is shown on the screen, written out."],[285.14697916666665,"sevens_3 is shown on the screen, written out."],[285.26697916666666,"sevens_4 is shown on the screen, written out."],[286.69497916666666,"rare is shown on the screen, written out."],[287.32197916666667,"rare (the \"0.05^4\" part) is emphasized."],[292.53497916666663,"rare (the \"0.05^4\" part) is no longer emphasized."],[292.53497916666663,"rare (the \"6 times 10^(-6)\" part) is emphasized."],[295.06597916666664,"sevens is hidden from the screen."],[295.06597916666664,"sevens_2 is hidden from the screen."],[295.06597916666664,"sevens_3 is hidden from the screen."],[295.06597916666664,"sevens_4 is hidden from the screen."],[295.06597916666664,"rare (the \"6 times 10^(-6)\" part) is no longer emphasized."]]},{"start":297.70897916666667,"say":"Now bring back the curve from the opening minute and lay it over the wall. This is the normal curve, and it hugs the bricks. A lopsided population, pushed through nothing but averaging, has manufactured the bell.","live":["pop","work3","rare","hist","heading1","bars","bars_2","bars_3","bars_4","bars_5","bars_6","bars_7","mu_line","bricks","bricks_2","bricks_3","bricks_4","bricks_5","bricks_6","bricks_7","bricks_8","bricks_9","bricks_10","bricks_11","bricks_12","bricks_13","bricks_14","bricks_15","bricks_16","bricks_17","bricks_18","bricks_19","bricks_20","bricks_21","bricks_22","bricks_23","bricks_24","bricks_25","bricks_26","bricks_27","bricks_28","bricks_29","bricks_30","bricks_31","bricks_32","bricks_33","bricks_34","bricks_35","bricks_36","bricks_37","bricks_38","bricks_39","bricks_40","bricks_41","bricks_42","bricks_43","bricks_44","bricks_45","bricks_46","bricks_47","bricks_48","bricks_49","bricks_50","bricks_51","bricks_52","bricks_53","bricks_54","bricks_55","bricks_56","bricks_57","bricks_58","bricks_59","bricks_60","bricks_61","bricks_62","bricks_63","bricks_64"],"does":[[300.71597916666667,"fit is shown on the screen, drawn."],[311.5019791666666,"heading1 is hidden from the screen — left the board."],[311.5019791666666,"pop is hidden from the screen — left the board."],[311.5019791666666,"bars is hidden from the screen — pop left the board."],[311.5019791666666,"bars_2 is hidden from the screen — pop left the board."],[311.5019791666666,"bars_3 is hidden from the screen — pop left the board."],[311.5019791666666,"bars_4 is hidden from the screen — pop left the board."],[311.5019791666666,"bars_5 is hidden from the screen — pop left the board."],[311.5019791666666,"bars_6 is hidden from the screen — pop left the board."],[311.5019791666666,"bars_7 is hidden from the screen — pop left the board."],[311.5019791666666,"mu_line is hidden from the screen — pop left the board."],[311.5019791666666,"rare is hidden from the screen — left the board."],[311.5019791666666,"work3 is hidden from the screen — left the board."]]},{"start":312.70197916666666,"say":"What we built deserves its name. The sample mean is itself random: a different sample gives a different average. Its distribution, across every sample you could possibly draw, is called the sampling distribution of the mean, and our histogram is a portrait of it.","live":["hist","bricks","bricks_2","bricks_3","bricks_4","bricks_5","bricks_6","bricks_7","bricks_8","bricks_9","bricks_10","bricks_11","bricks_12","bricks_13","bricks_14","bricks_15","bricks_16","bricks_17","bricks_18","bricks_19","bricks_20","bricks_21","bricks_22","bricks_23","bricks_24","bricks_25","bricks_26","bricks_27","bricks_28","bricks_29","bricks_30","bricks_31","bricks_32","bricks_33","bricks_34","bricks_35","bricks_36","bricks_37","bricks_38","bricks_39","bricks_40","bricks_41","bricks_42","bricks_43","bricks_44","bricks_45","bricks_46","bricks_47","bricks_48","bricks_49","bricks_50","bricks_51","bricks_52","bricks_53","bricks_54","bricks_55","bricks_56","bricks_57","bricks_58","bricks_59","bricks_60","bricks_61","bricks_62","bricks_63","bricks_64","fit"],"does":[[312.70197916666666,"heading2 is shown on the screen, written out."],[314.6869791666667,"defn is shown on the screen, written out."]]},{"start":330.39197916666666,"say":"And measure the portrait. Its centre sits at two point eight, the population mean, exactly. Its width is zero point nine, and zero point nine is one point eight divided by two: sigma, divided by the square root of four. The sample size has crept into the width, and that is the theorem knocking.","live":["hist","bricks","bricks_2","bricks_3","bricks_4","bricks_5","bricks_6","bricks_7","bricks_8","bricks_9","bricks_10","bricks_11","bricks_12","bricks_13","bricks_14","bricks_15","bricks_16","bricks_17","bricks_18","bricks_19","bricks_20","bricks_21","bricks_22","bricks_23","bricks_24","bricks_25","bricks_26","bricks_27","bricks_28","bricks_29","bricks_30","bricks_31","bricks_32","bricks_33","bricks_34","bricks_35","bricks_36","bricks_37","bricks_38","bricks_39","bricks_40","bricks_41","bricks_42","bricks_43","bricks_44","bricks_45","bricks_46","bricks_47","bricks_48","bricks_49","bricks_50","bricks_51","bricks_52","bricks_53","bricks_54","bricks_55","bricks_56","bricks_57","bricks_58","bricks_59","bricks_60","bricks_61","bricks_62","bricks_63","bricks_64","fit","defn","heading2"],"does":[[331.00697916666667,"stats3 is shown on the screen, written out."],[334.1889791666666,"stats3 (the \"2.8\" part) is 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board."],[350.7655625,"bricks_57 is hidden from the screen — hist left the board."],[350.7655625,"bricks_58 is hidden from the screen — hist left the board."],[350.7655625,"bricks_59 is hidden from the screen — hist left the board."],[350.7655625,"bricks_60 is hidden from the screen — hist left the board."],[350.7655625,"bricks_61 is hidden from the screen — hist left the board."],[350.7655625,"bricks_62 is hidden from the screen — hist left the board."],[350.7655625,"bricks_63 is hidden from the screen — hist left the board."],[350.7655625,"bricks_64 is hidden from the screen — hist left the board."],[350.7655625,"fit is hidden from the screen — hist left the board."],[350.7655625,"stats3 is hidden from the screen — left the board."]]}]},{"title":"The Theorem at Work","start":351.8072291666666,"end":480.6743958333333,"objects":{"bell16":"a FunctionPlot [yellow] drawn in bells (function=<function>)","bell4":"a FunctionPlot [blue] drawn in bells (function=<function>)","bell64":"a FunctionPlot [green] drawn in bells (function=<function>)","bells":"an Axes (x_range=(0.3, 5.3), y_range=(0.0, 2.0), x_ticks_every=1.0)","calc":"a Math [text] that says \"$frac(1.8, sqrt(4)) = frac(1.8, 2) = 0.9$\"","clt_math":"a Math [text] that says \"$sigma_(overline(x)) = frac(sigma, sqrt(n))$\"","heading1":"a Heading that says \"The Central Limit Theorem\"","heading2":"a Heading that says \"What Growing $n$ Buys\"","mu4":"a Line [text] drawn in bells (start=(2.8, 0.0), end=(2.8, 1.9), dashed=True)","n_live":"a Math [text] that says \"$n = 4$\"","se":"a VariableNumber (initial_value=0.9)","se_brace":"a Brace [red] labelled \"sigma_(overline(x))\" drawn in bells (x_start=2.8, x_end=(2.8 + se))","thm":"a Panel that says \"Draw samples of size $n$ from any population with mean $mu$ and finite standard deviation $sigma$. As $n$ grows, the distribution of $overline(x)$ approaches the normal curve with centre $mu$ and width $sigma / sqrt(n)$.\""},"beats":[{"start":351.8072291666666,"say":"Here it is then, in full: the Central Limit Theorem. Draw samples of size n from any population at all, provided it has a mean mu and a finite standard deviation sigma. As n grows, the distribution of the sample mean approaches the normal curve, centred on mu, with width sigma over the square root of n.","live":[],"does":[[351.8072291666666,"heading1 is shown on the screen, written out."],[354.95322916666663,"clt_math is shown on the screen, written out."],[356.40522916666663,"thm is shown on the screen, written out."]]},{"start":375.3607291666666,"say":"That one line is the quantitative heart. The width of the bell of averages is sigma over root n. The population's shape is not in it, and the skew is not in it. Only mu, sigma, and n survive the averaging.","live":["clt_math","thm","heading1"],"does":[[380.0622291666666,"clt_math (the \"frac(sigma, sqrt(n))\" part) is emphasized."],[383.0812291666666,"clt_math (the \"frac(sigma, sqrt(n))\" part) is no longer emphasized."],[390.6977291666666,"clt_math moves to a new place on the board."],[390.6977291666666,"heading1 is hidden from the screen — left the board."],[390.6977291666666,"thm is hidden from the screen — left the board."]]},{"start":391.89772916666664,"say":"Let us watch that square root earn its keep. Here is the bell for n equals four, centred on the dashed line at two point eight. The bracket underneath is one width of it: sigma of x bar, the quantity our formula computes.","live":["clt_math"],"does":[[391.89772916666664,"heading2 is shown on the screen, written out."],[392.5712291666666,"bells is shown on the screen, written out."],[395.6822291666666,"bell4 is shown on the screen, drawn."],[396.8312291666666,"n_live is shown on the screen, written out."],[398.0512291666666,"mu4 is shown on the screen, written out."],[400.6052291666666,"se_brace is shown on the screen, written out."]]},{"start":407.7762291666666,"say":"The computation sits beside it. Sigma is one point eight, the spread we measured back at the start. Root four is two. And one point eight over two is zero point nine.","live":["clt_math","n_live","bells","heading2","bell4","mu4","se_brace"],"does":[[408.2522291666666,"calc is shown on the screen, written out."],[411.4682291666666,"calc (the \"1.8\" part) is emphasized."],[414.5912291666666,"calc (the \"1.8\" part) is no longer emphasized."],[414.5912291666666,"calc (the \"sqrt(4)\" part) is emphasized."],[417.83022916666664,"calc (the \"frac(1.8, 2)\" part) is emphasized."],[417.83022916666664,"calc (the \"sqrt(4)\" part) is no longer emphasized."],[419.1072291666666,"calc (the \"0.9\" part) is emphasized."],[419.1072291666666,"calc (the \"frac(1.8, 2)\" part) is no longer emphasized."]]},{"start":420.4972291666666,"say":"Now buy more data. Quadruple the sample to sixteen: root sixteen is four, and the width drops to one point eight over four, zero point four five. Watch the bracket shrink to match the new bell.","live":["clt_math","n_live","calc","bells","heading2","bell4","mu4","se_brace"],"does":[[421.5772291666666,"calc (the \"0.9\" part) is no longer emphasized."],[423.4582291666666,"n_live becomes \"$n = 16$\"."],[424.6302291666666,"calc becomes \"$frac(1.8, sqrt(16)) = frac(1.8, 4) = 0.45$\"."],[429.7502291666666,"calc (the \"0.45\" part) is emphasized."],[430.9112291666666,"se_brace is redrawn as the numbers it depends on change."],[430.9112291666666,"se ticks to 0.45."],[432.98922916666663,"bell16 is shown on the screen, drawn."]]},{"start":434.58772916666663,"say":"Mark the price of that. Four times the data bought exactly half the width, not a quarter of it. The square root is the reason, and it never relents.","live":["clt_math","n_live","calc","bells","heading2","bell4","mu4","se_brace","bell16"],"does":[[435.5282291666666,"calc (the \"0.45\" part) is no longer emphasized."],[442.0072291666666,"clt_math (the \"sqrt(n)\" part) is emphasized."],[444.2012291666666,"clt_math (the \"sqrt(n)\" part) is no longer emphasized."]]},{"start":445.75322916666664,"say":"Quadruple again, to sixty-four. Root sixty-four is eight, and the width is zero point two two five: the bell has sharpened into a spike. It grows taller as it narrows because the area under every one of these curves is one, so the probability has nowhere to go but up.","live":null,"does":[[447.1692291666666,"n_live becomes \"$n = 64$\"."],[448.7482291666666,"calc becomes \"$frac(1.8, sqrt(64)) = frac(1.8, 8) = 0.225$\"."],[452.4982291666666,"calc (the \"0.225\" part) is emphasized."],[454.2752291666666,"se_brace is redrawn as the numbers it depends on change."],[454.2752291666666,"se ticks to 0.225."],[455.0882291666666,"bell64 is shown on the screen, drawn."],[458.51322916666663,"calc (the \"0.225\" part) is no longer emphasized."]]},{"start":464.6972291666666,"say":"So this is what averaging buys, and what it costs. It forgets the population's shape entirely, and it shrinks the noise by exactly root n. What remains is the fine print, because a theorem this strong has conditions.","live":["clt_math","n_live","calc","bells","heading2","bell4","mu4","se_brace","bell16","bell64"],"does":[[479.63272916666665,"bells is hidden from the screen — left the board."],[479.63272916666665,"bell4 is hidden from the screen — bells left the board."],[479.63272916666665,"mu4 is hidden from the screen — bells left the board."],[479.63272916666665,"se_brace is hidden from the screen — bells left the board."],[479.63272916666665,"bell16 is hidden from the screen — bells left the board."],[479.63272916666665,"bell64 is hidden from the screen — bells left the board."],[479.63272916666665,"calc is hidden from the screen — left the board."],[479.63272916666665,"clt_math is hidden from the screen — left the board."],[479.63272916666665,"heading2 is hidden from the screen — left the board."],[479.63272916666665,"n_live is hidden from the screen — left the board."]]}]},{"title":"The Fine Print, and Error Bars","start":480.6743958333333,"end":638.9268749999999,"objects":{"bar":"a Line [red] drawn in err_line (start=(2.9, 0.0), end=(3.3, 0.0))","bars":"a Polygon [blue] drawn in pop5 (vertices=((0.6, 0.0), (1.4, 0.0), (1.4, 0.32), (0.6, 0.32)), fill_opacity=0.5)","bars_2":"a Polygon [blue] drawn in pop5 (vertices=((1.6, 0.0), (2.4, 0.0), (2.4, 0.22), (1.6, 0.22)), fill_opacity=0.5)","bars_3":"a Polygon [blue] drawn in pop5 (vertices=((2.6, 0.0), (3.4, 0.0), (3.4, 0.16), (2.6, 0.16)), fill_opacity=0.5)","bars_4":"a Polygon [blue] drawn in pop5 (vertices=((3.6, 0.0), (4.4, 0.0), (4.4, 0.11), (3.6, 0.11)), fill_opacity=0.5)","bars_5":"a Polygon [blue] drawn in pop5 (vertices=((4.6, 0.0), (5.4, 0.0), (5.4, 0.08), (4.6, 0.08)), fill_opacity=0.5)","bars_6":"a Polygon [blue] drawn in pop5 (vertices=((5.6, 0.0), (6.4, 0.0), (6.4, 0.06), (5.6, 0.06)), fill_opacity=0.5)","bars_7":"a Polygon [blue] drawn in pop5 (vertices=((6.6, 0.0), (7.4, 0.0), (7.4, 0.05), (6.6, 0.05)), fill_opacity=0.5)","cap3":"a Tex [text] that says \"Halving the noise costs quadruple the data.\"","capL":"a Line [red] drawn in err_line (start=(2.9, -0.15), end=(2.9, 0.15))","capR":"a Line [red] drawn in err_line (start=(3.3, -0.15), end=(3.3, 0.15))","d16":"a PlotPoint [yellow] labelled \"0.45\" drawn in se_axes (target='se_curve', x=16.0)","d4":"a PlotPoint [blue] labelled \"0.9\" drawn in se_axes (target='se_curve', x=4.0)","d64":"a PlotPoint [green] labelled \"0.225\" drawn in se_axes (target='se_curve', x=64.0)","dot":"a Point [blue] drawn in err_line (location=(3.1, 0.0))","err_line":"a NumberLine labelled \"overline(x)\" (x_range=(2.5, 3.7), include_numbers=True, ticks_every=0.2)","h1":"a Heading that says \"The Fine Print\"","h2":"a Heading that says \"Why Error Bars Work\"","h3":"a Heading that says \"One Curve to Remember\"","meas":"a Math [text] that says \"$overline(x) = 3.1$\"","point":"a Point [yellow] drawn in pop5 (location=(6.8, 0.1))","point_2":"a Point [yellow] drawn in pop5 (location=(7.0, 0.07))","pop5":"an Axes (x_range=(0.0, 8.0), y_range=(0.0, 0.4), x_ticks_every=1.0)","quote":"a Math [text] that says \"$3.1 plus.minus 0.2$\"","rules":"a Block [text] that says \"The draws must be independent, and all from the same population. $sigma$ must be finite: a population with a wild enough tail breaks the theorem. How large $n$ must be depends on the shape: the more skewed the population, the more averagin…\"","se_axes":"an Axes (x_range=(0.0, 70.0), x_ticks_every=8.0, y_ticks_every=0.2)","se_chip":"a Math [text] that says \"$frac(sigma, sqrt(n)) = 0.2$\"","se_curve":"a FunctionPlot [text] drawn in se_axes (function=<function>, x_range=(3.4, 68.0))","wild":"a FunctionPlot [red] drawn in pop5 (function=<function>, x_range=(0.2, 7.8))"},"beats":[{"start":480.6743958333333,"say":"A theorem this useful earns its fine print, and there are three clauses. Beside them, the shop again, to test each clause against.","live":[],"does":[[480.6743958333333,"h1 is shown on the screen, written out."],[480.6743958333333,"pop5 is shown on the screen, written out."],[480.6743958333333,"bars is shown on the screen, written out."],[480.7143958333333,"bars_2 is shown on the screen, written out."],[480.75439583333326,"bars_3 is shown on the screen, written out."],[480.7943958333333,"bars_4 is shown on the screen, written out."],[480.8343958333333,"bars_5 is shown on the screen, written out."],[480.87439583333327,"bars_6 is shown on the screen, written out."],[480.9143958333333,"bars_7 is shown on the screen, written out."],[483.86739583333326,"pop5 moves to a new place on the board."],[483.86739583333326,"rules is shown on the screen, written out."]]},{"start":489.7148958333333,"say":"First: the draws must be independent, and all from one population. Our machine had that by construction. Real data, measured over time or across neighbours, has to earn it.","live":["rules","pop5","h1","bars","bars_2","bars_3","bars_4","bars_5","bars_6","bars_7"],"does":[[490.0633958333333,"rules (the \"independent\" part) is emphasized."],[496.5533958333333,"rules (the \"independent\" part) is no longer emphasized."]]},{"start":503.77139583333326,"say":"Second: sigma must be finite. Here is a population that breaks the rule: this red tail dies away so slowly that the spread is infinite, and averages drawn from it never settle into any bell. No sigma, no theorem.","live":null,"does":[[504.3283958333333,"rules (the \"finite\" part) is emphasized."],[510.4123958333333,"wild is shown on the screen, written out."],[511.73539583333326,"point is shown on the screen, grown."],[513.7353958333333,"point is hidden from the screen."],[515.6833958333333,"wild is hidden from the screen."],[515.6833958333333,"rules (the \"finite\" part) is no longer emphasized."]]},{"start":520.1138958333332,"say":"Third: how large n must be depends on the shape you start from. Our shop's mild tail was nearly gone by n equals four; a harsher skew takes more averaging. The theorem promises the bell in the limit, and the shape decides the price of getting there.","live":null,"does":[[520.5203958333333,"rules (the \"skewed\" part) is emphasized."],[525.7563958333333,"point_2 is shown on the screen, grown."],[527.7563958333333,"point_2 is hidden from the screen."],[534.3253958333332,"rules (the \"skewed\" part) is no longer emphasized."],[535.7648958333333,"h1 is hidden from the screen — left the board."],[535.7648958333333,"pop5 is hidden from the screen — left the board."],[535.7648958333333,"bars is hidden from the screen — pop5 left the board."],[535.7648958333333,"bars_2 is hidden from the screen — pop5 left the board."],[535.7648958333333,"bars_3 is hidden from the screen — pop5 left the board."],[535.7648958333333,"bars_4 is hidden from the screen — pop5 left the board."],[535.7648958333333,"bars_5 is hidden from the screen — pop5 left the board."],[535.7648958333333,"bars_6 is hidden from the screen — pop5 left the board."],[535.7648958333333,"bars_7 is hidden from the screen — pop5 left the board."],[535.7648958333333,"rules is hidden from the screen — left the board."]]},{"start":536.9648958333332,"say":"Now spend the theorem on the thing you meet everywhere: the error bar. A lab measures something noisy, averages n readings, and reports the average: three point one. An honest report also says how far off that number is likely to be.","live":[],"does":[[536.9648958333332,"h2 is shown on the screen, written out."],[541.8183958333333,"err_line is shown on the screen, written out."],[545.3243958333333,"err_line moves to a new place on the board."],[545.3243958333333,"meas is shown on the screen, written out."],[547.1003958333333,"dot is shown on the screen, written out."]]},{"start":553.3663958333333,"say":"The theorem is the answer. That reported average was one draw from a bell whose width is sigma over root n, and here that width is zero point two. So draw the bar: from two point nine to three point three, one width each side of the reading. That is the error bar, drawn.","live":["meas","err_line","h2","dot"],"does":[[558.8693958333333,"se_chip is shown on the screen, written out."],[563.0253958333333,"se_chip (the \"0.2\" part) is emphasized."],[564.6853958333332,"bar is shown on the screen, written out."],[564.6853958333332,"capL is shown on the screen, written out."],[564.8053958333333,"capR is shown on the screen, written out."],[570.7803958333333,"quote is shown on the screen, written out."],[570.7803958333333,"se_chip (the \"0.2\" part) is no longer emphasized."]]},{"start":572.8783958333333,"say":"So a quoted plus or minus is not decoration, and it is not a guess. It is this theorem, applied: the width of the normal curve the average came from. Wherever an error bar stands, a bell stands behind it.","live":["meas","se_chip","quote","err_line","h2","dot","bar","capL","capR"],"does":[[579.1243958333333,"A box is drawn around quote."],[583.6643958333333,"bar is indicated — a transient flash."],[587.6578958333333,"err_line is hidden from the screen — left the board."],[587.6578958333333,"dot is hidden from the screen — err_line left the board."],[587.6578958333333,"bar is hidden from the screen — err_line left the board."],[587.6578958333333,"capL is hidden from the screen — err_line left the board."],[587.6578958333333,"capR is hidden from the screen — err_line left the board."],[587.6578958333333,"h2 is hidden from the screen — left the board."],[587.6578958333333,"meas is hidden from the screen — left the board."],[587.6578958333333,"quote is hidden from the screen — left the board."],[587.6578958333333,"se_chip is hidden from the screen — left the board."]]},{"start":588.8578958333333,"say":"One curve carries the whole lecture. Here is the noise in the average, sigma over root n, plotted against the sample size. At n equals four it stood at zero point nine. At sixteen, zero point four five. At sixty-four, zero point two two five.","live":[],"does":[[588.8578958333333,"h3 is shown on the screen, written out."],[589.6943958333333,"se_axes is shown on the screen, written out."],[592.3293958333333,"se_curve is shown on the screen, drawn."],[598.5863958333333,"d4 is shown on the screen, written out."],[601.6753958333333,"d16 is shown on the screen, written out."],[604.9843958333333,"d64 is shown on the screen, written out."]]},{"start":609.0438958333333,"say":"The curve falls like one over root n, so each halving of the noise costs a quadrupling of the data. That is why large experiments are large, and why the last decimal place is always the expensive one.","live":["se_axes","h3","se_curve","d4","d16","d64"],"does":[[613.9553958333333,"cap3 is shown on the screen, written out."]]},{"start":623.2508958333333,"say":"And that is the Central Limit Theorem. Averages forget the shape they were drawn from, remember the mean, and close in on it like root n. The bell was never hiding in the population. Averaging builds it, every time.","live":["se_axes","cap3","h3","se_curve","d4","d16","d64"],"does":[[637.8852083333333,"cap3 is hidden from the screen — left the board."],[637.8852083333333,"h3 is hidden from the screen — left the board."],[637.8852083333333,"se_axes is hidden from the screen — left the board."],[637.8852083333333,"se_curve is hidden from the screen — se_axes left the board."],[637.8852083333333,"d4 is hidden from the screen — se_axes left the board."],[637.8852083333333,"d16 is hidden from the screen — se_axes left the board."],[637.8852083333333,"d64 is hidden from the screen — se_axes left the board."]]}]}]},"durationSeconds":639,"chapters":[{"title":"A Lopsided Population","startSeconds":0,"narration":"Every field that measures anything keeps meeting the same curve. Biologists find it in heights, physicists in noise, pollsters in polls: a symmetric bell, over and over, in data that has no business agreeing about anything. Today we find out where that bell comes from, and the answer is one theorem about averages. So let me build a world with no bell in it anywhere. Here is a population: the number of sales a small shop makes in a day. Most days are quiet, and one sale is the commonest day of all, at thirty-two percent. Two sales happen on twenty-two percent of days, and three on sixteen. Then the tail begins: four, five, six, and every so often a wild day of seven sales. Two numbers will follow us all lecture. The mean, mu, is two point eight, the balance point of the whole shape. And the standard deviation, sigma, is one point eight, the typical distance a single day lands from that mean. And look at the shape itself. It is lopsided: the tail runs off to the right, and a seven sits far from everything else. Nothing about this picture is symmetric, and nothing about it is a bell. Now the destination. This curve on the right is the normal curve: the bell that every one of those fields keeps finding, symmetric about its middle and thin in both tails. Keep the two pictures side by side, because together they are the whole lecture. The claim is that the lopsided shop on the left will manufacture the bell on the right, and the machine that does it is averaging. Take four days at random, average them, and keep the average. Do it again and again, and watch what the averages do."},{"title":"One Average at a Time","startSeconds":112.51533333333334,"narration":"Here is the machine we ended on, and its one rule: draw four days at random, average them, and keep nothing but the average. On the left, the population we draw from. On the right, an empty frame where the averages will live. First sample. The shop hands us a two, a one, a five, and a three, and there they are, sitting on the population. Now the arithmetic, in full. Two plus one plus five plus three is eleven, and eleven over four is two point seven five. That is the sample mean, written x bar, and it is the only thing we keep. Over it goes. The tick marks two point seven five on the new axis, and then the whole sample becomes one gray brick, standing in the bin it landed in. The four days that made it are gone. Again. A one, a four, and a pair of twos: the sum is nine, nine over four is two point two five, and a second brick lands just to the left of the first. Once more: a six, a one, a three, and a two. Twelve over four is three exactly, and a third brick lands. Now a lucky draw. A seven comes up, with a one, a one, and a two. Add them: eleven again, so the average is two point seven five, again. Even a wild day, averaged with three quiet ones, gets dragged back to the middle of the pile. From here, let it run. Sample, average, brick. Sample, average, brick. Five more spins of that loop, five more bricks, and already the pile stands tallest in the middle and thin at both ends. Nine averages hint at a shape without showing it. To see it properly we need not nine but dozens, so let the machine take over and run while we watch."},{"title":"The Shape of the Averages","startSeconds":232.73597916666665,"narration":"Sixty-four averages now: the machine kept sampling while we talked, and every average left its brick. Look at the wall they built. It stands tallest just short of three, and it falls away on both sides. Set the wall against where it came from. The population leans left and drags its tail out to seven. The averages have no tail worth the name: past four and a half, the bins are simply empty. Averaging erased the skew, and the question is how. So watch a lucky sample try to build a tail. A seven comes up, and with it a one, a two, and a three. Their average: thirteen over four, which is three and a quarter. The seven hauled it to the right, and the quiet days hauled it straight back. Averaging is a tug of war, and the middle wins. An average out at seven would need all four draws to be sevens at once, and each has probability zero point zero five. Multiply the four together: about six chances in a million. That is why the wall is dead long before seven. Now bring back the curve from the opening minute and lay it over the wall. This is the normal curve, and it hugs the bricks. A lopsided population, pushed through nothing but averaging, has manufactured the bell. What we built deserves its name. The sample mean is itself random: a different sample gives a different average. Its distribution, across every sample you could possibly draw, is called the sampling distribution of the mean, and our histogram is a portrait of it. And measure the portrait. Its centre sits at two point eight, the population mean, exactly. Its width is zero point nine, and zero point nine is one point eight divided by two: sigma, divided by the square root of four. The sample size has crept into the width, and that is the theorem knocking."},{"title":"The Theorem at Work","startSeconds":351.8072291666666,"narration":"Here it is then, in full: the Central Limit Theorem. Draw samples of size n from any population at all, provided it has a mean mu and a finite standard deviation sigma. As n grows, the distribution of the sample mean approaches the normal curve, centred on mu, with width sigma over the square root of n. That one line is the quantitative heart. The width of the bell of averages is sigma over root n. The population's shape is not in it, and the skew is not in it. Only mu, sigma, and n survive the averaging. Let us watch that square root earn its keep. Here is the bell for n equals four, centred on the dashed line at two point eight. The bracket underneath is one width of it: sigma of x bar, the quantity our formula computes. The computation sits beside it. Sigma is one point eight, the spread we measured back at the start. Root four is two. And one point eight over two is zero point nine. Now buy more data. Quadruple the sample to sixteen: root sixteen is four, and the width drops to one point eight over four, zero point four five. Watch the bracket shrink to match the new bell. Mark the price of that. Four times the data bought exactly half the width, not a quarter of it. The square root is the reason, and it never relents. Quadruple again, to sixty-four. Root sixty-four is eight, and the width is zero point two two five: the bell has sharpened into a spike. It grows taller as it narrows because the area under every one of these curves is one, so the probability has nowhere to go but up. So this is what averaging buys, and what it costs. It forgets the population's shape entirely, and it shrinks the noise by exactly root n. What remains is the fine print, because a theorem this strong has conditions."},{"title":"The Fine Print, and Error Bars","startSeconds":480.6743958333333,"narration":"A theorem this useful earns its fine print, and there are three clauses. Beside them, the shop again, to test each clause against. First: the draws must be independent, and all from one population. Our machine had that by construction. Real data, measured over time or across neighbours, has to earn it. Second: sigma must be finite. Here is a population that breaks the rule: this red tail dies away so slowly that the spread is infinite, and averages drawn from it never settle into any bell. No sigma, no theorem. Third: how large n must be depends on the shape you start from. Our shop's mild tail was nearly gone by n equals four; a harsher skew takes more averaging. The theorem promises the bell in the limit, and the shape decides the price of getting there. Now spend the theorem on the thing you meet everywhere: the error bar. A lab measures something noisy, averages n readings, and reports the average: three point one. An honest report also says how far off that number is likely to be. The theorem is the answer. That reported average was one draw from a bell whose width is sigma over root n, and here that width is zero point two. So draw the bar: from two point nine to three point three, one width each side of the reading. That is the error bar, drawn. So a quoted plus or minus is not decoration, and it is not a guess. It is this theorem, applied: the width of the normal curve the average came from. Wherever an error bar stands, a bell stands behind it. One curve carries the whole lecture. Here is the noise in the average, sigma over root n, plotted against the sample size. At n equals four it stood at zero point nine. At sixteen, zero point four five. At sixty-four, zero point two two five. The curve falls like one over root n, so each halving of the noise costs a quadrupling of the data. That is why large experiments are large, and why the last decimal place is always the expensive one. And that is the Central Limit Theorem. Averages forget the shape they were drawn from, remember the mean, and close in on it like root n. The bell was never hiding in the population. Averaging builds it, every time."}]}}