Generator

five values, unevenly weighted, and the mass outside 3 standard deviations

A generator in the probability library, called 41 times across 9 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

spread is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

five values, unevenly weighted, and the mass outside 3 standard deviations. A distribution drawn as bars, with the windows one and a half, two and three standard deviations wide marked. The probability outside each window is summed and compared with the bound that knows only the variance.

`show: "heavy"`

Averages of a heavy-tailed quantity, which never settle. Running averages of draws from a Cauchy distribution, which jump rather than converge, beside the cumulative distributions of averages of 1, 4 and 16 draws, which lie on top of one another.

`show: "average"`

7 running averages, and the band they stay inside. The running average of repeated draws, for several independent runs, with the band two standard errors wide. The runs wander early and settle onto the mean as the band closes.

`show: "tight"`

The distribution that meets the bound at 2 standard deviations. A three-valued distribution whose mass beyond k standard deviations is exactly the bound. It shows the inequality cannot be improved without assuming more than a variance.

`show: "fixed"`

A lopsided distribution added to itself, and the shape that returns. On the left, the exact distribution of a sum of copies of one lopsided distribution, standardised, for several counts: the shapes converge. On the right, the bell curve convolved with itself, which is the bell curve again.

`show: "sweep"`

Four distributions, and the one bound over all of them. The probability of landing at least k standard deviations from the mean, for four distributions, against the bound that knows only their variance. Every staircase stays under the curve.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

coin summed 2 times has probabilities adding to one ×7
coin summed 2 times has some spread to speak of ×7
skew summed 2 times has probabilities adding to one ×7
skew summed 2 times has some spread to speak of ×7
die summed 2 times has probabilities adding to one ×6
die summed 2 times has some spread to speak of ×6
the sum of 4 divided by 4 has probabilities adding to one ×6
the sum of 4 divided by 4 has some spread to speak of ×6
the sum of 4 divided by the root of 4 has probabilities adding to one ×6
the sum of 4 divided by the root of 4 has some spread to speak of ×6
the average of 1 draws still has the distribution of one draw ×5
the sum of 4 has probabilities adding to one ×4
the sum of 4 has some spread to speak of ×4
and doubling the draws moves it toward the computed one at 0.3 ×3
at 1.5 standard deviations the mass outside is within the bound ×3
the measured rate at 0.3 is never below the computed one ×3
there are exactly 91 possible histograms of 12 draws over three faces ×3
the largest sum is a power of two between 16 and 128 ×2
0 nine times in ten, 10 otherwise has probabilities adding to one ×1
0 nine times in ten, 10 otherwise has some spread to speak of ×1
0 nine times in ten, 10 otherwise stays under the bound at every width ×1
a die has probabilities adding to one ×1
a die has some spread to speak of ×1
a fair coin, ±1 has probabilities adding to one ×1
a fair coin, ±1 has some spread to speak of ×1
a fair coin, ±1 stays under the bound at every width ×1
a fair die has probabilities adding to one ×1
a fair die has some spread to speak of ×1
a fair die stays under the bound at every width ×1
and dividing by the root of n leaves it exactly where it started ×1
and increasing away from the mean ×1
and it comes within half the bound somewhere ×1
and its mean is exactly the level asked about ×1
and none is smaller than that divided by a polynomial in n ×1
and scaled so that one standard deviation is one unit ×1
and the more lopsided summand is further from the bell at every n ×1
and the rate at the mean itself is zero ×1
and the rate is the information distance from the original to the tilted law ×1
and the runs are further apart early than late ×1
and the second panel's settle on the height of the bell curve ×1
and the variance adds too ×1
between 2 and 12 runs are drawn ×1
between 2 and 3 summands are compared ×1
between 2 and 4 averages, each of at most 64 draws ×1
between 2,000 and 60,000 averages are formed ×1
between 3 and 40 draws ×1
between 3 and 6 sums, each of at most 200 copies ×1
between two and four levels are marked ×1
dividing by n shrinks the spread like 1/√n ×1
each distribution is one the family knows ×1
each n is between 2 and 200 ×1
each run is between 50 and 5000 draws ×1
each sum is closer to the bell curve than the last ×1
each sum is of between 1 and 400 copies ×1
each window is between half a standard deviation and six ×1
every distribution swept is one the family knows ×1
every level is between the mean and the largest value ×1
every run finishes inside four standard errors of the mean ×1
five values, unevenly weighted has probabilities adding to one ×1
five values, unevenly weighted has some spread to speak of ×1
five values, unevenly weighted stays under the bound at every width ×1
it is centred at zero ×1
no histogram is more likely than exp(−nD) ×1
sixteen dice has probabilities adding to one ×1
sixteen dice has some spread to speak of ×1
so one scaling is collapsing ×1
the average of two Cauchys has the density of one Cauchy, by the convolution integral ×1
the bell curve convolved with itself is the bell curve again, widened by √2 ×1
the bell curve's exponent differs from the rate by enough to show over the range drawn ×1
the bound is met at between 1.2 and 5 standard deviations ×1
the constrained region is not empty ×1
the constrained set has a probability strictly between nothing and everything ×1
the constraint asks for a mean above the true one and below the largest face ×1
the distribution has probabilities adding to one ×1
the distribution has some spread to speak of ×1
the distribution is one the family knows ×1
the event has positive probability at every n drawn ×1
the exact probability stays under the exponential the rate predicts ×1
the extremal distribution has probabilities adding to one ×1
the extremal distribution has some spread to speak of ×1
the first panel's curves grow taller as they narrow ×1
the gap for 0 nine times in ten, 10 otherwise falls like one over the square root of n ×1
the gap for a fair coin, ±1 falls like one over the square root of n ×1
the gap for a fair die falls like one over the square root of n ×1
the histograms' chances add to one ×1
the level asked about is above the mean and below the largest value the summand can take ×1
the level is between the mean and the largest value the summand takes ×1
the mass at exactly k standard deviations is the whole of the bound ×1
the mean of a sum is the sum of the means ×1
the measured decay rate agrees with the rate function computed from the summand alone ×1
the measured rate moves towards the smallest relative entropy in the region ×1
the measured rates come from between 16 and 96 draws ×1
the rate at a level away from the mean is positive ×1
the rate function is convex all the way along ×1
the scaled values are increasing ×1
the sweep includes a lopsided distribution ×1
the sweep runs to between 24 and 200 draws ×1
the tilted distribution has probabilities adding to one ×1
the tilted distribution has some spread to speak of ×1
the tilted weights are a distribution ×1
the view is one the family draws ×1
three face values ×1
three positive chances ×1
three positive chances adding to one ×1
while a light-tailed average narrows by the square root of the count ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Probability

An average that never settles

The average of many independent quantities is supposed to steady as their number grows. For one famous distribution it does not steady at all — the average of a thousand draws has exactly the same distribution as a single draw, and no amount of further averaging changes it.

Probability

How far from the average a thing can be

Knowing only an average and a spread — nothing about the shape, nothing about the number of outcomes, nothing about symmetry — the chance of landing three standard deviations out is at most one in nine. And there is a distribution that lands there exactly that often, so the bound cannot be improved.

Probability

How fast the bell arrives

The limit theorem says a standardised sum approaches the bell curve and says nothing about when. The rate is one over the square root of the number of terms, the constant in front is made of the third moment, and both are visible.

Probability

The average settles and the wobble does not

Two theorems are usually met a page apart and sound as though one is a sharper version of the other. They are the same sums looked at through two different magnifying glasses: divide by the number of them and everything collapses to a point, divide by its square root and a shape appears.

Probability

The error that does not care how many dimensions

A grid gets rapidly better in one dimension and hopelessly worse in twenty. Random points get better at the same slow rate whatever the dimension, which is why a method that is bad everywhere ends up being the only one that works.

Probability

The shape that averaging leaves alone

Adding independent quantities blurs their distributions together, and rescaling restores the width. Almost every shape is changed by that operation. Exactly one is returned unaltered, and that is why sums of unrelated things keep arriving at it.

Probability

The tail is not a bell

The limit theorem describes a window of width one over the root of n around the mean; ask instead for the chance that an average lands a fixed distance away and the answer falls exponentially, at a rate computed from the summand before any n is chosen.

Probability

The walk that becomes a curve

Shrink the steps of a random walk and it disappears. Shrink them while stretching the time in the right proportion — space by the square root of whatever time is divided by — and something is left behind, which is a curve nobody could draw.

Probability

When the whole histogram deviates

The rung below priced a rare average. Ask instead for the chance that the whole tally of outcomes comes out wrong, and the exponent is no longer a function of one number — it is a distance between two distributions, and every rare-average rate is a shadow of it.

The whole library · What the figures prove