five values, unevenly weighted, and the mass outside 3 standard deviations
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.
With nothing chosen
How much of a sphere lies near its equator
One coordinate of a random unit vector
Two random directions are nearly at right angles
A function on a high-dimensional sphere is nearly constant
Fatten half a sphere, and almost nothing is left over
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.
- the ratio at n = 1 ×100
- 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
- every one of the 6561 inputs at n = 8 is counted ×5
- the average of 1 draws still has the distribution of one draw ×5
- the margin for a town of 1250 ×4
- 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 bound holds at every deviation the distribution reaches, n = 8 ×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
- four products of between 1 and 60 factors ×2
- the largest sum is a power of two between 16 and 128 ×2
- the support is an odd number of points between 21 and 121 ×2
- ±½, even odds has mean 0 ×1
- ±½, even odds has variance 1/4 ×1
- −1, 0 or 1 has mean 0 ×1
- −1, 0 or 1 has variance 1/4 ×1
- 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 bound that says nothing belongs to a function one input nearly controls ×1
- a die has probabilities adding to one ×1
- a die has some spread to speak of ×1
- a distribution the figure knows ×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
- a sample that divides into k equal blocks ×1
- a sample with exactly one rare jump exists among the seeds tried ×1
- a uniform factor's coefficient is 1/√(1 + t²) ×1
- and a distribution that is actually bell-shaped is far inside both of them ×1
- and a useful bound to one where it does not ×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 it reaches it, so the bound is attained rather than merely true ×1
- and its average against any such distribution is exactly the bound ×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 binomial one stays under Hoeffding's lemma ×1
- and the median of block means ignores it where the mean does not ×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 typical value settles at √(2/π) ×1
- and the variance adds too ×1
- asking about one tail rather than two is worth something ×1
- at least one of the useful rows is not a sum, which is the point ×1
- Bennett's closed form ×1
- Bennett's exponent is never below Bernstein's ×1
- between 100 and 3,000 powers ×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 and 40 dice ×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 6 and 14 inputs ×1
- between two and four input counts, each from 6 to 16 ×1
- between two and four levels are marked ×1
- both have the same mean ×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 bound's exponent is at most the true one ×1
- every distribution swept is one the family knows ×1
- every extra factor brings the digits closer to the law ×1
- every level is between the mean and the largest value ×1
- every run finishes inside four standard errors of the mean ×1
- exact < Bennett ≤ Bernstein < Hoeffding ×1
- exact ≤ Bennett ≤ Bernstein at every threshold ×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
- Hoeffding's bound holds both ways ×1
- in 100 dimensions the coordinate's density is within 3% of the normal curve with standard deviation 1/√n ×1
- it is centred at zero ×1
- more dimensions, more pairs within ten degrees of a right angle ×1
- no histogram is more likely than exp(−nD) ×1
- on normal data the plain mean is the more accurate at every confidence ×1
- on the ordinary sphere a band's share is its width: Archimedes' hat-box theorem ×1
- one input can move the answer at all ×1
- Serfling's sharper bound holds without replacement ×1
- sixteen dice has probabilities adding to one ×1
- sixteen dice has some spread to speak of ×1
- so one scaling is collapsing ×1
- some distribution meets the constraints ×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 best distribution does not beat the bound ×1
- the block holding the jump has the most extreme block mean ×1
- the bound holds for n times the first flip ×1
- the bound holds for the longest non-decreasing run of choices ×1
- the bound holds for the longest run of heads in n flips ×1
- the bound holds for the total of n coin flips ×1
- the bound is met at between 1.2 and 5 standard deviations ×1
- the certificate lies above the indicator at every support point ×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 digit probabilities of a product of uniforms add to one ×1
- the distance never exceeds the Fourier bound ×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 chance that half the blocks fail is under e^(−k/8) ×1
- the exact probability stays under the exponential the rate predicts ×1
- the exhaustive sweep is feasible ×1
- the extremal distribution has probabilities adding to one ×1
- the extremal distribution has some spread to speak of ×1
- the far cap is never larger than e^(−nε²/2) ×1
- the first panel's curves grow taller as they narrow ×1
- the function is one of sum, longest, distinct, rising, dictator ×1
- the functions are among sum, longest, distinct, rising, dictator ×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 hypergeometric probabilities 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 grows with the number of inputs, so the panels differ ×1
- the mean of a sum is the sum of the means ×1
- the mean's error sits under Chebyshev's guarantee at every confidence ×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 one-sided answer is Cantelli's ×1
- the one-sided bound is strictly better than the two-sided one from the same two numbers ×1
- the powers of two follow the law closely ×1
- the product of the dice is closer to the law than their sum ×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 spread shrinks as the dimension grows ×1
- the support runs to between 3 and 12 deviations ×1
- the sweep includes a lopsided distribution ×1
- the sweep is feasible ×1
- the sweep runs to between 24 and 200 draws ×1
- the table separates the functions the bound says something about from the ones it does not ×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 two-point variable has the largest generating function ×1
- the two-sided answer is Chebyshev's bound ×1
- the uniform products approach the law by a factor |E Xⁱᵗ| per factor ×1
- the variance shrinks by exactly (N − n)/(N − 1) ×1
- the view is one the family draws ×1
- the window is between 1.2 and 4 deviations wide ×1
- the window is between 1.5 and 4 deviations ×1
- the window is two-sided or one-sided ×1
- the wrapped density flattens as factors are added ×1
- three face values ×1
- three positive chances ×1
- three positive chances adding to one ×1
- two-point: 1 or −¼ has mean 0 ×1
- two-point: 1 or −¼ has variance 1/4 ×1
- while a light-tailed average narrows by the square root of the count ×1
- with rare jumps the median of means is far more accurate at 1-in-1000 ×1
- without replacement the generating function is never larger (Hoeffding 1963) ×1
Where it is called
Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.
A sphere that is nearly all equator
On an ordinary globe, the band within a tenth of the radius of the equator holds a tenth of the surface. On a sphere in a thousand dimensions the same band holds 99.85% of it, and the band of a fifth holds all but about two parts in ten billion. Almost every point of a high-dimensional sphere is near every equator at once — and so any function that cannot change quickly is, over almost all of the sphere, almost constant.
ProbabilityAn 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.
ProbabilityCharged for the variance, not the range
Hoeffding's inequality knows one thing about each term of a sum: the interval it lies in. For ten thousand coins that each land heads once in a thousand, that makes it promise almost nothing — a 92% chance of thirty heads, when the truth is two in ten million. Tell the bound each term's variance as well and it changes character: Bernstein's and Bennett's inequalities decay like the normal curve while the deviation is small and like a Poisson tail beyond, and for rare events they are millions of times sharper.
ProbabilityDrawn without putting back
Every concentration bound on this shelf assumes the draws are independent. A real sample is not: a pollster does not ring the same person twice, and every ball taken from an urn changes what is left in it. The dependence runs the helpful way. Wassily Hoeffding proved in 1963 that a sample drawn without replacement is at least as concentrated as one drawn with it, for every convex measure of spread at once — and the variance falls by an exact factor that reaches zero when the whole urn is taken.
ProbabilityHow 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.
ProbabilityHow 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.
ProbabilityMultiplying makes the digit one common
Multiply a few random numbers together and the product starts with 1 about 30 per cent of the time and with 9 under 5 per cent — Benford's law, which no factor contains. The logarithm of a product is a sum, the central limit theorem spreads that sum across many powers of ten, and once it is spread its fractional part is uniform. The approach is geometric, at a rate fixed by a single number for each kind of factor; sums never get there, and the powers of two get there with no randomness at all.
ProbabilityNo single input can move it far
Independence was never the hypothesis doing the work. A quantity built from many separately drawn inputs concentrates whenever changing one of them moves it only a little — and that covers quantities which are not sums of anything and have no formula at all.
ProbabilityThe 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.
ProbabilityThe bound is the answer to a search
Chebyshev's inequality is not a clever estimate that happens to be sharp. It is the exact answer to a maximisation over all distributions with a stated mean and variance, and the polynomial that proves nothing beats it is the certificate a search of that kind always produces.
ProbabilityThe 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.
ProbabilityThe median of many small averages
Knowing only that a quantity has a finite spread, the plain average of n samples can be promised to within σ/√(nδ) with confidence 1 − δ, and no better — Chebyshev's bound is tight, and rare large jumps achieve it. Cut the same samples into a dozen blocks, average each block, and take the median of the averages, and the promise improves to within about σ√(log(1/δ)/n). Nothing about the data has been assumed beyond the spread; only the way of combining it has changed.
ProbabilityThe 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.
ProbabilityThe 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.
ProbabilityThe 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.
ProbabilityWhen the whole histogram deviates
A rare average has a price, an exponent that grows with the number of trials. 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.