spread
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
show: "tight"
show: "sweep"
show: "average"
show: "scaling"
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.
- at 1.5 standard deviations the mass outside is within the bound ×3
- the sum of 4 divided by 4 has probabilities adding to one ×3
- the sum of 4 divided by 4 has some spread to speak of ×3
- the sum of 4 divided by the root of 4 has probabilities adding to one ×3
- the sum of 4 divided by the root of 4 has some spread to speak of ×3
- 0 nine times in ten, 10 otherwise stays under the bound at every width ×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 it comes within half the bound somewhere ×1
- and scaled so that one standard deviation is one unit ×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
- between 2 and 12 runs are drawn ×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 of between 2 and 200 copies ×1
- each window is between half a standard deviation and six ×1
- every distribution swept is one the family knows ×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
- so one scaling is collapsing ×1
- the bound is met at between 1.2 and 5 standard deviations ×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 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 mass at exactly k standard deviations is the whole of the bound ×1
- the scaled values are evenly spaced and increasing ×1
- the sweep includes a lopsided distribution ×1
- the view is one the family draws ×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.
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.
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.