Every essay
The distribution itself
Sums converge on it, which is the most-quoted theorem in the subject. What the quoting leaves out is the rate: the middle converges quickly and the tail does not, and the tail is where the approximation is actually read.
Sums of almost anything
The theorem says sums converge on one shape whatever they are sums of, which is remarkable and true. Watching it happen from a one-sided skewed source, with the rate of convergence predicted in advance, is more convincing than watching the shape appear.
The tail converges last
The central limit theorem is usually shown as a shape arriving. What the demonstration leaves out is the rate — and the rate is wildly different in the middle and in the tail, which is where every approximation in the subject is actually read.
What normal actually looks like
A single quantile plot of forty normal points wanders enough to look suspicious. Twenty of them, all genuinely normal, show what the noise looks like — and any single panel a reader would have rejected is in there.
The shape, and where its mass is
68, 95, 99.7 is recited more often than any other set of numbers in the subject. They are integrals of a specific curve, they are worth computing rather than remembering, and the third one is the one people misuse.
Intervals, counted
An interval that claims 95% is making a checkable statement about a procedure. Build every possible sample and count. The interval taught first fails, the failure is worst where proportions are most often reported, and more data does not monotonically help.
What the 95% refers to
An interval that claims 95% is making a checkable statement about a procedure, not about the interval in front of you. Build every possible sample and count, and the interval taught first turns out to cover 87.6% of the time.
More data is not monotonically better
Coverage of an interval for a proportion does not improve smoothly as the sample grows. It oscillates, and there are larger samples that cover materially worse than smaller ones — a sample of twenty covers twelve points worse than a sample of nineteen.
Twenty intervals and one expected miss
The 95% belongs to the procedure, not to the interval in front of you. Twenty intervals from twenty samples make that visible in a way no definition does, and the one that misses is not a mistake.
The shortest interval is the one that misses
Four intervals for the same data, with their widths and their coverage measured together. The narrowest is the one that fails its stated level, which is exactly why it looks the most appealing.
Where the bootstrap lies
Resampling is the most generally useful trick in the subject and it has a failure mode that is easy to state: it cannot see past the data. For a statistic that lives at the edge of the sample, coverage collapses from 95% to almost nothing.
The correction for not knowing the spread
The t distribution exists because the standard deviation is estimated rather than known. At eight observations, using the normal instead makes every interval 12% too short — and the coverage that follows can be measured rather than argued about.
Tests, and the second number
A p-value alone cannot be read: the same 0.04 means different things at different sample sizes, and nothing at all without knowing how many analyses were available. Every figure here carries the number that makes it interpretable.
A p-value that is not flat is not a p-value
Under a true null, p-values are uniform. That is stronger than saying the test rejects 5% of the time, it constrains the whole distribution rather than one point of it, and it catches implementation errors that a rejection rate sails past.
What a p-value does not say
The same p of 0.04 corresponds to a large effect in ten observations and a negligible one in two thousand. A p-value alone cannot be interpreted, and the number that makes it interpretable is almost never printed beside it.
The winner's curse
Filter honest studies down to the ones that reached significance and the effects they report are systematically too large. At low power the inflation is a factor of two, nobody has done anything wrong, and the selection did all of it.
Twenty analyses of nothing
Twenty honest, correct analyses of data with no effect in it find something significant 58% of the time. Nobody p-hacked, every individual p-value is right, and the reported one is the smallest of twenty.
Reversals that are not errors
Simpson's reversal, the base rate, regression to the mean. Each is normally taught with one famous table. A table is a point; these are swept, so how much of the space behaves that way and how large it can get both have answers.
Simpson's reversal is a region, not a table
The treatment wins in both groups and loses overall. That is normally shown with one famous table, which cannot answer the two questions a reader has — how often, and how large. Swept, it turns out to occupy 31% of the allocation space.
What a positive test is worth
A test that is 90% sensitive and 95% specific sounds accurate. For a condition affecting one person in a thousand, 98% of its positive results are wrong, and a worse test on a commoner condition beats a better test on a rare one.
Regression to the mean
Select the worst performers, measure them again, and they improve. Select the best and they decline. No intervention is required for either, the size of the apparent effect is predictable from the correlation alone, and it is the reason so many things appear to work.
What makes it checkable
A seeded generator, so a figure is the same on every build. A closed form beside every simulation, so there are two routes to each number. And p-values checked for uniformity, which catches errors no rejection rate would reveal.
The seed is part of the figure
Every other site in this fleet draws from a deterministic rule, so a figure either is or is not what it claims. Here the figures are samples, and a sample can be right by luck. That changes what a figure has to carry.
Two routes to every number
A site about probability that only simulates has one route to each answer and no way to tell a right one from a plausible one. Every important number here is computed twice, by arithmetic that shares nothing, and the two are required to agree.