Concept

Student's t — where it appears

The distribution of a sample mean's departure from the population mean divided by the sample's own estimated standard error. It is wider than a normal at every degrees of freedom, and it is exact only because the mean and the variance of a normal sample are independent.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Which side a 95% t interval misses on, exponential source. Both tails should be 2.5%. At 8 observations the interval falls short of the mean on 9.75% of samples and overshoots on 0.31%. At 500 they are 3.31% and 2.05%, and the total is 5.36% — which a coverage table reports as very nearly right.

Where the two tails disagree

A 95% t interval on an exponential source at 120 observations covers 94.81%, which reads as very nearly right. It misses below the mean on 4.08% of samples and above on 1.11% — one tail 63% too heavy and the other 56% too light, and the total is the statistic that hides it.

tails · Student

The pooled two-sample test's size, with a true null everywhere. Forty units split between two groups, with the second group's variance a stated multiple of the first's, and the two population means equal. A 5% test should reject 5% of the time. The pooled test runs from 0.55% to 18.91% across this region; Welch's runs from 4.63% to 5.51%.

A degrees of freedom that is not a count

The pooled two-sample test's size runs from 0.55% to 18.91% across forty units split five ways against five variance ratios, with a true null in every cell. Welch's runs from 4.63% to 5.51% — bought with a degrees of freedom that is a function of the data, not an integer, and not a count of anything.

tails · Student

What the plot says, and what the interval does, at n = 40. For each source: how often a quantile plot of the data leaves its pointwise band, and how often the 95% t interval for the mean misses. The two-lump source leaves the band on 100% of samples and its interval covers 94.80%; the t on three degrees of freedom leaves it on 57% and covers 95.73%, the best of the five.

The plot is about the wrong quantity

A t interval needs the sampling distribution of the mean to be normal, not the data. A two-lump source leaves its quantile band on 100% of samples of forty and its interval covers 94.80%; a t on three degrees of freedom leaves it on 57% and covers 95.73%, the best of five sources.

lineup · Qq

Four 95% intervals for the mean of 30 exponential observations, split by the side they miss on. Both bars should read 2.5%. The t interval misses below the mean on 6.38% of samples. Widened until its total is exactly 5%, it misses below on 4.69% and above on 0.30%. Hall's transformation misses below on 3.31% and above on 1.96%.

The side a bound is read from

On thirty exponential observations the upper limit of a 95% t interval is exceeded by the true mean 6.38% of the time, against the 2.5% a safety margin set from it assumes. Widen the interval until its total coverage is exactly 95% and the upper limit is still exceeded 4.69% of the time. A symmetric repair fixes the number that is reported and not the one that is used; Hall's transformation, which bends the interval, takes the same rate to 3.31%.

tails · Student

Student's t on 5 degrees of freedom, against the normal. The two-sided 95% critical value is 2.571 for t(5) and 1.960 for the normal — 31% wider. Using the normal at this sample size makes every interval too short by that much.

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.

intervals · Student

Named alongside it

The objects these essays reach for when they reach for this one.

Coverage Estimated variance Sample size Skewness Normality Central limit theorem Conditional coverage Degrees of freedom Behrens–Fisher Error rate Model diagnostics Q–Q plot

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