What the top percentile means in centimetres

Under a normal distribution the 99th percentile sits about 2.33 standard deviations above the mean. On the clinician-measured erect figures in Veale et al. (2015), that is roughly 16.99 cm - under 17 cm, not the much larger figures that circulate online as "top percentile".

The phrase gets used loosely, as a synonym for "very large", and doing the arithmetic once is more useful than repeating it.

The z-value

Under a normal distribution, the 99th percentile sits at a z-score of approximately 2.33 - meaning 2.33 standard deviations above the mean. This is a fixed property of the standard normal curve, not something specific to this dataset: the NIST/SEMATECH e-Handbook's standard normal table gives 0.49010 of the area between the mean and z = 2.33, so 0.99010 below it. It says that 99 percent of a normally distributed population falls below that point and 1 percent falls above it.

Applying it once

Veale et al. (2015), BJU International, reports a pooled mean erect length of 13.12 cm with a standard deviation of 1.66 cm, measured by health professionals on 692 men - the erect row of a pool that reached 15,521 men for other measurements.

99th percentile = mean + (2.33 × SD) = 13.12 + (2.33 × 1.66) = 13.12 + 3.87 = approximately 16.99 cm

That is where the normal approximation puts the top one percent of the erect length distribution. The full table of Veale means and spreads across states carries the flaccid, stretched and erect figures this calculation could equally be run against, with the same method each time.

Two caveats that matter here

The normal approximation is least trustworthy exactly where this calculation is asking it to work hardest. Veale's own nomograms share the assumption rather than escaping it: the authors built them by simulating 20,000 observations from the normal distribution using each pooled mean and SD. Why the tails of a normal distribution are thin is worth reading for the general shape of the problem; this post is only doing the one specific sum for the 99th percentile and not repeating that explanation. Second, this is a population statistic about where measured men sit relative to each other, not a statement about the precision of any single reading near that point - the studies' own measurement error does not vanish at the extreme end, and if anything a reading that far from the mean deserves more scrutiny of its method, not less.

Running the same arithmetic on girth

The formula does not care which row of the table it is applied to. Erect girth, mean 11.66 cm and SD 1.10 cm in the same pooled data, gives a 99th-percentile figure of 11.66 + (2.33 × 1.10) = 11.66 + 2.56 = approximately 14.22 cm. Notice how much closer that sits to the mean, in relative terms, than the length figure does - girth's smaller standard deviation compresses the whole distribution, top percentile included, which is the same reason a small girth reading error swings a percentile further than the same error in length would. The arithmetic is identical in both cases; only the mean and SD that go into it change.

Other percentiles, same method

The same z-value table that gives 2.33 for the 99th percentile gives other fixed multipliers for other thresholds: about 1.28 for the 90th percentile, about 1.64 for the 95th, about 2.58 for the 99.5th. Multiply whichever value applies by the SD of the row being examined, add it to that row's mean, and the result is that threshold in centimetres. There is nothing special about the 99th percentile in this respect - it gets asked about more often because it functions as informal shorthand for "extreme," not because the arithmetic behind it differs from any other point on the curve.

What this arithmetic is not doing

It is not adjudicating anyone's specific claim about themselves or anyone else. It is one calculation, from one published mean and standard deviation, under one standard assumption, producing one number. Change the input distribution - self-reported figures rather than clinician-measured ones, for instance - and the same arithmetic produces a different, larger, less trustworthy answer, which is one more reason self-report inflation matters upstream of any percentile claim rather than only at the level of the mean.

This kind of arithmetic has nothing to do with a subjective rating, and the two should not be substituted for each other. A percentile from Veale describes a position among clinician-measured centimetres; a score on Rate Cock describes an impression of a photograph, assessed by a rating system built for that purpose or by a person, and an AI estimate from an image is answering neither question with the precision this arithmetic assumes, for reasons that are geometric rather than statistical.