The scale

The IQ scale explained

Two numbers define the whole scale: a mean of 100 and a standard deviation of 15. Everything else — the bands, the percentiles, the rarity figures — falls out of those two choices.

The two numbers that define everything

A deviation IQ scale is set by two parameters. The mean is where the middle of the comparison group sits, and it is placed at 100. The standard deviation is how spread out the scores are, and on most English-language tests it is 15.

Neither is a discovery. They are conventions chosen because they produce readable numbers, and they have the consequence that any score can be converted directly into a percentile without knowing anything else about the test.

The conversion works because the scores are fitted to a normal distribution. A score of 115 is exactly one standard deviation above the mean, which on a normal curve is the 84th percentile. Always, on any test using the same parameters.

What a standard deviation actually does

A standard deviation is a unit of spread. On this scale it is worth 15 points, and the normal curve puts fixed proportions of the population inside each step.

That last row is the reason scores above 160 should be treated with suspicion. To calibrate a score four standard deviations out, you would need a norm sample containing enough people that far out to measure them — and in a sample of ten thousand you would expect roughly one. Scores at the extremes are extrapolations from a curve, not measurements against real people.

Why the same person gets different numbers on different tests

Not every test uses a standard deviation of 15, and the difference produces numbers that look wildly different while meaning the same thing.

This is the single most common source of confusion about high scores. Someone reporting 148 may be at the 98th percentile or the 99.9th depending entirely on which scale they were measured on, and those are very different claims. An IQ figure without its scale is not interpretable.

It is also where the Mensa myth comes from. Mensa's requirement is the 98th percentile, which is about 130 on a Wechsler scale and about 148 on Cattell. The figure of 140 that circulates belongs to neither.

The bands and what they are called

Test publishers attach descriptive labels to ranges. The current Wechsler labels, with the proportion of the population each covers:

The percentages are not survey findings. They are what a normal distribution produces, which means any test reporting a noticeably different distribution is either not normed on a representative sample or is not really using a fifteen-point standard deviation.

The labels themselves have changed repeatedly, and the history is not flattering. We trace it here.

Converting a score to a percentile

This is the conversion that makes the scale useful, because a percentile is a statement people can interpret without knowing any psychometrics.

The full table from 40 to 160, with a calculator for anything in between, is on the IQ percentile chart.

The scale moves, quietly

One property of a deviation scale that almost nobody is told about: the mean is reset every time a test is renormed.

Raw performance on IQ tests rose through the twentieth century at roughly three points per decade. Publishers responded by renorming, which redefines the current average as 100. The rise is therefore invisible in published scores and visible only when an old test is given to a modern sample.

This has a practical consequence. A score measured against 1985 norms is not comparable to one measured against 2015 norms, and the difference can exceed the thresholds used for educational placement. Which edition of a test was used, and when it was normed, is a real question. The full story of the Flynn effect.