IQ Standard Deviation 15 vs 16: Why One Percentile Has Three Numbers

Two people compare IQ scores. One says 130, the other says 132, and a third says 148. All three can be describing exactly the same standing in exactly the same population. Nobody has exaggerated and nothing has gone wrong. They have read one position off three different rulers, and none of them mentioned which ruler they used. That single omission is the most common source of confused IQ numbers on the internet, and it is entirely fixable with two lines of arithmetic.

The standard deviation is the ruler

An IQ score is not a count of anything. Nobody holds 130 units of reasoning the way they hold 130 pounds or 130 pence. What a modern test produces is a ranking. The test is given to a large sample chosen to represent a population, everyone's raw performance is put in order, and each person's position in that order is then printed on a scale the publisher settled on in advance. Building that reference sample and fixing that scale is called norming, and it is the step that turns a pile of right and wrong answers into a number anyone can interpret. Our explainer on how IQ tests work covers the rest of that pipeline.

A scale of this kind needs only two settings. The first is where the middle sits, and on essentially every modern test it is 100. The second is how widely the printed numbers are spread out, and that is fixed by the standard deviation — a measure of the typical distance between one individual result and the average, expressed in the same units the results are printed in. Choose 100, choose a standard deviation, and every other number on the scale is decided for you.

So a score of 115 where the standard deviation is 15 sits exactly one standard deviation above average. So does 116 where the standard deviation is 16, and 124 where it is 24. Those are three printings of one position, in the way that 20 degrees Celsius and 68 degrees Fahrenheit are two printings of one temperature. The position itself has a name that survives the translation: the percentile, meaning the share of the reference sample scoring at or below that point. One standard deviation above the mean is the 84.1st percentile whichever of the three scales prints it; only the printed number changes.

Which scale a test uses is a publishing decision, and the field never fully converged on one. The spread below is not exotic history — every one of these scales has scores in circulation today, and readers routinely compare numbers across them without noticing.

• Standard deviation 15 — the Wechsler scales, meaning the WAIS for adults and the WISC for children, plus the great majority of instruments in current professional use. The Stanford-Binet Fifth Edition also reports on 15, having moved there from the 16 its predecessors used. If a modern report names no scale, this is the likeliest one.
• Standard deviation 16 — Stanford-Binet forms up to the fourth edition, including the Form L-M that is the source of a good many of the very high childhood figures still quoted in profiles of gifted children. Several group-administered school ability tests use 16 as well, among them the Otis-Lennon School Ability Index and the standard age score on the Cognitive Abilities Test.
• Standard deviation 24 — the Cattell scales, covering the Culture Fair series and the Cattell III B long used by British Mensa for its supervised test. Numbers here look dramatically inflated against everything else for the same standing, and they are not — the Cattell scales simply spread their printed numbers over a wider range, so every distance from 100 is stretched.
• Mean 10, standard deviation 3 — the scaled scores for individual subtests inside a Wechsler test. A 13 there is one standard deviation above average, the same standing as 115 on the composite scale. Two different scales appear inside a single report.
• No fixed standard deviation at all — the obsolete ratio IQ, mental age divided by chronological age and multiplied by 100. Its spread changes with age, which is one of the reasons it was abandoned.

The conversion, in two steps

Because the scales differ only in their spread, translating between them is a matter of stripping the scale out of a score and putting a different one back. Step one: subtract 100, then divide by the standard deviation the score was measured on. What comes out is called a z score — simply the number of standard deviations the result sits above or below average, with no scale attached to it. Step two: multiply that z score by the standard deviation of the scale you want, and add 100 back on.

Four worked examples, using the numbers people actually argue about:

• 132 on a standard-deviation-16 scale. (132 − 100) ÷ 16 = 2.0. Then 100 + (2.0 × 15) = 130 on a standard-deviation-15 scale.
• 148 on the standard-deviation-24 Cattell scale. (148 − 100) ÷ 24 = 2.0. Then 100 + (2.0 × 15) = 130 again. The three numbers in the opening paragraph were never in disagreement; they are one result written three ways.
• 120 on a standard-deviation-15 scale. (120 − 100) ÷ 15 = 1.33, which prints as roughly 121 where the standard deviation is 16 and roughly 132 on the standard-deviation-24 scale.
• 120 on the standard-deviation-24 scale. That is 0.83 of a standard deviation, which converts to exactly 112.5 on a standard-deviation-15 scale — the 79.8th percentile rather than the 90.9th. Two identical printed numbers, two clearly different standings.

If the z score is a step too many, there is a shortcut that does the same job: multiply the distance from 100 by the ratio of the two standard deviations. Going from 16 to 15 means multiplying by 15 ÷ 16, or 0.9375. Going from 24 to 15 means multiplying by 15 ÷ 24, or 0.625. A 32-point advantage on the standard-deviation-16 scale becomes a 30-point advantage on 15; a 48-point advantage on the Cattell scale becomes the same 30. Doing it by hand is a useful exercise once, after which the IQ score converter will do it faster and without arithmetic slips.

Run a percentile through the same machinery and the reverse trick works. The 98th percentile sits about 2.05 standard deviations above the mean, which is roughly 131 on a standard-deviation-15 scale, roughly 133 on 16 and roughly 149 on 24. That is the whole reason the Mensa entry requirement circulates as three incompatible numbers when it has only ever been one rule. Exactly two standard deviations, meanwhile, is the 97.72nd percentile rather than the 98th, and prints as 130, 132 and 148 on the same three scales — which is where the popular figures come from and why they are all a fraction low.

The most damaging case is the one that needs no conversion at all, because the same number appears on both scales and means something different on each. Take a reported 140. On a standard-deviation-15 scale that is 2.67 standard deviations above average, the 99.6th percentile, about one person in 260. On the standard-deviation-24 Cattell scale the identical figure is 1.67 standard deviations up, the 95.2nd percentile, about one person in 21. Same printed number, and a twelve-fold difference in how unusual it is. You can watch both positions on the same curve using the bell curve visualiser, and get the exact figure for any score with the percentile calculator.

Where the missing scale actually costs something

This is not a curiosity for pedants. Almost every recurring dispute about IQ numbers online reduces to two people quoting different rulers at each other, and a few of the failures are expensive.

• Comparing yourself to a published figure. A childhood score from a Stanford-Binet Form L-M is on the standard-deviation-16 scale; your adult Wechsler result is on 15. Setting them side by side without converting overstates the older number by roughly a sixteenth of its distance from 100.
• High-IQ society thresholds. Societies state entry as a percentile and then publish a qualifying figure per accepted test. Reading one society's Cattell figure as though it were a Wechsler score makes the requirement look far stricter than it is.
• Historical and celebrity claims. Numbers attached to famous names are frequently retellings of a childhood estimate, on an unstated scale, sometimes from the ratio formula that has no fixed standard deviation at all. There is often nothing to convert.
• Online tests that print a number and nothing else. If a result arrives without a named scale and a stated reference sample, no conversion is possible, because the first input to the arithmetic is missing.
• Small differences that are not differences. Even within one scale, a reported score is an estimate. Test manuals publish a standard error of measurement — an estimate of how far a reported score bounces around a person's true standing between sittings — and on a full-scale score it is typically around two to three points, which puts a 95 per cent range several points either side of the figure reported. A converted 130 and a directly measured 132, both on the standard-deviation-15 scale, are not distinguishable results.

That last point is the one most often lost in a conversion argument. Getting the scale right moves a number by two or three points; the confidence range around the number is usually wider than that. Both facts matter, and they matter in that order — you cannot reason about the error bar until you know what scale the number is printed on. Our piece on what a single reported score does and does not carry takes that second half further, and how accurate IQ tests are sets out where the error comes from.

What to ask when a number arrives without one

When someone quotes a score, three questions recover everything the number left out, and if any of them cannot be answered the figure should be treated as decoration rather than data.

• Which test, and what standard deviation does it report on? The test name usually settles it, and any properly written report states the scale explicitly.
• Measured against whom, and when? A percentile is a position in a particular reference sample. An old sample no longer describes the current population, so a percentile computed against it has quietly drifted.
• What was the range around it? A single figure quoted with no confidence interval — the range within which the true score most likely sits — is one day's best guess presented as a certainty.

Answer the first and the conversion is mechanical. Answer all three and you have something you can actually compare with another person's result — which is, in the end, the only reason anyone wants a converted number. If you want a current score to run through this yourself, take a properly built assessment under proper conditions on our IQ test and keep the scale, the percentile and the range together with the number.

The habit worth taking away is small and it costs nothing. When a number arrives, ask what it was measured on before you ask what it means. A score with its scale attached is a precise, checkable statement about where somebody stood in a named sample on a named day. The same score with the scale stripped off is not a weaker version of that statement. It is not a statement at all.

Common questions