Standard deviation in IQ

Each step out from the middle is rarer than the last. On the Wechsler scale a score of 115 is one SD above average. About 16% of people score higher. A score of 130 is two SD above, and about 2.3% score higher, or 1 person in 44. A score of 145 is three SD above, and about 1 in 741 score higher.

The same steps work below the mean. A score of 85 is one SD below, and 70 is two SD below. The IQ classifications are built from these cut points.

Different test authors picked different spreads. David Wechsler introduced the deviation IQ in 1939 and his scales settled on 15. The Stanford-Binet of 1960 to 1986 used 16, and the fifth edition of 2003 moved to 15. Raymond Cattell built a standard deviation of 24 into his Culture Fair tests.

The result is that the same rank gets different numbers. Two standard deviations above average is 130 on the Wechsler, 132 on the older Stanford-Binet and 148 on the Cattell. All three are the top 2.3%, which is why American Mensa lists those three numbers for its cut-off. See the Mensa eligibility checker for the full table.

It also means a Cattell 148 is not "higher" than a Wechsler 130. The scales are rulers with different units. If you quote a score, name the test: our pages on the Stanford-Binet and Cattell tests show the history of each scale.

Use the z-score. It counts how many standard deviations a score sits from the mean: z = (score - 100) / SD. Then turn the z-score into a score on the target scale: new score = 100 + z x new SD.

Example 1. A Cattell score of 148 has z = (148 - 100) / 24 = 2. On the Wechsler scale that is 100 + 2 x 15 = 130.

Example 2. A score of 120 on the older Stanford-Binet has z = 20 / 16 = 1.25. On the Wechsler scale it is 100 + 1.25 x 15 = 118.75, so about 119. That is the 89th percentile.

Example 3. A Wechsler score of 118 has z = 18 / 15 = 1.2. About 88.5% of people score below it. On a scale with SD 24 the same z-score gives 100 + 1.2 x 24 = 128.8, about 129.

The percentile to IQ converter does the same job in the other direction, from a percentile to a score.

The two are easy to confuse. The standard deviation describes how far apart different people are. The standard error of measurement (SEM) describes how far one person's observed score might be from their true score on a given test. Reliability links them: SEM = SD x the square root of (1 - reliability).

Worked example. The WAIS-5 Full Scale IQ is reported with a reliability of .97. The SEM is 15 x the square root of 0.03, which is about 2.6 points. A 95% band is roughly twice that, about 5 points either side. So a reported 120 is best read as about 115 to 125. A test with a reliability of .90 would have an SEM of 4.7 and a 95% band of about 9 points either side.

This is why short online tests, which tend to have lower reliability, should not be read to the nearest point. The IQ confidence interval calculator does the sum for any score and reliability.

The normal curve is a model. Real norm samples are finite. The WAIS-5, for example, was standardised on about 2,000 people. In a sample of that size roughly three people would be expected to land beyond +3 SD, so a score of 160 rests on extrapolation, not on a count of people who scored that high. Treat scores far from 100 as approximate.

Every score also has a margin of error. On a full professional test the 95% band is a few points either side, which is about a third of a standard deviation. Two people scoring 128 and 131 have not been shown to differ. The IQ test validity guide covers reliability and error in detail.

To see a score as a percentile, use the IQ percentile calculator. To see how rare it is, use the IQ rarity calculator. For the verbal labels attached to each band, see what a good IQ is.