How is an IQ score calculated? Mean, standard deviation and percentiles

If you answer 22 of 30 puzzles correctly, what is your IQ? The honest answer is: it depends on everyone else. An IQ-style score is not a count of correct answers. It describes where a result sits compared with a reference group. Understanding that one idea makes most of the mystery around IQ numbers disappear.

From “mental age” to a comparison with others

The first intelligence tests, developed in France at the beginning of the 20th century, were meant to find children who needed extra help at school. Soon after, the idea of a “quotient” appeared: a child’s “mental age” divided by their actual age, multiplied by 100. A ten-year-old who performed like a typical twelve-year-old would get 120.

That ratio works poorly for adults – performance on most reasoning tasks does not keep rising year after year – so modern tests use a different approach. They compare your result with the results of a large reference group, often called the norm group, and express the comparison on a fixed scale. This is sometimes called the deviation IQ.

The two numbers behind the scale: 100 and 15

The scale is defined by two numbers:

• the mean (average) is set to 100;
• the standard deviation, a measure of how spread out the results are, is set to 15.

The conversion itself is simple. First you calculate how far a raw result is from the reference group’s average, measured in standard deviations. Statisticians call this a z-score:

z = (your raw score − average raw score of the group) ÷ standard deviation of the group

Then you put it on the IQ scale: score = 100 + 15 × z.

An example with made-up numbers: suppose the reference group solved 20 of 30 puzzles on average, with a standard deviation of 4. If you solved 24, you are one standard deviation above the average (z = 1), which gives 100 + 15 = 115. If you solved 16, you are one standard deviation below (z = −1): 85.

From score to percentile

Because results on such tests are roughly bell-shaped (a normal distribution), each score corresponds to a percentile – the share of the reference group that scored lower.

Score	Approximate percentile
70	2
85	16
100	50
115	84
130	98
145	99.9

Put differently: about 68 % of people score between 85 and 115, and about 95 % between 70 and 130. A percentile of 84 does not mean you got 84 % of the questions right; it means you did better than about 84 of 100 people in the reference group.

Why a single number promises too much

No test measures perfectly. Your result on a given day depends on concentration, sleep, familiarity with the puzzle format and simple luck on a few questions. Psychologists describe this with the standard error of measurement: the amount a score would typically vary if you could take an equivalent test again.

That is why careful reports give a range – for example 112–122 – rather than pretending that the last digit is meaningful. A difference of a few points between two people, or between two attempts, often says more about measurement error than about ability. The shorter the test, the wider that range should be.

Reference groups get old

A score is only as good as the group it is compared with. During the 20th century, average performance on many intelligence tests rose noticeably from one generation to the next – the so-called Flynn effect, documented across 14 nations in a well-known 1987 paper. A test normed decades ago would therefore make today’s test-takers look better than they are. Serious test makers renew their norms regularly for this reason.

What an IQ score does not tell you

The 1996 report “Intelligence: Knowns and unknowns”, written by a task force of the American Psychological Association, summarised what scores do and do not capture. Scores on such tests are related to outcomes like school performance, but they leave out much of what people mean by being “smart” – creativity, wisdom, practical sense and social sensitivity, among others.

So a score is a summary of performance on one kind of task, compared with one group of people, measured with some error. It is not a measure of a person’s worth, and it is not a diagnosis.

How IQ Arion applies this

In the IQ Arion app, the same logic applies with honest limits:

• Your answers are turned into an estimated score on the scale with mean 100 and standard deviation 15 by a statistical model (item response theory); the same answers always give the same score.
• Until about 3,000 tests have been completed, the norms are preliminary and scores are marked “beta”; when calibration ends, these results are recalculated once and certificates are updated free of charge.
• The unlocked result shows an estimated score with its 90% range – about ±9 points during the beta. Scores are shown between 55 and 160; beyond that, the result reads “≤ 55” or “≥ 160”. It is an estimate, not a clinical assessment or a diagnosis.

You can read the details on our page about how scoring works – and if you want to see what kinds of puzzles are involved, the free short test takes about three minutes.