How Is IQ Calculated? The Deviation IQ Formula Explained

Updated 24 September 2026. About 11 minutes to read.

The IQ formula in one line

Every modern intelligence test, from the Wechsler scales to the Stanford-Binet, uses the same basic calculation. It is called a deviation IQ, because it measures how far your score deviates from the average.

The part in brackets, divided by the standard deviation, is called a z-score. It simply counts how many standard deviations you are from the middle. A z-score of 0 is exactly average, +1 is one step above and −1 is one step below.

The rest of the formula just moves that z-score onto the familiar IQ scale. Multiplying by 15 sets the size of each step, and adding 100 puts the average in the middle. Nothing about the answer changes; it is only relabelled in units people recognise.

A worked example, step by step

Imagine a reasoning test with 60 questions. The publisher gave it to a large sample of 30-year-olds and found that their average raw score was 35, with a standard deviation of 7. You are 30, and you answer 42 questions correctly.

1. Find the gap from average. 42 − 35 = 7 points above the average for your age.
2. Turn it into a z-score. 7 ÷ 7 = 1.0, so you are exactly one standard deviation above average.
3. Convert to the IQ scale. 100 + (15 × 1.0) = 115.
4. Read the percentile. On a normal bell curve, one standard deviation above the mean beats about 84% of people, so your IQ of 115 sits at roughly the 84th percentile.

Now take someone else of the same age who scores 30. Their gap is −5, their z-score is −5 ÷ 7 = −0.71, and their IQ is 100 + (15 × −0.71) ≈ 89. That is around the 24th percentile, well within the normal range.

Where IQ came from: the old ratio formula

The word "quotient" is a clue to how IQ was first calculated. In 1905 the French psychologists Alfred Binet and Théodore Simon published the first practical intelligence test, designed to spot schoolchildren who needed extra help. Their results were expressed as a mental age: the age at which an average child would perform at the same level.

In 1912 the German psychologist William Stern suggested dividing a child's mental age by their actual age. In 1916 Lewis Terman at Stanford University multiplied the result by 100 to remove the decimals, called it the "intelligence quotient" and built it into his new Stanford-Binet test.

For children, this was intuitive. A child exactly on track scored 100, a child ahead scored above 100 and a child behind scored below it.

Why the ratio IQ broke down

The formula has two serious flaws. The first is that mental age stops rising in late adolescence, while real age keeps going up. Under the ratio formula, an average 40-year-old would appear to have half the IQ of an average 20-year-old, which is obviously nonsense.

The second is that the spread of ratio scores differed from one age to the next, so a score of 130 could mean something quite rare at one age and less rare at another. It also explains the extraordinary childhood IQs of 180 or 200 you sometimes read about. They come from the ratio method, where a very young child who is far ahead can post huge numbers, and they cannot be compared with modern scores.

Wechsler's fix: the deviation IQ

The American psychologist David Wechsler solved both problems with his Wechsler-Bellevue scale in 1939. Instead of mental age, he compared each person with a sample of people of the same age, and expressed the result as a standard score with an average of 100 and a standard deviation of 15.

That approach, the formula at the top of this page, is how every major test works today. It kept the familiar number 100 for "average", so the new scores looked recognisable to people used to the old ones, while giving them a meaning that works for adults as well as children.

The choice of 100 and 15 is a convention, not a law of nature. What gives the score meaning is the percentile underneath it.

How a professional test gets from your answers to an IQ

A real assessment such as the WAIS-5 follows the same logic, but in several layers, because it measures several abilities rather than one.

1. Raw scores. Each subtest, such as vocabulary or matrix reasoning, is scored by counting points earned.
2. Scaled scores. Each raw score is converted, using tables for your age band, into a scaled score with an average of 10 and a standard deviation of 3.
3. Sum of scaled scores. The scaled scores for the subtests that make up each index, and the full scale, are added together.
4. Composite scores. Those sums are converted, again using norm tables, into index scores and a Full Scale IQ, each with an average of 100 and a standard deviation of 15.
5. Percentile and confidence range. Each score is reported with a percentile and a 95% confidence interval showing where the true score most likely lies.

Pearson's own published sample WAIS-5 report shows what that looks like in practice: a Full Scale IQ of 111, at the 77th percentile, with a 95% confidence range of 106 to 116. Even the gold standard presents IQ as a range, not a single exact point.

What IQ scores mean on the bell curve

Because IQ is built on the normal distribution, every score maps onto a fixed percentile. About 68% of people score between 85 and 115, about 95% between 70 and 130, and about 99.7% between 55 and 145. For descriptions of each band, see our guide to what counts as a good IQ score.

For a closer look at two of the most searched scores, read what an IQ of 120 means and what an IQ of 130 means.

Same person, different numbers: SD 15, 16 and 24

All IQ scales put the average at 100, but not all use 15 points per standard deviation. The Wechsler tests and the current Stanford-Binet (fifth edition) use 15. Older Stanford-Binet editions used 16, and the Cattell Culture Fair test is often reported on a scale of 24.

That means one person, at one point on the bell curve, can be given very different numbers. Two standard deviations above average works out as:

These are exactly the entry scores Mensa lists for those tests: 130 on a Wechsler scale, 132 on the older Stanford-Binet and 148 on the Cattell. It is also a reason to be wary of eye-catching IQ numbers. A "160" on an SD 24 scale is about 137 on the standard scale. Famous figures like the IQ of 160 attributed to Einstein often travel without any mention of the scale, or of whether a test was ever taken.

To convert any score, find its z-score on the original scale, then multiply by the new standard deviation and add 100. The converter in the calculator above does this for you.

Why age matters so much

Deviation IQ compares you with people your own age, so the same raw performance can produce different IQs at different ages. A 70-year-old and a 25-year-old who answer the same number of questions correctly will not necessarily receive the same IQ, because the average raw score in each age group is different.

That is why professional tests are normed in narrow age bands and why the WAIS-5 covers adults from 16 to 90 with separate tables along the way. An IQ tells you where you stand among your peers, not how many problems you can solve in absolute terms.

Why norms go out of date

The comparison group is fixed at the time a test is standardised. Over the 20th century, average raw scores on intelligence tests rose from one generation to the next, a pattern known as the Flynn effect. A test normed decades ago compares you with a group that would score lower today, which quietly inflates your IQ.

Publishers deal with this by re-standardising their tests roughly once a decade. It is one of several reasons that scores from casual online tests, which often have little or no norm group at all, drift away from reality. We cover those reasons in our guide to whether online IQ tests are accurate.

Frequently asked questions

IQScore tests are for education and entertainment. Scores are estimates and are not a clinical or diagnostic result. If you have concerns about memory, attention or learning, please speak to a qualified professional.