The IQ formula: how an IQ score is calculated

Short answer: Most IQ tests today use the deviation formula: IQ = 100 + 15 × z. Here z is how far your raw score sits from the mean of people your age, in standard deviations. The result sits on a scale with an average of 100 and a standard deviation of 15 (Voncken and colleagues, 2019). The older ratio formula divided mental age by chronological age, as William Stern proposed in 1912. Lewis Terman (1916) multiplied it by 100: IQ = mental age ÷ chronological age × 100. The APA's 1995 intelligence report (published 1996) says that ratio procedure "is no longer used."

The two IQ formulas

• Deviation IQ, used today: IQ = 100 + 15 × z. In its simplest form, z = (X − M) ÷ SD.
  • X is your raw score, the points you earned on the test.
  • M is the mean raw score of the norm group, the people your age the test was standardized on.
  • SD is the standard deviation of that norm group's raw scores.
• Ratio IQ, historical: IQ = mental age ÷ chronological age × 100.

The z step is the standard score from statistics. David Lane's Online Statistics Education textbook (Rice University, read November 2, 2026) gives it as Z = (X − μ) ÷ σ. Voncken, Albers and Timmerman (2019) describe the second step. The z score "can be transformed to an IQ score by multiplying the z score by the standard deviation of the desired distribution (i.e., 15)", and then adding 100.

A worked example: deviation IQ

The person and every number in this example are invented. Real means and standard deviations come from a test publisher's norm tables.

1. The norm group. An invented test-taker is 34. On an invented test, people in that age band score a mean of 40 raw points, with a standard deviation of 10.
2. The raw score. The test-taker earns 52 raw points.
3. The z-score. z = (52 − 40) ÷ 10 = 1.2. The score sits 1.2 standard deviations above the age group's mean.
4. The IQ. IQ = 100 + 15 × 1.2 = 100 + 18 = 118.
5. The percentile. By the normal curve, 118 sits at about the 88.5th percentile.

The original ratio IQ: mental age and Stern's quotient

The formula began with Alfred Binet's tests for children. Terman's 1916 book says Binet's first scale of 1905 "consisted merely of 30 tests, arranged roughly in order of difficulty." The 1908 revision grouped tasks by age, an "age-grade scale", Terman writes. That grouping produced the mental age, which Stern (1914) defined as the age whose average performance a child's results match.

Mensa International says the term IQ "was coined in 1912 by the psychologist William Stern" (its What is IQ? page, read November 2, 2026). Stern's proposal, in the 1914 English translation, was "the ratio of mental to chronological age". His own example: "An 8-year old child with a mental age of six has, then, a mental quotient of 6/8 = 0.75."

Lewis Terman published the Stanford-Binet in 1916. In The Measurement of Intelligence (1916) he explains: "To find it we simply divide mental age (expressed in years and months) by real age (also expressed in years and months)." On his scale, an IQ of 100 meant exactly average intelligence. Mensa says Terman was "keen to multiply the result by 100". Our average IQ guide explains why the average is always 100.

Two worked ratio examples, by our calculation from the sources' cases:

1. Mensa's example: a 10-year-old with a mental age of 12. IQ = 12 ÷ 10 × 100 = 120.
2. Stern's example (1914 translation): an 8-year-old with a mental age of 6. IQ = 6 ÷ 8 × 100 = 75.

The mental age formula, run backward

Rearrange the ratio formula and you get mental age = IQ × chronological age ÷ 100. By our calculation, a ratio IQ of 75 at age 8 gives 75 × 8 ÷ 100 = 6, Stern's child again.

This only works on a ratio IQ. A modern deviation IQ is not built from a mental age, so the backward step gives an age the test never measured.

Why the ratio IQ fails for adults

Terman (1916) wrote that intelligence "appears to improve but little after the age of 15 or 16 years". So "any person over 16 years of age, however old, is for purposes of calculating I Q considered to be just 16 years old." His example: "If a youth of 18 and a man of 60 years both have a mental age of 12 years, the I Q in each case is 12 ÷ 16, or .75." Terman wrote the quotient both as a decimal and as a whole number, so his .75 is the 75 of the × 100 form.

David Wechsler made the case against this in The Measurement of Adult Intelligence (third edition, 1944). Mental-age equivalents become impossible once "the mean scores made on the test cease to increase with advancing chronological age." He rejected the fixed adult divisor too. Dividing a 60-year-old's mental age by 15, he wrote, "is as incorrect as to obtain an I.Q. for a boy of 12 by dividing his M.A. score by 15."

A second problem hit children. Wechsler (1944) used Burt's data on a child two standard deviations below the mean. A child "at age 6 attaining that rank would get an I.Q. of 76; at age 10, he would get an I.Q. of 81; at age 14, an I.Q. of 84." The same standing gave three different numbers.

Wechsler's deviation IQ in 1939

Wechsler's answer was the Wechsler-Bellevue scale, described in the first edition of his book, April 1939. The Gale Encyclopedia of Psychology says he "introduced the deviation quotient, a new calculation that compared individuals with their peer group".

Wechsler's own words (third edition, 1944) state the idea. "An I.Q. merely tells you how much better or worse, or how much above or below the average any individual is, when compared with persons of his own age." Each age group, in his method, set its own denominator.

His original scale was not stated as a standard deviation of 15. He set an IQ of 90 at one probable error below the mean of 100. One probable error is 0.6745 standard deviations, so 10 points per probable error is 10 ÷ 0.6745, or about 14.8 points per standard deviation, by our calculation. That is close to today's 15.

The Stanford-Binet followed. Denisova and Lin (2022) date the Stanford-Binet's deviation IQs, with a mean of 100 and an SD of 16, to its 1960 third revision. Wikipedia's Stanford-Binet article adds that in that edition "the use of the mental age scale and ratio IQ were not eliminated". Our Stanford-Binet guide and WAIS guide cover both tests today. The APA's 1995 report describes the modern scale as one "in which the mean is 100 and the standard deviation is 15".

How a real test applies the formula

No examiner types your raw score into 100 + 15 × z. Publishers do that work once, on a norm group, and print the results as norm tables by age band.

Traditionally, test developers norm each age group in one of three ways, a 2024 review by Urban and colleagues explains: linear transformation, percentiles or area transformation. A linear transformation uses the age group's mean and SD, the (X − M) ÷ SD above, and keeps the raw scores' shape. Percentiles express your rank in the age group. An area transformation takes that rank and uses the normal curve to find z. Voncken, Albers and Timmerman (2019) describe that route: percentile, then z, then 100 + 15 × z.

Whatever the method, the result reaches the examiner as a table. Even continuously normed tests "usually present classical norm tables in which a norm score is assigned to each raw score", Urban and colleagues write.

On the WAIS-IV, subtest raw scores become scaled scores with a mean of 10 and an SD of 3. The summed scaled scores then become the composite, with a mean of 100 and an SD of 15 (Albayrak Rasborg Hartogsohn and colleagues, 2025). Our Full Scale IQ guide walks through Pearson's sample WAIS-5 report.

The norm group matters. The WAIS-5 was standardized from February 2023 to January 2024 on 2,020 people aged 16 to 90, Winter, Trudel and Kaufman reported in 2024. Norms also age: the APA's 1995 report puts the average gain at "about three IQ points per decade", the Flynn effect.

IQ scales with SD 15, 16 and 24, and how to convert

Not every test uses 15 points. Mensa International says the standard deviation on some tests was defined as "16 or 24 IQ points". To convert a score between scales, go through z:

1. z = (IQ − 100) ÷ the old SD.
2. The new IQ = 100 + the new SD × z.

Worked conversions, by our calculation:

• An IQ of 140 on an SD 24 scale: z = 40 ÷ 24 ≈ 1.67. On an SD 15 scale, that is 100 + 15 × 40 ÷ 24 = 125.
• An IQ of 130 on an SD 15 scale: z = 30 ÷ 15 = 2. On an SD 16 scale, that is 100 + 16 × 2 = 132.

Mensa's cut-offs show the recipe at work. American Mensa's qualifying scores page (read November 2, 2026) lists "Stanford Binet IQ 132", "Stanford Binet 5 IQ 130" and 130 Full Scale IQ on the Wechsler scales. The 132 fits the Stanford-Binet editions with an SD of 16, which Denisova and Lin date from the 1960 third revision. All three sit two standard deviations above the mean: (132 − 100) ÷ 16 = (130 − 100) ÷ 15 = 2. Its 148 on the Cattell fits a scale with an SD of 24, so check which scale your report names. Our IQ percentile guide lists more cut-offs, and our Mensa IQ test guide covers how to qualify.

A conversion changes the scale only. It cannot make two different tests, normed on different groups, give the same result. Our Wonderlic to IQ guide shows how that plays out for one workplace test.

How item response theory scoring differs

Item response theory (IRT) estimates an ability score, θ, from your answers. Klein and Kovacs (2024) write that θ "is estimated based on the string of correct and incorrect responses provided by the examinee". Antoniou and colleagues (2022) give the simplest IRT model, the one-parameter logistic model. Your chance of answering an item correctly depends on your ability, θ, and the item's difficulty, b: P = e^(θ − b) ÷ (1 + e^(θ − b)).

On Klein and Kovacs's adaptive vocabulary test, "θ is a standardized z-score". It "compares test-takers to the norming sample and expresses performance in standard deviation units". Some clinical tests report IRT scores under their own names. Farmer and colleagues (2023) list "W-scores on the Woodcock-Johnson" and "Change Sensitive Scores on the Stanford-Binet Intelligence Scale".

When θ is a z-score like this, the last step can look like the deviation formula. If θ has a mean of 0 and an SD of 1 in the norming sample, IQ = 100 + 15 × θ, and a θ of 1.2 would give 118. Each test sets its own mapping, so this shows the idea only.

Why the result is a range

Every test has measurement error, so a score is an estimate. Schmukle (2025) gives the standard error of measurement (SEM) as SD × √(1 − reliability). The traditional 95% interval is the score plus or minus 1.96 × SEM. If a test had the WAIS-5's mean FSIQ reliability of 0.97 (Winter and colleagues, 2024), the SEM would be about 2.6 points. On the traditional interval, the invented 118 would read as 118 plus or minus about 5.1 points at 95%, by our calculation.

Schmukle notes that this traditional interval "has been criticized", because, with imperfect measurement, true scores sit closer to the average than observed scores. He recommends a regression approach with a scale correction instead.

Scores also move between sittings. Winter, Trudel and Kaufman (2024) also report a WAIS-5 stability coefficient of 0.93 for 201 people tested twice, so a retake can land on a different score. Our IQ percentile guide shows how a range carries into percentiles, and how accurate IQ tests are covers other sources of error.

From IQ score to percentile

Your percentile is the share of the normal curve below your z, so it does not depend on the SD a test uses. By the normal curve, z = 2 sits at the 97.7th percentile, whether the report says 130 on an SD 15 scale or 132 on an SD 16 scale. Our IQ bell curve guide shows the whole distribution.

WhatIQ's homepage (read November 2, 2026) describes its test as 24 picture puzzles in about 15 minutes, with IRT scoring. It reports a score on the standard IQ scale with the range it most likely sits in, as an estimate rather than a clinical assessment.

Frequently asked questions

Who gave the IQ formula?

William Stern proposed dividing mental age by chronological age in 1912, and Mensa International credits him with coining the term IQ. Lewis Terman's 1916 Stanford-Binet wrote the quotient on a scale where 100 is average. In 1939, the Wechsler-Bellevue from David Wechsler replaced the ratio with a comparison against people of the same age. That is the deviation IQ that most modern tests use.

How do I calculate my IQ?

You cannot calculate a real IQ from a formula alone, because you need the norm group's mean and standard deviation for your age. Test publishers keep those in norm tables, built on large samples such as the WAIS-5's 2,020 people (Winter and colleagues, 2024). A normed test applies the deviation formula for you. Read its result with its range.

How is IQ calculated for adults?

Adult IQ is a deviation IQ. Your raw score is compared with people in your age band and turned into a z-score through a norm table. That z is placed on a scale with a mean of 100 and an SD of 15 (APA, 1995). Wechsler built the Wechsler-Bellevue of 1939 for adults because mental age stops rising in adulthood, which breaks the ratio formula.

What is the mental age formula?

Under the ratio IQ, mental age = IQ × chronological age ÷ 100. By our calculation, an 8-year-old with a ratio IQ of 75 has a mental age of 6, which is Stern's own example run backward. The forward formula is IQ = mental age ÷ chronological age × 100. Modern deviation IQs do not use a mental age.

Is the ratio IQ still used?

Not for IQ tests in general. The APA's 1995 report says the mental-age procedure "is no longer used." Some developmental research still divides an age equivalent by chronological age: a 2022 Scientific Reports paper by Locatelli and colleagues computed IQ that way. Modern intelligence tests such as the WAIS-5 report deviation scores through norm tables.

Can I convert a ratio IQ to a deviation IQ?

Not from the score alone. Ratio IQs did not have the same spread at every age, so one ratio score meant different standings at different ages. In Wechsler's 1944 example from Burt's data, a child two standard deviations below the mean got 76 at age 6, 81 at age 10 and 84 at age 14. On a deviation scale with an SD of 15, that child would score 70 at every age, by our calculation.