How is IQ calculated?

Modern IQ tests calculate your score by comparing your performance with a large reference group of people your age. Your result is converted to a z-score and then to the IQ scale with the formula IQ = 100 + 15 × z. This is called the deviation IQ.

The original formula: ratio IQ

The term IQ, intelligence quotient, comes from an early formula proposed by psychologist William Stern in 1912 and popularized by Lewis Terman's Stanford–Binet test:

A 10-year-old who performed like a typical 12-year-old would score 12 ÷ 10 × 100 = 120. This works for children, but breaks down for adults: reasoning ability stops rising in early adulthood, so the ratio would make everyone's IQ fall each year. Ratio IQ is no longer used for adults.

The modern formula: deviation IQ

David Wechsler replaced the ratio with a comparison to people of the same age, starting with the Wechsler–Bellevue scale in 1939. Every major test now uses this method.

1. Score the test. Count correct answers to get a raw score.
2. Compare with the norm group. The test publisher has given the test to a large, representative sample of each age group and recorded the mean and standard deviation.
3. Compute a z-score. z = (raw score − norm mean) ÷ norm SD
4. Convert to the IQ scale. IQ = 100 + 15 × z

Worked example

You answer 46 items correctly. People your age average 40 with a standard deviation of 8.

An IQ of 111 is higher than about 77% of your age group.

Why some tests use 16 or 24 instead of 15

The mean of 100 is standard, but not every test uses a standard deviation of 15. Older Stanford–Binet editions used 16, and the Cattell Culture Fair test is often reported with 24. The same performance gets a different number on each scale. The 98th percentile is:

Always check which scale a score uses before comparing two results.

Norming and the Flynn effect

Scores only mean something relative to the norm group. Average raw scores on IQ tests rose by roughly 3 points per decade through much of the 20th century, a pattern called the Flynn effect. That is why publishers re-norm their tests every 10 to 20 years; an outdated norm makes everyone look smarter than they are.

Item response theory: weighting hard questions

Counting correct answers treats every question as equal. Item response theory (IRT) gives each question its own difficulty and estimates the ability level that best explains your full pattern of answers. Getting a hard question right moves your estimate more than getting an easy one right. Computer-based tests such as the GRE use IRT.

The Lattice Reasoning Test uses a three-parameter IRT model that also allows for lucky guesses. It estimates your ability θ on a z-score scale and reports IQ = 100 + 15 × θ, along with a 95% confidence interval.

Related: what is a good IQ score? and IQ test practice questions.