IQ Percentile Calculator

What an IQ score is

An intelligence quotient (IQ) is a standardised score from a test of reasoning and problem solving. Early tests literally divided a child's mental age by their chronological age and multiplied by 100, which is where the word quotient comes from. Modern tests no longer do this. They compare your performance with that of a large, representative sample of people your age and report how far above or below the average you fall.

Scores are rescaled so that the average is 100. The standard deviation (SD) describes how widely scores spread around that average, and for most current tests it is 15. That convention is what lets a number like 130 mean the same thing across tests: it is two standard deviations above the mean.

A single score is a statistical summary, not a diagnosis or a measure of a person's worth. This calculator describes where a score sits on a normal curve. For an assessment that matters, consult a qualified psychologist, and see our disclaimer.

How the calculator works

The tool assumes scores follow a normal distribution with a mean of 100 and the standard deviation you enter. First it computes the z-score, then the share of the distribution lying below that point:

z = (IQ − 100) ÷ SD

percentile = Φ(z) × 100Φ is the cumulative distribution function of the standard normal curve.

The result has three lines. The percentile says what percentage of the population scores at or below you. The second line shows the share of people on the other side of your score, labelled top when you are above 100 and bottom when you are at or below it. The rarity is the reciprocal of that share, rounded to a whole number, so a tail of 2.275% becomes roughly 1 in 44.

Worked example: an IQ of 130 on a test with SD 15 has z = (130 − 100) ÷ 15 = 2. About 97.72% of people score lower, so the percentile is the 97.72nd. The tail above is 100 − 97.725 = 2.2750%, and 1 ÷ 0.02275 ≈ 44, so such a score is roughly 1 in 44. With an SD of 16, the same 130 is only 1.875 standard deviations out, which is a lower percentile.

The normal curve is evaluated through the complementary error function, which keeps the tail accurate to about 15 significant digits even far from the mean. The percentage line shows four decimals and the percentile two. The 1-in-N figure is written out in full up to about one in a trillion; beyond that only its first twelve digits are meaningful, so the rest are printed as zeros. Once a score lies more than about 37.5 standard deviations above the mean (an IQ near 660 on a 15-point scale) the tail is smaller than the smallest number a computer can store, and the rarity reads 1 in ∞. Real populations are not perfectly normal in the far tails either, so treat such rarities as illustrative.

IQ percentiles and rarity (SD 15)

A useful memory aid is the 68-95-99.7 rule. About 68% of people score within one SD of the mean (85 to 115), about 95% within two (70 to 130) and about 99.7% within three (55 to 145). For other normal-curve questions, see the normal distribution calculator, the z-score calculator and the z-score to p-value calculator.

How scores are described

Test publishers attach descriptive labels to score ranges. The names differ between publishers and between editions of the same test, so the labels below are a common pattern rather than a standard. The shares assume a normal curve with SD 15.

Types of IQ tests and why the SD matters

Clinical psychologists mostly use individually administered tests such as the Wechsler scales (WAIS for adults, WISC for children) and the Stanford-Binet. Other widely used instruments include the Woodcock-Johnson tests of cognitive abilities and nonverbal reasoning tests such as Raven's Progressive Matrices. Online quizzes are usually unstandardised and their results should not be read as real IQ scores.

Most current tests set the SD to 15, but older editions of the Stanford-Binet used 16 and some historic scales used 24. A score of 132 means two SDs above the mean on a test with SD 16, yet only about 2.1 SDs on SD 15. This is why the calculator asks for the standard deviation: the same number can represent different percentiles on different scales. If the standard deviation of your test is not stated, 15 is the safest assumption.

How reliable is an IQ score?

• Measurement error. Full-scale scores on major tests are quite consistent from one sitting to the next, but every score has a margin of error of a few points, and published confidence intervals are commonly about five to seven points either side.
• Age. Scores of young children are less stable than those of adults, and a child's score can change noticeably as they develop.
• Norms age. Average raw performance rose through the twentieth century, a pattern called the Flynn effect, so publishers re-norm their tests periodically. A score from an old edition may not match the same person's result on a new one.
• Circumstances. Tiredness, illness, motivation and test anxiety all move a result by a few points.

For these reasons, differences of a handful of points between two people, or between two attempts, carry little meaning. Read a score as a range rather than a precise value.

Can you raise your IQ?

Intelligence test performance is shaped by both genes and environment, and the research on how much it can be changed is mixed. Studies of schooling suggest that each extra year of education is associated with a gain of a few IQ points. Good nutrition and health in childhood, adequate sleep, and avoiding exposure to toxins such as lead also help children reach their potential. Practising a particular test raises scores on that test, but that largely reflects familiarity.

Commercial brain-training programs have so far shown limited evidence of improving general reasoning beyond the trained tasks. Learning new skills, reading widely and staying mentally active are worthwhile for their own sake, even if they do not lift a number.