IQ Percentile Calculator

What an IQ percentile says

A percentile answers a ranking question, not a score question: it is the share of people in the comparison group who scored at or below a given result. An IQ at the 90th percentile beats 90 percent of the group and trails the other 10 percent. The score itself, the number on the report, only becomes meaningful once it is placed against that group.

Percentiles compress a wide range of scores into 1 to 99, which makes them easy to read and easy to compare across tests. They also hide distance: the gap between the 50th and 60th percentile is a few IQ points, while the gap between the 98th and 99th is about 7 points on the standard scale. That asymmetry comes from the shape of the score distribution, and it is the single most useful thing to understand before reading any table on this page.

The calculator above works in both directions. Enter a score to get its percentile, the z score behind it, the share of people above it and the rarity as 1 in N. Switch the mode to enter a percentile and read the IQ that matches it.

The curve underneath: mean 100, SD 15

Modern IQ tests are built so that the population mean is 100 and the standard deviation is 15. Standard deviation is the unit that turns a raw difference into a comparable distance: a score of 115 is one standard deviation above the mean, 130 is two, and 85 is one below. Because scores pile up near the middle and thin out at the edges, those equal steps in points are not equal steps in population share.

The bands explain why a score of 130 feels rare and a score of 115 does not. Roughly two thirds of people sit within one standard deviation of the mean, so 115 is unremarkable in a school hallway. Only about one person in 44 clears 130, and about one in 741 clears 145.

These shares are properties of the model, not of any individual test session. A test estimates where a person sits on the curve; the curve itself is a convention that test makers agree on so scores from different instruments can be read on one scale.

The math, one score at a time

Two steps turn a score into a percentile. First convert the score to a z score, which counts standard deviations from the mean: z = (score - 100) / 15. Then feed z to the normal cumulative distribution, the function that returns the share of the curve below that point. The function has no closed form in elementary terms, so tables and software use the error function: the share below z is 0.5 x (1 + erf(z / 1.4142)).

Worked example with a score of 130. The z score is (130 - 100) / 15 = 2. The error function at 2 / 1.4142 = 1.4142 is 0.9545, so the share below is 0.5 x (1 + 0.9545) = 0.9772, or the 97.7th percentile. The share above is 1 - 0.9772 = 0.0228, which is 2.28 percent, and 1 / 0.0228 = 43.9, so about one person in 44 scores higher.

A second example below the mean: a score of 85. The z score is (85 - 100) / 15 = -1, and symmetry gives a share below of 0.1587, the 15.9th percentile. About 84 percent of people score higher, so this is roughly one person in 6.3 from the bottom, or one in 1.2 from the top, which is why rarity language only makes sense for the upper tail.

The NIST Digital Library of Mathematical Functions section on the error function carries the exact definition and series, and any statistics text with a normal table produces the same numbers to four decimals.

Percentiles for the scores people ask about

The table below uses mean 100 and standard deviation 15. Percentile means the share at or below the score, and the last column is the share strictly above it.

Read the top and bottom rows together and the compression becomes visible. Ten points from 120 to 130 moves a person from the 91st percentile to the 98th, a jump of 7 points of population. Ten more points to 140 adds less than 2 points of percentile, and five more to 145 adds a quarter of one. Above 130 each IQ point buys a smaller share of the population than the point before it.

Rarity is the same fact written as a headcount. At 130 the share above is 2.3 percent, so 1 / 0.023 = 43.5 and the answer lands near 1 in 44. At 145 the share above is 0.13 percent, so 1 / 0.0013 = 769, close to the exact 1 in 741 once the unrounded 0.13499 percent is used. Small rounding in the percentage moves the 1 in N figure by tens at the tail, which is why the calculator keeps full precision and rounds only for display.

Different tests, different numbers

An IQ of 130 is not a fixed quantity across instruments. The number depends on the standard deviation the test publisher chose, so the same underlying performance prints as a different score on each scale. The percentile, which is scale free, stays put.

Every row describes the same rank: two standard deviations above the mean, roughly the top 2 percent. The Wechsler 130, the Stanford-Binet 132 and the Cattell 148 are the same percentile wearing three different numbers. Mensa accepts any of them, which is why the society publishes a conversion table rather than a single cutoff.

When someone quotes an IQ without naming the test, the number is incomplete. A 140 on a scale with a standard deviation of 24 is a far more common result than a 140 on a scale of 15. Convert to a percentile before comparing two scores, or ask which instrument produced them.

Reading it backward, from a percentile to an IQ

The inverse direction is the same curve read from the other axis. The percentile fixes the area under the curve, and the score that cuts off that area is the answer. A few landmarks cover most of the questions people ask:

Note that the 98th percentile lands at 131 rather than the round 130, because 130 is the 97.7th percentile exactly. The difference is small in points and large in the way people talk: a 130 score is usually described as the 98th percentile, and a true 98th percentile score is a touch above it.

The upper tail compresses fast in the other direction too. Going from the 99th to the 99.9th percentile costs about 11 IQ points, from 135 to 146, while going from the 50th to the 60th costs about 4 points. The calculator's percentile mode does this conversion for any input, including values like 99.5 that fall between the landmarks.

What the percentile does not tell you

A percentile is a comparison, and comparisons carry limits worth stating plainly. Every test has measurement error: a score of 130 on a Wechsler scale usually comes with a confidence interval of roughly 3 to 4 points, so the true value sits somewhere around 127 to 133. In percentile terms that band runs from about the 96th to the 99th, which is a wide spread for one number on a report.

Scores also drift. Performance on the same test on a different day moves a few points with sleep, practice and mood, and the Flynn effect means the raw difficulty of a test shifts across decades, which is why norms are restandardized. A percentile is only valid against the norm group and the year the test was normed.

Finally, an IQ percentile is not a measure of knowledge, effort or worth. It ranks performance on a set of reasoning tasks against a sample of people the same age. The OpenStax introduction to the standard normal distribution covers the percentile logic in a general setting if you want the statistics without the testing context, and the same cautions apply to any score read off a normal curve.

Used with those limits in mind, the percentile is a compact and honest summary: it says where a score falls in a crowd, and it says nothing more. For decisions that matter, a full assessment with a qualified examiner carries far more information than any single number.