IQ Percentile Calculator

What Is an IQ Percentile Calculator?

An IQ percentile calculator converts an intelligence quotient (IQ) score into a percentile rank. The percentile tells you what fraction of the population scores at or below a given IQ. For example, an IQ at the 84th percentile means the score is higher than about 84% of people.

IQ tests are designed so that scores follow a normal (bell-shaped) distribution. By convention the distribution has a mean of 100. The standard deviation depends on the test: most modern scales (such as the Wechsler tests) use a standard deviation of 15, while the older Stanford–Binet scale uses 16.

How Does the Calculator Work?

The calculator assumes IQ scores are normally distributed with a mean of 100 and a standard deviation you choose (15 or 16). It first converts the IQ score into a standard score, or z-score, which measures how many standard deviations the score is from the mean. It then applies the standard normal cumulative distribution function (CDF), written $\Phi$, to find the proportion of the population below that z-score.

Formulas

The z-score is:

The percentile is the standard normal CDF of the z-score, expressed as a percentage:

Where:

• IQ is the score you enter.
• $\mu$ is the mean, fixed at 100.
• $\sigma$ is the standard deviation (15 or 16).
• $\Phi(z)$ is the probability that a standard normal variable is less than or equal to $z$.

The calculator evaluates $\Phi(z)$ with the Abramowitz–Stegun approximation of the error function, which is accurate to within a few thousandths of a percentile.

Worked Examples

These use a standard deviation of 15.

Example 1: IQ 100

An IQ of 100 sits exactly at the 50th percentile — the middle of the distribution.

Example 2: IQ 115

An IQ of 115 is one standard deviation above the mean, at roughly the 84th percentile.

Example 3: IQ 130

An IQ of 130 is two standard deviations above the mean, at about the 98th percentile — the threshold many societies use for “gifted”.

Example 4: IQ 85

An IQ of 85 is one standard deviation below the mean, at roughly the 16th percentile.

Practical Notes

• The percentile depends on the standard deviation. The same raw IQ produces a slightly different percentile on a scale with $\sigma = 16$ than on one with $\sigma = 15$, so always match the scale your test reports.
• The “1 in N people” figure describes the rarer tail of the distribution. For an IQ of 130 it is roughly 1 in 44 people.
• Real test scores are only approximately normal, and percentiles in the extreme tails are sensitive to small modelling differences. Treat very high or very low percentiles as estimates.
• To turn a percentile back into a range of plausible scores, use the confidence interval calculator. To average several test results, use the average calculator.