What is the IQ Percentile Calculator?
IQ scores are designed around a normal (bell-curve) distribution with a mean of 100. The percentile rank tells you what fraction of the population scores at or below a given IQ. For example, a percentile of 84 means you scored higher than about 84% of people. This calculator converts any IQ score into its exact percentile using the standard normal cumulative distribution function.
How to use it
Enter the IQ score and pick the standard deviation that matches the test. Most modern tests (Wechsler, WAIS, WISC) use SD = 15. The Stanford-Binet uses SD = 16, and the Cattell scale uses SD = 24. The calculator returns the percentile rank, the share of people who score higher, and the underlying z-score.
The formula explained
First the IQ is converted to a z-score: \(z = \frac{\text{IQ} - 100}{\text{SD}}\). This expresses how many standard deviations the score sits from the mean. The percentile is then
$$\text{percentile} = 100 \times \Phi\!\left(\frac{\text{IQ} - 100}{\text{SD}}\right)$$where \(\Phi\) is the standard normal CDF — the area under the bell curve to the left of \(z\). We compute \(\Phi\) using a high-accuracy error-function approximation, accurate to about \(\pm 1.5\times 10^{-7}\).
Worked example
Take an IQ of 115 on the SD = 15 scale. The z-score is
$$z = \frac{115 - 100}{15} = 1.0$$The standard normal CDF at \(z = 1\) is about 0.8413, so the percentile is
$$100 \times 0.8413 \approx 84.1$$This person scores higher than roughly 84% of the population, and about 16% score higher.
FAQ
What does the 50th percentile mean? An IQ of exactly 100 is the population average and sits at the 50th percentile.
Why does the standard deviation matter? The same IQ number maps to different percentiles on different scales. An IQ of 130 is the 97.7th percentile at SD 15 but only the 89.4th percentile at SD 24.
Are these scores exact? The percentile reflects an idealized normal distribution. Real test norms can differ slightly, especially at the extreme tails.
