Use this interactive calculator to convert any IQ score on the standard deviation 15 scale into its exact population percentile, z-score, and rarity ranking. The tool uses the Abramowitz and Stegun polynomial approximation of the Gaussian cumulative distribution function, accurate to within 0.01 percentile points.
Exact population median (higher than 50% of the population, 1 in 2 people)
Wechsler Classification: Average (0.00 SD)
How the normal CDF algorithm calculates percentiles
In statistical psychometrics, the probability density function (PDF) for a standard normal distribution is given by:
f(z) = (1 / √(2π)) · e^(-z² / 2)
Where z = (Score - 100) / 15. To find the cumulative percentile, our engine calculates the integral of this function from -∞ to z using the Abramowitz and Stegun rational approximation (Formula 26.2.17), achieving maximum error |ε(x)| < 7.5 × 10⁻⁸.
Reference benchmark percentiles
IQ Score
Z-Score
Percentile
Population Standing
145
+3.00
99.87th
Top 0.13% (1 in 740)
130
+2.00
97.72nd
Top 2.28% (Mensa cutoff)
115
+1.00
84.13th
Top 15.87% (High Average)
100
0.00
50.00th
Exact Median (50 out of 100)
85
-1.00
15.87th
Bottom 15.87%
70
-2.00
2.28th
Bottom 2.28%
Frequently Asked Questions
How accurate is this percentile calculation?
It uses the standard Gaussian normal distribution formula (Abramowitz & Stegun approximation) accurate to within 0.01 percentile points for any score on the standard SD 15 scale.
What percentile is an IQ of 130?
An IQ of 130 corresponds to the 97.72nd percentile (+2.00 standard deviations), meaning it ranks in the top 2.28% of the population.
What percentile is an average IQ of 100?
An IQ of 100 is exactly at the 50.00th percentile — the precise median and mean of the population.
Researchers, educators, and content publishers may reference this analysis using the citation hyperlink below:
<a href="https://iq-test.icu/iq-scores/iq-percentile-calculator" title="https://iq-test.icu/iq-scores/iq-percentile-calculator">APEX IQ Test — IQ SCORES IQ PERCENTILE CALCULATOR</a>
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