Quick answer
IQ percentile = Φ((IQ − mean) ÷ SD) × 100. With a mean of 100 and SD of 15, an IQ of 115 has a z-score of 1 and a percentile of 84.13%.
Enter an IQ score, the test's mean and standard deviation. The calculator returns the percentile and z-score.
Related tools: Percentile Calculator: The value at a given percentile of a data set. Binomial Distribution Calculator: Binomial probabilities for n trials. Sampling Distribution of the Mean Calculator: Probability for a sample mean using the central limit theorem.
How to use the IQ Percentile Calculator
- Enter iQ score.
- Enter mean. It starts at 100.
- Enter standard deviation (above 0). It starts at 15.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| IQ score | A number | — |
| Mean | A number | starts at 100 |
| Standard deviation | A number | > 0, starts at 15 |
Formula and method
Formula
percentile = Φ((IQ − mean) ÷ sd) × 100
IEEE-754 double precision; shown to at most 6 decimal places.
Assumes a normal distribution with the stated mean and standard deviation (commonly 100 and 15; some tests use 16).
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: IQ score = 115; Mean = 100; Standard deviation = 15.
| Result | Value |
|---|---|
| Percentile | 84.1345% |
| z-score | 1 |
Example 2
Inputs: IQ score = 130; Mean = 100; Standard deviation = 15.
| Result | Value |
|---|---|
| Percentile | 97.725% |
| z-score | 2 |
Example 3
Inputs: IQ score = 100; Mean = 100; Standard deviation = 15.
| Result | Value |
|---|---|
| Percentile | 50% |
| z-score | 0 |
The first line is the headline result (percentile). Below it the iQ Percentile Calculator lists z-score, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Reading the percentile
84.13% means the score is higher than about 84 of 100 people in the norm group. An IQ of 130 (z = 2) is the 97.7th percentile and 70 (z = −2) is the 2.3rd.
Test-dependent
The percentile depends on the test's own scale: most modern tests use mean 100 and SD 15, but some use SD 16 or 24. A single score has a margin of error of several points. IQ tests measure certain cognitive skills, not overall worth.
How the result changes with iQ score
Holding every other input at the first worked example, this table changes only iQ score. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| IQ score | Percentile | z-score |
|---|---|---|
| 57.5 | 0.2303% | -2.833333 |
| 115 | 84.1345% | 1 |
| 230 | 100% | 8.666667 |
| 460 | 100% | 24 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Standard deviation: entering "0" shows "Standard deviation must be greater than 0." beside that field and no result.
Choosing between the IQ Percentile Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Percentile Calculator](/statistics/percentile-calculator/) | data set and percentile | 90th percentile |
| [Binomial Distribution Calculator](/statistics/binomial-distribution/) | trials, success probability and successes | P(X = 5) |
| [Sampling Distribution of the Mean Calculator](/statistics/normal-probability-sampling-distributions/) | population mean, population SD and sample size | P(x̄ < value) |
| [Empirical Rule Calculator](/statistics/empirical-rule/) | mean and standard deviation | ≈68% (μ ± σ) |
| [Poisson Distribution Calculator](/statistics/poisson-distribution-calculator/) | average rate and events | P(X = 3) |
Background: probability and distributions
A probability model only applies when its conditions hold. The binomial model needs a fixed number of independent trials with two outcomes and a constant success probability; its mass function gives the chance of exactly k successes. The normal density is f(x) = 1 ÷ (σ√(2π)) × e^(−(x − μ)² ÷ (2σ²)).
A cumulative probability adds up everything at or below a value, a density or mass function gives the value at a point. For a standard normal variable, z = −1.96 leaves 0.025 in the lower tail, which is why ±1.96 bounds a 95% interval.
Assumptions and limitations
- Depends on the test's own scale
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for standard deviation to confirm this.
Frequently asked questions
What IQ is the 90th percentile?
The 90th percentile is z = 1.2816, so IQ = 100 + 1.2816 × 15 ≈ 119.
What is average IQ?
The average IQ is 100 by design, because tests are scaled to a population mean of 100 and an SD of 15.
Is the distribution normal?
It is constructed to be approximately normal.
What does the IQ Percentile Calculator calculate?
Percentile of an IQ score on a normal scale. It returns Percentile and z-score.
What formula does the IQ Percentile Calculator use?
percentile = Φ((IQ − mean) ÷ sd) × 100. Assumes a normal distribution with the stated mean and standard deviation (commonly 100 and 15; some tests use 16).
What do I need to enter in the iQ Percentile Calculator?
You enter IQ score, Mean and Standard deviation.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Depends on the test's own scale.
What is an example of a iQ Percentile result?
With iQ score 115, mean 100 and standard deviation 15, the percentile is 84.1345%.
Related calculators
- Percentile Calculator — The value at a given percentile of a data set.
- Binomial Distribution Calculator — Binomial probabilities for n trials.
- Sampling Distribution of the Mean Calculator — Probability for a sample mean using the central limit theorem.
- Empirical Rule Calculator — 68–95–99.7 intervals for a normal distribution.
- Poisson Distribution Calculator — Poisson probabilities for a given mean rate.
- Normal Distribution Calculator — Probabilities for a normal distribution.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods (itl.nist.gov/div898/handbook)
- Standard textbook definitions (e.g. Moore, McCabe & Craig, Introduction to the Practice of Statistics)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.