What this calculator does
IQ scores are defined by their position in a normal distribution rather than by any absolute quantity. A score of 100 is set at the mean, and every other score is a statement about how far from that mean it sits.
The standard deviation is what converts a score into a rank. On the usual 15 point scale, 115 is one deviation above the mean and lands at the 84.13th percentile, while 130 is two deviations up and reaches the 97.72nd.
The formula
The score is converted to a z-score by subtracting the mean and dividing by the standard deviation. The standard normal cumulative distribution then gives the percentile.
| Term | Meaning |
|---|---|
| z-score | How many standard deviations a score sits from the mean. |
| Percentile rank | The share of the population scoring at or below that value. |
| Standard deviation | 15 on most modern tests, but 16 or 24 on some older ones. |
The inputs explained
| Field | What to enter |
|---|---|
| IQ score | The IQ score to convert. |
| Population mean | The population mean, which is 100 by definition on almost every scale. |
| Standard deviation | The standard deviation the test uses. This must match the test or the percentile will be wrong. |
When to use it
Interpreting a test result
A percentile is far more meaningful than a score on an arbitrary scale.
Comparing scores across tests
Different tests use different standard deviations, so raw scores are not comparable.
Understanding the normal distribution
IQ is the most familiar example of a deliberately normalised scale.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What percentile does each score reach?
Six scores across the range.
| IQ score | Percentile rank | z-score |
|---|---|---|
| 70 | 2.28th percentile | -2.00 |
| 85 | 15.87th percentile | -1.00 |
| 100 | 50.00th percentile | 0.00 |
| 115 | 84.13th percentile | 1.00 |
| 130 | 97.72th percentile | 2.00 |
| 145 | 99.87th percentile | 3.00 |
Questions
Why does the standard deviation matter so much?
Because it sets the scale. A score of 130 is the 97.72nd percentile on a 15 point scale but only the 89.44th on a 24 point one. Quoting an IQ without the deviation is genuinely ambiguous.
Why is 100 always the mean?
Because the scale is defined that way and periodically renormalised to keep it there. Raw performance on IQ tests has risen substantially over the last century, the Flynn effect, and tests are restandardised to hold the mean at 100.
Are extreme percentiles meaningful?
Less than they appear. Very high and very low scores fall in the tails where few people were sampled during standardisation, so the precision implied by a figure like 99.87 is greater than the test can really support.
Does this tell me anything about a person?
It tells you where a number sits in a distribution, which is a statistical statement rather than a description of anyone. What an IQ score means, and how much it should be relied on, is a much larger and more contested question.
For the normal distribution generally, see the z-score calculator. For percentile work in statistics, see the percentile rank calculator.