IQ Percentile Calculator
An IQ score of 115 has about 84.1% of this mean-100, SD-15 normal-curve model below it. This IQ percentile calculator converts a score into an approximate z-score and percentile using a selectable standard deviation of 15 or 16. The output is a mathematical distribution estimate, not an official test interpretation, diagnosis, measure of personal worth, or complete account of cognitive ability. Published tests use specific age norms, standardization samples, score composites, confidence intervals, and rounding rules. Use the official score report and a qualified psychologist for decisions that affect education, health, accommodations, or services.
Quick answer
This tool assumes an IQ mean of 100.
What this tells you
- •This tool assumes an IQ mean of 100.
- •The percentile comes from a normal distribution, often called a bell curve.
- •Most modern IQ scales use SD 15, while some older or alternate norms use SD 16.
- •The result is an approximation, not an official score report.
- •A percentile estimates the share of the modeled distribution below the score, not the percent of questions answered correctly.
- •The z-score equals score minus 100, divided by the selected standard deviation.
- •Changing SD 15 to SD 16 changes the percentile even when the score stays the same.
- •The display rounds percentile to one decimal and caps extreme nonzero outputs at 0.1% or 99.9%.
- •The calculator does not use a test name, age, subtest pattern, confidence interval, or testing conditions.
How to Use
- 1Enter the reported score you want to model, checking that it belongs to the intended test and norm table.
- 2Choose SD 15 or SD 16 to match the scale stated by the publisher or score report.
- 3Calculate to see the modeled percentile, rounded z-score, and percentages below and above.
- 4Read percentile as relative position in the assumed curve, not as test accuracy.
- 5Compare the estimate with the official report, which may use discrete tables and publisher-specific rounding.
- 6Do not combine scores from different tests or ages as though they share identical norms.
- 7Use a qualified psychologist for diagnostic interpretation or decisions about services, placement, or accommodations.
How It Works
Formula
z = (IQ score - 100) ÷ standard deviation
Approximate percentile = standard normal CDF of z × 100Subtract the fixed mean of 100 from the score and divide by the selected SD. For 115 on SD 15, z = (115-100)/15 = 1. The calculator approximates the standard normal cumulative distribution with an error-function formula, producing about 84.1%. It rounds z to two decimals and percentile to one decimal. This continuous curve can differ from an official publisher table based on empirical norms, discrete score bands, age groups, or another rounding convention.
Calculation note: values are processed in the order shown above, using the current input units.
Worked Examples
Common SD-15 example
The z-score is (115 - 100) ÷ 15 = 1.00. A z-score of 1.00 maps to about 84.1%, so roughly 84 out of 100 people in the reference group score lower.
Higher score on an SD-16 scale
The z-score is (130 - 100) ÷ 16 = 1.875. On a normal curve that is about the 96.96th percentile, which rounds to about the 97th percentile.
Score at the modeled mean
The z-score is (100 - 100) / 15 = 0. A standard normal curve has 50% of its area below the mean.
Score one SD below the mean
The z-score is (85 - 100) / 15 = -1. The modeled percentile is about 15.9%, the mirror of 84.1% at +1.
Score two SD above the mean
The z-score is (130 - 100) / 15 = 2. The modeled percentile is about 97.7%, leaving about 2.3% above.
Reading a reported score of 112
Jordan's school report lists a full-scale score of 112 on a test that uses SD 15. The z-score is (112 - 100) / 15 = 0.80, which the model places at 78.8%, so about 79 of 100 people in the modeled reference group score lower and about 21 score higher. On a Wechsler scale the descriptive label for that range is High Average, a band label rather than a judgment about ability. The percentile is not a percent of questions answered correctly, and it ignores the confidence interval printed on the report. A 3-point swing in either direction, ordinary measurement error, moves it to about the 73rd or 84th percentile (scores 109 and 115), so treat it as a range rather than a point.
IQ score, percentile, and Wechsler classification
Modeled percentiles for scores from 55 to 145 on a mean-100, SD-15 normal curve, beside the descriptive classification published for the Wechsler Adult Intelligence Scale, Fourth Edition. Each percentile is a position in a statistical model, not a measure of a person.
| IQ score (SD 15) | z-score | Percentile below | WAIS-IV descriptive classification |
|---|---|---|---|
| 55 | -3.00 | 0.1% | Extremely Low (69 and below) |
| 60 | -2.67 | 0.4% | Extremely Low (69 and below) |
| 65 | -2.33 | 1.0% | Extremely Low (69 and below) |
| 70 | -2.00 | 2.3% | Borderline (70 to 79) |
| 75 | -1.67 | 4.8% | Borderline (70 to 79) |
| 80 | -1.33 | 9.1% | Low Average (80 to 89) |
| 85 | -1.00 | 15.9% | Low Average (80 to 89) |
| 90 | -0.67 | 25.2% | Average (90 to 109) |
| 95 | -0.33 | 36.9% | Average (90 to 109) |
| 100 | 0.00 | 50.0% | Average (90 to 109) |
| 105 | 0.33 | 63.1% | Average (90 to 109) |
| 110 | 0.67 | 74.8% | High Average (110 to 119) |
| 115 | 1.00 | 84.1% | High Average (110 to 119) |
| 120 | 1.33 | 90.9% | Superior (120 to 129) |
| 125 | 1.67 | 95.2% | Superior (120 to 129) |
| 130 | 2.00 | 97.7% | Very Superior (130 and above) |
| 135 | 2.33 | 99.0% | Very Superior (130 and above) |
| 140 | 2.67 | 99.6% | Very Superior (130 and above) |
| 145 | 3.00 | 99.9% | Very Superior (130 and above) |
Percentiles computed with the formula above (standard normal CDF of z, rounded to one decimal and display-capped at 0.1% and 99.9%). The mean-100, SD-15 convention and the descriptive classifications are from the Wechsler Adult Intelligence Scale, Fourth Edition (WAIS-IV), 2008. Other publishers use different labels, and some use a different standard deviation, so match the table to the test named on the score report.
Why an official report can differ
A published assessment is standardized on a defined sample and may provide age-specific tables. Its percentile can come from empirical score distributions rather than a perfectly smooth normal curve. A generic formula cannot reproduce every publisher's conversion.
Scores also carry measurement uncertainty. An official report may show a confidence interval because performance can vary with test conditions and ordinary measurement error. Converting only the central score hides that uncertainty.
A full evaluation considers the referral question, test validity, language, education, disability, health, behavior, and patterns across subtests. One composite score or percentile cannot diagnose a condition or summarize a person's abilities and needs.
Common mistakes
- Treating percentile as the percent of questions answered correctly
- Choosing SD 15 when the score report or manual is based on SD 16, or the other way around
- Reading a model-based percentile as an exact clinical, school, or employment decision threshold
- Treating percentile as a fixed trait that cannot vary with test version, age norms, measurement error, or testing conditions
- Ignoring the confidence interval printed on an official score report
- Comparing scores from different publishers without checking their scales and norm groups
- Using a calculator output to label, diagnose, rank, or make a high-stakes decision about a person
Limitations
This calculator assumes a continuous normal distribution with mean 100 and the selected standard deviation, the convention of the Wechsler scales (Wechsler Adult Intelligence Scale, Fourth Edition, 2008, mean 100 and SD 15). Not every test shares it. The Cattell Culture Fair scale reports scores with an SD of 24, and Stanford-Binet editions before the fifth used an SD of 16, so a 124 on the first and a 115 on a Wechsler scale describe the same relative position, and the SD 16 option here covers only the Stanford-Binet case. Every score also carries a standard error of measurement of a few points, which is why official reports print a confidence interval, so a single score stands for a range of plausible percentiles rather than one number. Norms drift as well. Flynn (1987, Psychological Bulletin) documented large gains in raw test performance across 14 nations over a generation, which is why publishers renorm their tests and why a score compared against older norms reads higher than the same performance compared against current ones. The error-function approximation and display rounding used here can differ from a publisher's empirical percentile table, and extreme values are display-capped at 0.1% and 99.9% when the modeled probability is nonzero. The result does not identify the test, edition, age band, norm sample, language, composite type, floor or ceiling effects, practice effects, accommodations, validity indicators, or testing conditions. A percentile from this page is not a clinical assessment and cannot assess intellectual functioning, diagnose disability or giftedness, determine eligibility, or replace a standardized evaluation by a qualified professional.
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