How this instrument works
IQ tests are constructed so that scores follow an approximately normal (bell-curve) distribution across the population, with a mean of 100 and a standard deviation of 15. That construction means any single IQ score can, in principle, be converted into a percentile — the share of the reference population expected to score at or below it — just from where it sits on that curve.
This calculator first converts the entered score into a z-score, the number of standard deviations above or below the mean: z = (IQ − 100) / 15. It then converts that z-score to a percentile using a logistic function, percentile ≈ 100 / (1 + e^(−1.702z)), rather than the exact normal cumulative distribution function. The constant 1.702 was derived by Haley (1952) specifically to make the logistic curve hug the normal curve as closely as possible, and its use is explained in detail by Camilli (1994) — it's a well-documented approximation, not a shortcut invented for this calculator.
Two honesty points worth stating plainly. First, this is an approximation: the logistic curve with that scaling constant differs from the exact normal CDF by at most roughly one percentage point at any given z-score, which is small but not zero. Second, real IQ test publishers (Wechsler scales, Stanford-Binet, and others) derive their own percentile tables from actual normative testing samples, which are close to — but not perfectly — normally distributed in practice; a published test's own percentile for a given score can therefore differ slightly from this calculator's pure statistical estimate. Treat this tool as a way to understand roughly where a score falls on the standard normal curve, not as a replacement for a test publisher's official norms, and IQ scores in general should not be treated as a clinical or diagnostic measure without proper testing and interpretation by a qualified professional.
- Enter the IQ score.
- Read the z-score — how many standard deviations the score sits from the mean of 100, using a standard deviation of 15.
- Read the approximate percentile rank, calculated from the z-score using the logistic approximation to the normal distribution.
Worked example — IQ 115, 100, and 130
An IQ score of 115: z = (115 − 100) / 15 = 1.0, exactly one standard deviation above the mean. Plugging into the logistic approximation gives a percentile of about 84.6 — so a score of 115 is estimated to sit near the 85th percentile of the reference population.
An IQ score of exactly 100: z = 0, and the logistic approximation gives exactly the 50th percentile — the approximation is exact at z=0 by symmetry, since a score right at the mean should split the population evenly regardless of which curve (normal or logistic) is used.
An IQ score of 130: z = (130 − 100) / 15 = 2.0, two standard deviations above the mean, giving an approximate percentile of about 96.8 — meaning a score this high or higher is estimated to occur in only around 3% of the reference population.
Questions
Is this the exact percentile, or an approximation?
An approximation. This calculator uses the logistic function (with the 1.702 scaling constant from Haley, 1952) as a stand-in for the exact normal cumulative distribution function, because it's simpler to compute and very close to the true curve — differing from the exact normal-CDF percentile by at most roughly one percentage point at any given score. For most purposes that gap is negligible, but it means the number shown is a close estimate, not a mathematically exact percentile.
Why might a real IQ test report a different percentile for the same score?
Because this calculator assumes a perfectly normal distribution with mean 100 and SD 15, while an actual test publisher (Wechsler, Stanford-Binet, and others) derives its percentile tables from real normative samples — large groups of test-takers whose actual score distribution is close to, but not perfectly, normal, especially out at the extreme tails. A test's official percentile table, built from its own norming data, should be treated as more authoritative for that specific test than this general statistical estimate.
Where does the 1.702 constant come from?
It was derived by Haley in a 1952 Stanford technical report as the scaling factor that makes a logistic curve most closely match the normal (probit) curve, minimizing the maximum difference between the two. Camilli's 1994 paper in the Journal of Educational and Behavioral Statistics walks back through and explains that original derivation for a modern audience; the two are commonly cited together as the origin and the explanation of the same constant.
Is this calculator a diagnostic or clinical tool?
No. It performs a general statistical conversion from a score to an approximate percentile assuming a standard normal distribution — it does not administer, score, or interpret an IQ test, and it is not a substitute for testing and interpretation by a qualified psychologist or other professional, particularly for any clinical, educational placement, or diagnostic purpose.
Why is the mean set to 100 and the standard deviation to 15?
Those are the standard scaling conventions used by most modern IQ tests, including the Wechsler scales — scores are calibrated so the population mean comes out to 100 with a standard deviation of 15. This calculator adopts that same convention so its z-score and percentile outputs line up with how most commonly reported IQ scores are already scaled.
What does a percentile actually mean here?
A percentile of, say, 84.6 means the score is estimated to be higher than about 84.6% of the reference population (and lower than about 15.4%) — not that the person answered 84.6% of questions correctly, and not a percentage grade of any kind.
References
- Camilli G — Teacher's Corner: Origin of the Scaling Constant d = 1.7 in Item Response Theory. J Educ Behav Stat. 1994;19(3):293-295
- Haley DC — Estimation of the Dosage Mortality Relationship When the Dose is Subject to Error. Technical Report No. 15, Applied Mathematics and Statistics Laboratory, Stanford University, 1952
Read this first: This instrument computes a screening figure from population formulas — it is not a diagnosis, and it cannot see the whole picture a clinician can. Use it to inform a conversation, not to replace one.