Convert IQ ↔ Z-score
How the conversion works
IQ and Z-score scores are normalised on bell-curve distributions. The conversion maps a person's standing in one distribution to the equivalent standing in the other.
- IQ: mean 100, standard deviation 15.
- Z-score: mean 0, standard deviation 1. Z = (raw score − mean) / SD. Z-scores express how many standard deviations a value is above or below the mean of its distribution.
IQ-equivalent = (target − meantarget) / SDtarget × 15 + 100
target-equivalent = (IQ − 100) / 15 × SDtarget + meantarget
Common IQ ↔ Z-score reference table
| IQ | Z-score | Percentile |
|---|---|---|
| 70 | −2.0 | ~2nd |
| 85 | −1.0 | ~16th |
| 100 | 0.0 | ~50th |
| 115 | +1.0 | ~84th |
| 130 | +2.0 | ~98th |
| 145 | +3.0 | ~99.9th |
| 160 | +4.0 | ~99.997th |
Important caveats
- IQ and Z-scores are mathematically equivalent representations of the same standing on a normal distribution. The conversion is exact (Z = (IQ − 100) / 15), not an estimate, because both use the same underlying scale.
- Extreme-tail conversions are less reliable because both tests have fewer calibration cases there.
- No score entered here is stored, submitted, or connected to an account.
What is a Z-score, and how does it relate to IQ?
A Z-score, also called a standard score, tells you how many standard deviations a value sits above or below the mean of its distribution. By definition the Z-score scale has a mean of 0 and a standard deviation of 1, which makes it the most fundamental standard score in all of statistics. An IQ converts to a Z-score with one simple formula, z = (IQ - 100) / 15, because the standard IQ scale uses a mean of 100 and a standard deviation of 15.
The Z-score is the universal common currency for any normally distributed measure, and IQ is one of the cleanest examples because the modern deviation IQ scale was deliberately built on the normal curve. An IQ of 100 is exactly average and converts to z = 0. An IQ of 115 is one standard deviation above the mean and converts to z = +1. An IQ of 85 is one standard deviation below and converts to z = -1. Because the conversion is a fixed linear formula rather than a statistical estimate, the relationship between IQ and Z-score is exact, not a correlation. Once you have the Z-score you can read off the percentile directly from the standard normal distribution, which is why psychologists, educators, and researchers lean on Z-scores to compare scores across tests that use different means and standard deviations.
The exact formula (and how to reverse it)
Converting between IQ and Z-score is pure arithmetic, with no estimation involved, because both scales describe the same underlying normal distribution.
To go from IQ to Z-score, subtract the mean of 100 and divide by the standard deviation of 15.
- z = (IQ - 100) / 15
To go back from a Z-score to IQ, multiply by 15 and add 100.
- IQ = (z * 15) + 100
A few worked examples make the pattern clear.
- IQ 100 gives z = 0 (dead center)
- IQ 115 gives z = +1.00
- IQ 130 gives z = +2.00
- IQ 145 gives z = +3.00
- IQ 85 gives z = -1.00
- IQ 70 gives z = -2.00
The converter on this page runs both directions for you and carries full decimal precision, so you can drop in any IQ value, including non-integer scores, and get the matching Z-score instantly.
From Z-score to percentile
The reason the Z-score matters is that it unlocks the percentile. Every Z-score corresponds to one fixed point on the standard normal curve, so once you know z you know exactly what share of the population scores below that point.
The most useful anchor values are worth memorizing.
The well known 68-95-99.7 rule of the normal distribution falls straight out of this. About 68 percent of people fall within one standard deviation of the mean (IQ 85 to 115, or z between -1 and +1), about 95 percent within two standard deviations (IQ 70 to 130), and about 99.7 percent within three. This is exactly why an IQ of 130, at z = +2, is the common cutoff for the top 2 percent.
Why the Z-score is the master scale
Almost every standardized score you will encounter is just a rescaled Z-score. T-scores use a mean of 50 and a standard deviation of 10, so a T-score is z * 10 + 50. Stanines, scaled subtest scores, and SAT sections are all built the same way, by stretching and shifting the Z-score onto a more reader friendly range.
That is the practical power of converting IQ to a Z-score first. Once an IQ score is expressed as z, you can translate it into any other standard score, or into a percentile, without needing a separate lookup table for each format. The Z-score is the hub, and every other scale is a spoke.
One caution applies. The IQ to Z-score conversion assumes the standard deviation of 15 used by Wechsler and most modern tests. A handful of older or specialized tests use a standard deviation of 16 (historically associated with Stanford-Binet) or even 24. If your score comes from one of those, swap the 15 in the formula for the test's actual standard deviation, otherwise the Z-score and percentile will be slightly off.
Frequently asked questions
What Z-score does an IQ of 130 equal?
An IQ of 130 equals a Z-score of exactly +2.00. Using z = (130 - 100) / 15 gives 30 / 15 = 2. That places the score two standard deviations above the mean, at roughly the 98th percentile, which is the common threshold for the top 2 percent and many gifted programs.
How do I convert IQ to a Z-score by hand?
Subtract 100 from the IQ score and divide the result by 15. For example, an IQ of 112 becomes (112 - 100) / 15 = 12 / 15 = 0.80, so the Z-score is +0.80. To reverse it, multiply the Z-score by 15 and add 100. The converter on this page does both directions automatically with full precision.
Is the IQ to Z-score conversion exact or an estimate?
It is exact. Unlike converting IQ to the SAT or ACT, which rely on statistical correlations, IQ and Z-score describe the same normal distribution, so the conversion is a fixed linear formula. The only assumption is the standard deviation, which is 15 on the standard Wechsler scale used by most modern IQ tests.
What is the difference between a Z-score and an IQ score?
They measure the same thing on different scales. A Z-score uses a mean of 0 and a standard deviation of 1, while the standard IQ scale uses a mean of 100 and a standard deviation of 15. An IQ of 115 and a Z-score of +1.00 are the same point on the distribution, both one standard deviation above average.
How does a Z-score give me a percentile?
Each Z-score maps to one fixed point on the standard normal distribution, and the percentile is the percentage of people scoring below that point. A Z-score of 0 is the 50th percentile, +1.00 is about the 84th, and +2.00 is about the 98th. Converting your IQ to a Z-score first is the cleanest way to find its percentile.
Does the standard deviation matter when converting IQ to a Z-score?
Yes. The standard formula uses a standard deviation of 15, which most modern tests including the Wechsler scales use. Some older or specialized tests use 16 or 24. If your score comes from one of those, replace the 15 in z = (IQ - 100) / 15 with the test's actual standard deviation, otherwise the resulting Z-score and percentile will be slightly inaccurate.
Related tools
IQ percentile calculator · IQ to T-score converter · IQ to Stanine converter · IQ score chart
Cite this converter
Editorial content and curation are released under CC BY-SA 4.0. This converter is part of the What's Your IQ educational resources.