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An IQ z-score counts how many standard deviations a score sits from 100, which is what lets you compare scores from different tests. Here are the formula, a lookup table from IQ 55 to 160, and the point where the table stops being data.
An IQ z-score is the number of standard deviations an IQ sits above or below the average. On a test scored with a mean of 100 and a standard deviation (SD) of 15, which covers the Wechsler scales and most modern tests, the formula is z = (IQ – 100) ÷ 15. So IQ 130 has a z-score of +2.00, IQ 85 has -1.00 and IQ 100 has 0. The standard deviation is a measure of spread: about two-thirds of people score within one SD of the mean, about 95% within two and about 99.7% within three. A z-score says where in that spread a person landed, in units that do not depend on which test was used.
Subtract the test’s mean from the score, then divide by the test’s standard deviation. Going the other way, IQ = 100 + 15 × z. For an IQ of 118 on an SD-15 test, (118 – 100) ÷ 15 gives z = +1.20. For a z-score of -1.5, 100 + 15 × (-1.5) gives an IQ of 77.5. The one thing you must know is the scale, because a bare IQ number has no z-score until you know its mean and standard deviation. Most modern tests use 100 and 15, Cattell’s scales use an SD of 24 and some older Stanford-Binet editions used 16. Our explainer on SD 15 versus SD 16 shows how the same score lands at different percentiles on the two.
Once you have z, the percentile comes from the normal curve: the percentile is the share of the norming group scoring below that point. The list gives the z-score, percentile and rarity for common scores on an SD-15 scale, with rarity counted from the nearer end of the curve.
These are values from the normal curve, not counts of people. The percentile calculator returns the same figures for any score, the bell curve page shows the shape behind them, and our piece on how rare a high IQ score is puts the top of the table in plain numbers.
A z-score strips out the scale, which makes it the right tool for comparing tests. Take one person who scores 120 on an SD-15 test and 136 on an SD-24 test. The second number looks bigger, but 120 is z = +1.33 (90.9th percentile) and 136 is z = +1.50 (93.3rd percentile), so the two describe similar standing. The larger figure is larger only because the scale is wider. On a standard-deviation-16 scale, z = +2.00 is an IQ of 132; on an SD-24 scale it is 148.
Mensa’s published cut-offs work the same way. The Wechsler scales and Stanford-Binet 5 at 130 and Cattell’s scale at 148 are all exactly z = +2.00, the 97.7th percentile. Mensa describes its bar as the top 2%, which is the 98th percentile, z = +2.05, or 130.8 on an SD-15 scale; our Mensa explainer covers that gap.
The z-score is what stays put when the scale changes.
Every standard score is the same z-score rescaled: new score = mean + SD × z. A Wechsler subtest scaled score has a mean of 10 and an SD of 3, so a scaled score of 13 is z = +1.00, and the reported range of 1 to 19 is exactly z = -3 to +3. A T-score has a mean of 50 and an SD of 10. A normal curve equivalent (NCE), used in some US school reports, has a mean of 50 and an SD of 21.06.
That is also why the difference between a Wechsler full-scale score and its index scores is one of scale reading rather than mystery. Our explainer on full-scale IQ versus index scores shows where each sits, and the score converter does the arithmetic for you.
With a score in cell A1, =(A1-100)/15 returns the z-score and =NORM.S.DIST((A1-100)/15,TRUE) returns the percentile as a fraction, so 130 gives 0.977. To go the other way, =100+15*NORM.S.INV(0.98) returns 130.8, the IQ at the 98th percentile on an SD-15 scale. Change the 100 and 15 to match the test if it uses another scale.
A table that runs to IQ 160 looks like observation. It is not. The WAIS-IV, the fourth edition of the Wechsler adult scale, was standardised on about 2,200 people aged 16 to 90, and the Stanford-Binet 5 on about 4,800. In a sample of 2,200 the normal curve predicts about 50 people above z = +2.00 (IQ 130), about 3 above z = +3.00 (IQ 145) and 0.07 of a person above z = +4.00 (IQ 160). A norm sample cannot contain a z = +4 person to count, so any conversion out there comes from extending a fitted curve rather than from counting people. The 1-in-31,574 figure for IQ 160 is a claim about a model, not a headcount. Our piece on the highest IQ score a test can report covers the ceilings.
That spread is why a single z-score should be read as a range. Our piece on the IQ margin of error shows how to do it, and the same idea explains why the exact 1-in-N figures near the top of any table should be treated as approximate.
Use it to compare, not to rank. Convert each score to z when tests use different scales, read the result as a percentile range rather than a point, and be sceptical of any rarity figure beyond z = +3 that comes without a stated norming sample. To see how it works on a real result, take the IQ test and then convert the score with the steps above.
Curious where your own score lands in standard units? Take the IQ Metrics test and read the result as a z-score and a percentile.
Find your IQ score now! →For more on how the scale is built, see whether IQ is linear or logarithmic: the score is linear in z, but the rarity behind it is not.
Subtract 100 and divide by the test’s standard deviation, which is 15 on most modern tests. IQ 115 gives z = +1.00, IQ 130 gives +2.00 and IQ 85 gives -1.00.
z = +2.00 on a standard-deviation-15 test, the 97.7th percentile, with about 1 person in 44 scoring higher. On Cattell’s standard-deviation-24 scale the same z-score is an IQ of 148.
z = +3.00 on an SD-15 scale, the 99.87th percentile, about 1 person in 741 scoring higher. That figure comes from the normal curve, not from a count of people in a norming sample.
No. A z-score is a distance from the mean in standard deviations. A percentile is the share of the norm group scoring below. The normal curve converts one into the other, so z = +1.00 is the 84.1st percentile.
Yes, if you know each test’s mean and standard deviation. A 136 on an SD-24 scale is z = +1.50, higher than a 120 on an SD-15 scale at z = +1.33, even though 136 looks far larger.
Corrections: spotted an error? Email corrections@iqmetrics.org and we will update this story and note the change here.
Wechsler scales use a standard deviation of 15, older Stanford-Binet forms used 16, and the Cattell scales use 24. One standing prints as three different numbers — and one printed 140 is a one-in-260 result on one of those scales and a one-in-21 result on another.
On a mean-100, standard-deviation-15 scale, 130 sits at the 97.72nd percentile, which leaves about one person in 44 above it. Here is the rarity at every threshold — and why the figures stop describing real people long before they stop being calculable.
No IQ number qualifies anyone for US disability benefits on its own. Social Security writes 70 and 75 into its intellectual-disorder listing, but each sits inside a three-part test, and a rule many pages still describe was removed in 2017.
Our IIF-certified assessment reports your score with its scale, percentile and confidence range — and a breakdown of the cognitive domains behind it.
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