IQ Z-Score: Formula, Table and Worked Examples
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, worked answers for an IQ of 75 and 120, a z-table, a percentile for every IQ from 70 to 145, and the point where the table stops being data.
What you need to know
- An IQ z-score is (IQ – 100) ÷ 15 on any test scored with a mean of 100 and a standard deviation of 15. IQ 75 is z = -1.67 (4.8th percentile), IQ 120 is +1.33 (90.9th), IQ 130 is +2.00 (97.7th) and IQ 145 is +3.00 (99.87th, about 1 in 741).
- The z-score is what carries across tests, the IQ number is not. An IQ of 130 on a standard-deviation-15 test, 132 on an older standard-deviation-16 Stanford-Binet and 148 on Cattell’s standard-deviation-24 scale are all exactly z = +2.00.
- Every standard score is a z-score rescaled: T = 50 + 10z, scaled score = 10 + 3z, NCE = 50 + 21.06z. A Wechsler subtest scaled score of 13, a T-score of 60 and an IQ of 115 all mean z = +1.00, and the subtest range of 1 to 19 is exactly z = -3 to +3.
- Above z = +3 an IQ table is a fitted curve, not a headcount. A norming sample of about 2,200 people (the WAIS-IV) would hold about 3 people above z = +3 and 0.07 of a person above z = +4 (IQ 160), so the rarity figures out there are model output.
An IQ z-score is the number of standard deviations an IQ sits above or below the average. On any 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 an IQ of 75 has a z-score of -1.67, an IQ of 120 has +1.33, IQ 130 has +2.00 and IQ 85 has -1.00. 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.
How do you convert an IQ score to a z-score?
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.
Worked examples: what is the z-score for an IQ of 75 or 120?
These are the problems people type into search boxes, solved with the formula. Each answer is rounded the way the question asks, and the percentile comes from the normal curve.
- What is the z-score for an IQ score of 75? z = (75 – 100) ÷ 15 = -25 ÷ 15 = -1.67 to two decimal places, which is the 4.8th percentile
- What is the z-score for an IQ score of 120? z = (120 – 100) ÷ 15 = +1.33 to two decimal places, which is the 90.9th percentile
- What is the IQ score of someone with a z-score of 1.3? IQ = 100 + 15 × 1.3 = 119.5, which rounds to 120 (the 90.3rd percentile)
- An IQ is measured at 102 on an SD-15 scale and described as the 55th percentile. Check: z = (102 – 100) ÷ 15 = +0.13, which is the 55.3rd percentile, so the description is right to the nearest whole percentile
- IQ scores are normally distributed with a mean of 100 and an SD of 15. What percentage of people score above 118? z = (118 – 100) ÷ 15 = +1.20, and 11.5% of people score higher (the 88.5th percentile)
- What percentile is an IQ of 85? z = -1.00, which is the 15.9th percentile
- What percentile is an IQ of 131? z = +2.07, which is the 98.1st percentile
A printed z-table rounds z to two decimal places, so it can differ from a calculator in the last digit. For IQ 75 the table row -1.6 and column .07 read 0.0475, the value at exactly z = -1.67, while a calculator using the unrounded z of -1.6667 returns 0.0478, a gap of 0.03 percentage points. The percentiles on this page use the unrounded z.
What is the z-score for each IQ, and how rare is it?
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.
- IQ 55: z = -3.00, 0.135th percentile, about 1 in 741 score lower
- IQ 70: z = -2.00, 2.3rd percentile, about 1 in 44
- IQ 85: z = -1.00, 15.9th percentile, about 1 in 6
- IQ 100: z = 0.00, 50th percentile
- IQ 115: z = +1.00, 84.1st percentile, about 1 in 6 score higher
- IQ 120: z = +1.33, 90.9th percentile, about 1 in 11
- IQ 125: z = +1.67, 95.2nd percentile, about 1 in 21
- IQ 130: z = +2.00, 97.7th percentile, about 1 in 44
- IQ 135: z = +2.33, 99.0th percentile, about 1 in 102
- IQ 140: z = +2.67, 99.6th percentile, about 1 in 261
- IQ 145: z = +3.00, 99.87th percentile, about 1 in 741
- IQ 150: z = +3.33, 99.96th percentile, about 1 in 2,331
- IQ 160: z = +4.00, 99.997th percentile, about 1 in 31,574
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. For what a single score means in plain language, see the blog guides to an IQ of 85, 115 and 145.
IQ to z-score to percentile: every whole score from 70 to 145
The list above gives selected scores. This one gives every whole score on an SD-15 scale from 70 to 145 with its z-score and percentile, so a search for a particular number, say 117 or 131, lands on its own row. Each percentile is the share of the norming group scoring below that score, taken from the normal curve; the IQ percentile calculator returns the same figures. Scores outside the range follow the same arithmetic, with the caveat in the section on the tails below.
- IQ 70: z = -2.00, 2.3rd percentile
- IQ 71: z = -1.93, 2.7th percentile
- IQ 72: z = -1.87, 3.1st percentile
- IQ 73: z = -1.80, 3.6th percentile
- IQ 74: z = -1.73, 4.2nd percentile
- IQ 75: z = -1.67, 4.8th percentile
- IQ 76: z = -1.60, 5.5th percentile
- IQ 77: z = -1.53, 6.3rd percentile
- IQ 78: z = -1.47, 7.1st percentile
- IQ 79: z = -1.40, 8.1st percentile
- IQ 80: z = -1.33, 9.1st percentile
- IQ 81: z = -1.27, 10.3rd percentile
- IQ 82: z = -1.20, 11.5th percentile
- IQ 83: z = -1.13, 12.9th percentile
- IQ 84: z = -1.07, 14.3rd percentile
- IQ 85: z = -1.00, 15.9th percentile
- IQ 86: z = -0.93, 17.5th percentile
- IQ 87: z = -0.87, 19.3rd percentile
- IQ 88: z = -0.80, 21.2nd percentile
- IQ 89: z = -0.73, 23.2nd percentile
- IQ 90: z = -0.67, 25.2nd percentile
- IQ 91: z = -0.60, 27.4th percentile
- IQ 92: z = -0.53, 29.7th percentile
- IQ 93: z = -0.47, 32.0th percentile
- IQ 94: z = -0.40, 34.5th percentile
- IQ 95: z = -0.33, 36.9th percentile
- IQ 96: z = -0.27, 39.5th percentile
- IQ 97: z = -0.20, 42.1st percentile
- IQ 98: z = -0.13, 44.7th percentile
- IQ 99: z = -0.07, 47.3rd percentile
- IQ 100: z = 0.00, 50.0th percentile
- IQ 101: z = +0.07, 52.7th percentile
- IQ 102: z = +0.13, 55.3rd percentile
- IQ 103: z = +0.20, 57.9th percentile
- IQ 104: z = +0.27, 60.5th percentile
- IQ 105: z = +0.33, 63.1st percentile
- IQ 106: z = +0.40, 65.5th percentile
- IQ 107: z = +0.47, 68.0th percentile
- IQ 108: z = +0.53, 70.3rd percentile
- IQ 109: z = +0.60, 72.6th percentile
- IQ 110: z = +0.67, 74.8th percentile
- IQ 111: z = +0.73, 76.8th percentile
- IQ 112: z = +0.80, 78.8th percentile
- IQ 113: z = +0.87, 80.7th percentile
- IQ 114: z = +0.93, 82.5th percentile
- IQ 115: z = +1.00, 84.1st percentile
- IQ 116: z = +1.07, 85.7th percentile
- IQ 117: z = +1.13, 87.1st percentile
- IQ 118: z = +1.20, 88.5th percentile
- IQ 119: z = +1.27, 89.7th percentile
- IQ 120: z = +1.33, 90.9th percentile
- IQ 121: z = +1.40, 91.9th percentile
- IQ 122: z = +1.47, 92.9th percentile
- IQ 123: z = +1.53, 93.7th percentile
- IQ 124: z = +1.60, 94.5th percentile
- IQ 125: z = +1.67, 95.2nd percentile
- IQ 126: z = +1.73, 95.8th percentile
- IQ 127: z = +1.80, 96.4th percentile
- IQ 128: z = +1.87, 96.9th percentile
- IQ 129: z = +1.93, 97.3rd percentile
- IQ 130: z = +2.00, 97.7th percentile
- IQ 131: z = +2.07, 98.1st percentile
- IQ 132: z = +2.13, 98.4th percentile
- IQ 133: z = +2.20, 98.6th percentile
- IQ 134: z = +2.27, 98.8th percentile
- IQ 135: z = +2.33, 99.0th percentile
- IQ 136: z = +2.40, 99.2nd percentile
- IQ 137: z = +2.47, 99.3rd percentile
- IQ 138: z = +2.53, 99.4th percentile
- IQ 139: z = +2.60, 99.5th percentile
- IQ 140: z = +2.67, 99.6th percentile
- IQ 141: z = +2.73, 99.7th percentile
- IQ 142: z = +2.80, 99.7th percentile
- IQ 143: z = +2.87, 99.8th percentile
- IQ 144: z = +2.93, 99.8th percentile
- IQ 145: z = +3.00, 99.87th percentile
Read each row as the middle of a range, not a point. An IQ of 117 is the 87.1st percentile, but a measurement error of 5 points either way runs from IQ 112 (the 78.8th percentile) to IQ 122 (the 92.9th percentile). Our explainer on the IQ margin of error shows how large that error is on the main tests.
How do you read a z-score table?
A standard normal table, or z-table, lists for each z-score the proportion of a normal distribution that lies below it. To use one, split the z-score into two parts: the row is the z-score to one decimal place and the column is the second decimal place. For z = +1.33, take the row 1.3 and the column .03, which reads 0.9082, so 90.8% of scores lie below it. The unrounded z of 1.3333 gives 0.9088, which is why tables and calculators can differ in the second decimal place. The list below gives the table value at each tenth from 0.0 to 3.0.
- z = 0.0: 0.5000
- z = 0.1: 0.5398
- z = 0.2: 0.5793
- z = 0.3: 0.6179
- z = 0.4: 0.6554
- z = 0.5: 0.6915
- z = 0.6: 0.7257
- z = 0.7: 0.7580
- z = 0.8: 0.7881
- z = 0.9: 0.8159
- z = 1.0: 0.8413
- z = 1.1: 0.8643
- z = 1.2: 0.8849
- z = 1.3: 0.9032
- z = 1.4: 0.9192
- z = 1.5: 0.9332
- z = 1.6: 0.9452
- z = 1.7: 0.9554
- z = 1.8: 0.9641
- z = 1.9: 0.9713
- z = 2.0: 0.9772
- z = 2.1: 0.9821
- z = 2.2: 0.9861
- z = 2.3: 0.9893
- z = 2.4: 0.9918
- z = 2.5: 0.9938
- z = 2.6: 0.9953
- z = 2.7: 0.9965
- z = 2.8: 0.9974
- z = 2.9: 0.9981
- z = 3.0: 0.9987
For a negative z-score, subtract the table value for the matching positive z-score from 1: z = -1.00 is 1 – 0.8413 = 0.1587, the 15.9th percentile. To run the table backwards, find the percentile inside it and read off the z-score: the 90th percentile falls at z = +1.28, which is an IQ of 119.2 on an SD-15 scale. The same z-score can be read as a percentile, an IQ or a T-score, which is the point of the next section.
Why do z-scores matter when tests use different scales?
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.
Other scales are z-scores in different clothes
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.
- z = -2.00: IQ 70 (SD 15), scaled score 4, T-score 30, IQ 52 on an SD-24 scale, NCE 8
- z = +1.00: IQ 115 (SD 15), scaled score 13, T-score 60, IQ 124 on an SD-24 scale, NCE 71
- z = +2.00: IQ 130 (SD 15), scaled score 16, T-score 70, IQ 148 on an SD-24 scale, NCE 92
To convert a T-score to a z-score, subtract 50 and divide by 10: a T-score of 65 is z = +1.50, and a T-score of 40 is z = -1.00. To go the other way, T = 50 + 10 × z, so z = -0.50 is a T-score of 45. The same one-line conversion works for a scaled score (scaled score = 10 + 3 × z, so z = +2.00 is 16) and for a normal curve equivalent (NCE = 50 + 21.06 × z). On an SD-15 IQ scale, a T-score of 65 is an IQ of 122.5 and a scaled score of 7 is an IQ of 85.
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.
How do you calculate an IQ z-score in Excel or Google Sheets?
With a score in cell A1, =(A1-100)/15 returns the z-score, and so does =STANDARDIZE(A1,100,15), which takes the mean and standard deviation as its second and third arguments. =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.
Where does the table stop being data?
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.
What should you do with an IQ z-score?
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.
Where would your own score land?
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.
Common questions
How do you convert an IQ score to a z-score?
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.
What is the z-score of an IQ of 130?
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.
What is the z-score for an IQ of 145?
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.
Is a z-score the same as a percentile?
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.
Can you compare IQ scores from different tests using z-scores?
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.
What is the z-score for an IQ of 75?
z = (75 – 100) ÷ 15 = -1.67 to two decimal places on a standard-deviation-15 scale. That is 1.67 standard deviations below the mean, the 4.8th percentile.
What is the z-score for an IQ of 120?
z = (120 – 100) ÷ 15 = +1.33 to two decimal places. That is the 90.9th percentile, so about 1 person in 11 scores higher.
How do you read a z-score table?
Split the z-score into its first decimal, which is the row, and its second decimal, which is the column. The cell gives the share of scores below it: for z = 1.33 the row 1.3 and column .03 read 0.9082. For a negative z-score, subtract the value for the matching positive z from 1.
How do you convert a T-score to a z-score?
Subtract 50 and divide by 10: z = (T – 50) ÷ 10. A T-score of 65 is z = +1.50 and a T-score of 40 is z = -1.00. The reverse is T = 50 + 10 × z.
What is a high z-score?
Above +2.00 is usually called high, because it marks the top 2.3% of a normal distribution. +1.00 is the top 16% and +3.00 the top 0.13%. On a scale with a mean of 100 and an SD of 15 those points are IQ 130, 115 and 145. A negative z-score means below average, not an error.
Sources for this story
- Wechsler Adult Intelligence Scale, Fourth Edition (WAIS-IV): Technical and Interpretive Manual — Pearson, 2008
- Stanford-Binet Intelligence Scales, Fifth Edition: Technical Manual (G. H. Roid) — Riverside Publishing, 2003
- Qualifying test scores for membership (Wechsler, Stanford-Binet 5, Cattell) — Mensa
- The normal distribution, in the e-Handbook of Statistical Methods — NIST/SEMATECH
- NORM.S.DIST and NORM.S.INV functions — Microsoft Support
- STANDARDIZE function — Microsoft Support
- Iowa Assessments score descriptions (percentile rank, normal curve equivalent, stanine) — Riverside Insights, 2025
Corrections: spotted an error? Email corrections@iqmetrics.org and we will update this story and note the change here.
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