113 IQ Statistics: Percentile, z-Score and Test Scales
Statistically, an IQ of 113 is 0.87 standard deviations above the mean, which is the 80.7th percentile on the standard scale (mean 100, SD 15) used by the Wechsler tests and Stanford-Binet 5. About 19.3% of people score higher, roughly 1 in 5.2. On older scales the same number means less: about the 79th percentile with SD 16 and about the 71st with SD 24.
How the 113 IQ Percentile Is Calculated
IQ tests are normed so that scores follow a normal distribution. To find the percentile of any score you convert it to a z-score and look up the normal curve:
- z-score: (113 − 100) ÷ 15 = 0.867
- Area below z = 0.867: 0.807, so 113 is the 80.7th percentile
- Area above: 1 − 0.807 = 0.193, so 19.3% of people score higher (about 1 in 5.2)
Some publishers round to whole score bands, so a test manual may list 113 as the 81st percentile. Either way the answer is "about 81%". For a plain-language verdict on whether that is good, see Is 113 IQ good?
| Statistic | Value for IQ 113 (SD 15) |
|---|---|
| Points above mean | 13 |
| z-score | 0.87 |
| Percentile | 80.7 |
| Share scoring higher | 19.3% |
| Out of 1,000 people, expected to score higher | About 193 |
| Classification (Wechsler) | High Average (110–119) |
IQ Percentile Table Around 113
Percentiles for nearby scores on the SD 15 scale, calculated from the normal distribution:
| IQ | z-score | Percentile | Share scoring higher |
|---|---|---|---|
| 100 | 0.00 | 50.0 | 50.0% |
| 105 | 0.33 | 63.1 | 36.9% |
| 108 | 0.53 | 70.3 | 29.7% |
| 110 | 0.67 | 74.8 | 25.2% |
| 112 | 0.80 | 78.8 | 21.2% |
| 113 | 0.87 | 80.7 | 19.3% |
| 114 | 0.93 | 82.5 | 17.5% |
| 115 | 1.00 | 84.1 | 15.9% |
| 118 | 1.20 | 88.5 | 11.5% |
| 120 | 1.33 | 90.9 | 9.1% |
| 130 | 2.00 | 97.7 | 2.3% |
113 on Different IQ Scales (SD 15, 16 and 24)
Not every test uses a standard deviation of 15. The same number therefore corresponds to a different percentile depending on the scale, which is a common reason two "IQ 113" results don't mean the same thing.
| Scale | Typical use | Percentile of a 113 | Same percentile on SD 15 |
|---|---|---|---|
| SD 15 | WAIS, WISC, Stanford-Binet 5, Woodcock-Johnson | About 81st (80.7) | 113 |
| SD 16 | Stanford-Binet editions before SB5 | About 79th | About 112 |
| SD 24 | Cattell-type scales | About 71st | About 108 |
Going the other way, a 113 on the standard SD 15 scale corresponds to about 114 on an SD 16 scale and about 121 on an SD 24 scale. The conversion is: score on new scale = 100 + (z × new SD), where z = (113 − 100) ÷ 15.
Classification of 113 Across Major Tests
| Test | Band containing 113 | Notes |
|---|---|---|
| WAIS-IV / WISC-V (Wechsler) | High Average (110–119) | Full Scale IQ built from verbal, visual-spatial/perceptual, fluid reasoning, working memory and processing speed indexes |
| Stanford-Binet 5 | High Average (110–119) | Five factors: fluid reasoning, knowledge, quantitative reasoning, visual-spatial processing, working memory |
| Woodcock-Johnson | High Average (111–120) | Uses slightly shifted bands |
| Raven's Progressive Matrices | Grade II, "definitely above average" | Often reported as percentiles; Grade II covers roughly the 75th–94th percentile |
Theoretical share of people in the High Average band (110–119) is about 16%, the same as in the Low Average band (80–89).
Measurement Error: Why 113 Is Really a Range
No test measures ability perfectly. Clinical reports therefore give a confidence interval around the full-scale score. For well-built tests the 95% interval is roughly ±5 points, so a measured 113 is consistent with a true score of about 108–118, which spans roughly the 70th to the 88th percentile.
- Retesting: taking the same test again often gives a slightly higher score because of practice effects, so retests are normally spaced well apart.
- Different tests: two well-validated tests can differ by several points for the same person because they weight abilities differently.
- Norm age: raw performance on IQ tests rose over the 20th century (the Flynn effect), so a test with old norms can produce inflated scores. Publishers renorm tests periodically for this reason.
- Online tests: most free online tests are not normed on representative samples, so their "113" may not correspond to the 81st percentile of anything.
Index Scores: Same 113, Different Profiles
A Full Scale IQ is a composite. On the Wechsler tests it summarizes index scores such as verbal comprehension, visual-spatial, fluid reasoning, working memory and processing speed. Two people with a 113 can have very different profiles: one may score around 125 on verbal comprehension and around 100 on processing speed, another the reverse. When index scores differ a lot, psychologists interpret the profile rather than the single number. For what these abilities mean for school and work, see what a 113 IQ means in practice.
Frequently Asked Questions
What is the exact percentile of a 113 IQ?
On a scale with mean 100 and standard deviation 15, an IQ of 113 has a z-score of 0.867 and sits at the 80.7th percentile. That means about 80.7% of people score lower and about 19.3% score higher.
What percentage of people have an IQ over 113?
About 19.3% of people score above 113 on the standard SD 15 scale, roughly 1 in 5. Around 9.1% score 120 or above and about 2.3% score 130 or above.
Why does one source say 113 is the 79th percentile and another the 81st?
Usually because of the scale. On the standard SD 15 scale 113 is the 80.7th percentile (about 81st). On an SD 16 scale, used by older Stanford-Binet editions, it is the 79th percentile, and on an SD 24 scale it is only about the 71st.
What is 113 IQ on the Cattell scale?
A 113 on the standard SD 15 scale is about 121 on an SD 24 (Cattell-type) scale. Conversely, a 113 on an SD 24 scale is only about 108 on the standard scale, around the 71st percentile.
How accurate is an IQ score of 113?
A professionally administered test typically has a 95% confidence interval of about plus or minus 5 points, so a 113 most likely reflects a true score between about 108 and 118. Online tests are far less reliable.
Is 113 in the top 20%?
Yes, just. At the 80.7th percentile, a 113 is slightly inside the top 20% of scores on the standard scale. A score of about 112.6 marks the exact 80th percentile.