The IQ bell curve is the normal distribution that IQ scores follow: most people score near the average of 100, and scores become steadily rarer the further they are from it. With a standard deviation of 15, about 68 percent of people score between 85 and 115, about 95 percent between 70 and 130, and about 99.7 percent between 55 and 145.
What is a normal distribution?
A normal distribution is a symmetric, bell-shaped curve in which the mean, median, and most common value are the same. Two numbers fully describe it: the mean, which sets where the peak sits, and the standard deviation, which sets how spread out the curve is. Height within one sex, measurement errors, and many test scores all come close to this shape.
Modern IQ tests are scaled so that, within each age group, the mean is 100 and the standard deviation is 15. Once those two numbers are fixed, every score corresponds to a specific position on the curve, which is why an IQ can be translated directly into a percentile.
What a standard deviation means in IQ points
The standard deviation is the typical distance of scores from the mean. On the IQ scale, one standard deviation is 15 points. To see how many standard deviations a score sits from average, compute its z-score:
z = (IQ − 100) ÷ 15
A score of 124 gives z = 1.6, which on the normal curve is higher than about 94.5 percent of people. A score of 85 gives z = −1, higher than about 16 percent. Psychologists think in z-scores because they make results from different scales comparable: a z of +2 means the same rarity whether the scale is IQ, the SAT, or height.
The 68-95-99.7 rule with IQ scores
The 68-95-99.7 rule (sometimes called the empirical rule) summarizes how much of a normal distribution falls within one, two, and three standard deviations of the mean. Applied to IQ:
| Standard deviations from 100 | IQ range | Share of people inside the range | Share above the top of the range |
|---|---|---|---|
| ±1 SD | 85–115 | 68.3% | 15.9% (about 1 in 6) |
| ±2 SD | 70–130 | 95.4% | 2.3% (about 1 in 44) |
| ±3 SD | 55–145 | 99.7% | 0.13% (about 1 in 740) |
Because the curve is symmetric, the same share falls below the bottom of each range. For example, about 2.3 percent of people score below 70, mirroring the 2.3 percent above 130. All figures here are the mathematics of the normal curve with mean 100 and standard deviation 15; real test samples match it well near the middle and less well at the extremes.
From IQ score to percentile
A percentile rank is the percentage of the reference group scoring at or below a given score. It is not the percentage of questions you got right. Some common reference points on the normal curve:
| IQ score | z-score | Percentile rank |
|---|---|---|
| 70 | −2.00 | 2.3 |
| 80 | −1.33 | 9.1 |
| 90 | −0.67 | 25.2 |
| 100 | 0.00 | 50.0 |
| 110 | +0.67 | 74.8 |
| 115 | +1.00 | 84.1 |
| 120 | +1.33 | 90.9 |
| 125 | +1.67 | 95.2 |
| 130 | +2.00 | 97.7 |
| 140 | +2.67 | 99.6 |
Notice how unevenly percentiles move. Going from 100 to 110 passes about 25 percent of the population, because so many people are packed near the middle. Going from 130 to 140 passes fewer than 2 percent. That is why a few points matter a lot to your percentile near the average and very little near the top, and why small differences near 100 are easy to overinterpret. Every score from 55 to 145 is listed in the IQ scale table, and the bands and their labels are explained in IQ score ranges.
Reading your own score on the curve
A real score report gives a range, not a point. Suppose your full-scale IQ is 112 with a 95 percent confidence interval of 106 to 117, a typical width for a clinical test. On the curve, 112 sits at about the 79th percentile, but the interval stretches from about the 66th to the 87th. Both ends are plausible, so the fair summary is "above average, probably in the High Average range," not "79th percentile exactly."
The same curve also links IQ to other score scales. Many tests report subtest results as scaled scores with a mean of 10 and a standard deviation of 3, and some report T-scores with a mean of 50 and a standard deviation of 10. A scaled score of 13, a T-score of 60, and an IQ of 115 all sit one standard deviation above the mean and mean the same thing: higher than about 84 percent of the reference group.
Why are IQ scores normally distributed?
There are two honest answers, and both are true.
The first is that traits influenced by many small, independent factors tend toward a bell shape, a consequence of the central limit theorem in statistics. Intelligence is shaped by thousands of genetic variants of tiny effect plus many environmental influences, so a roughly normal distribution is what you would expect.
The second is that the bell is partly built in. Test developers choose items that spread people out across the range, and when they norm a test they convert raw scores into standard scores, often using a step that forces the result into a normal shape even if raw scores were somewhat skewed (Anastasi & Urbina, 1997). The curve you see in a score report is therefore partly a property of nature and partly a convention of measurement. The details are in how IQ scores are calculated.
Where the bell curve breaks down
The fit is weakest at the tails:
- The low end. Researchers have long noted that more people score in the lowest ranges than a pure normal curve predicts (Dingman & Tarjan, 1960). Many of these cases have a specific cause, such as a chromosomal condition, birth complication, or brain injury, rather than reflecting the ordinary sum of small influences.
- The high end. Norming samples typically include a few thousand people, so very few score above 145. Percentiles that far out rest on extrapolation, and the standard norms of major batteries stop at about 160. Claims of IQs far above that are not measurements in any meaningful sense; see what counts as a genius IQ.
- Measurement error. Each score is an estimate with a margin of a few points. The curve describes the population, not the exact position of any one person on it.
Whose bell curve is it?
An IQ of 100 means "average for the norm group," and the norm group matters. Clinical tests in the United States are normed on large samples matched to census data. Norms also age: raw scores rose substantially across much of the 20th century, by 5 to 25 points in a single generation in the 14 nations Flynn (1987) examined, so an old test can place people higher on the curve than a freshly normed one would. Comparisons between countries are even harder, because compilations of national averages have relied on small and unrepresentative samples in many cases (Wicherts et al., 2010).
For the same reason, the estimated IQ our own test shows is only a conversion: it takes your percentile among the people who have taken the test and expresses it on the mean-100 curve, labelled as an estimate, because the test has not been normed on a representative sample. You can take the free test and see where you fall among those respondents.
"The Bell Curve" the book
If you searched for the phrase and found a book, that is The Bell Curve by Richard Herrnstein and Charles Murray (1994), which argued that intelligence plays a large role in social and economic outcomes in the United States and also discussed average differences between groups. It became one of the most controversial social science books of its decade. In response to the public debate, the American Psychological Association convened a task force whose report, "Intelligence: Knowns and Unknowns" (Neisser et al., 1996), summarized what the research did and did not support at the time. That report remains a useful, widely cited starting point for anyone who wants the mainstream scientific picture rather than the polemics on either side.