IQ Percentile Calculator

percentile = Φ((IQ − 100) ÷ 15) × 100% · z = (IQ − 100) ÷ 15 · 1 in N = 1 ÷ min(Φ, 1 − Φ)

IQ

The score being placed. The scale is defined so that 100 is the average and 15 points is one standard deviation, so this is the only number you have to supply

z

How many standard deviations the score sits from 100 — negative below the average, positive above. On this scale it is just (IQ − 100) ÷ 15, and the sign carries the direction while the size carries how unusual the score is

Φ

The share of a normal curve that falls at or below z. It is what turns a distance into a percentile, and it is the one quantity in this formula that has to be looked up rather than worked out in your head

percentile

That share written as a percentage — the reading most people came for. A score of 115 sits at about 84, meaning roughly 84% of scores fall at or below it

1 in N

How many people there are per person scoring at least that far from 100 in the same direction. It exists because the percentile column runs out of room at the ends of the range: at a score of 145 the percentile is 99.87 and has almost nowhere left to go, while the rarity reading, 1 in 741, is still an ordinary number

Use it when you have a single IQ score and want to know where it sits in the population — the ranking is what most people are after when they look a score up. What it is not for is a distribution of your own: 100 and 15 are built in, so a class average, a set of measurements or a scale with a different spread will not fit it, and the z-score or normal distribution pages are the ones that take an average and a standard deviation as inputs. It is not a converter between tests either. Two different intelligence tests both report on a scale with an average of 100 and a standard deviation of 15, but they were normed on different samples in different years, so one score does not turn into the other by arithmetic. Read the two readings side by side: the percentile is the one to quote in a sentence, and the rarity figure is the one that still says something at the extremes, where the percentile has run out of decimals.

1. An IQ of 130, two standard deviations above the average

   1. Distance from the average: 130 − 100 = 30 points
   2. In standard deviations: 30 ÷ 15 = 2, so z = 2
   3. Read the normal curve at z = 2: 0.97725 of scores fall at or below it, which is the percentile, 97.72%
   4. The share at or above it is 1 − 0.97725 = 0.02275, and 1 ÷ 0.02275 = 43.96 — about 44 people per person

   This is the score most people arrive with, because it is the round number two standard deviations out. The rarity reading is what makes it concrete: roughly one person in 44. Notice how the two readings diverge as the score climbs — the percentile is already at 97.72 and has only 2.28 points of room left, while the rarity figure is still a number you can picture. The textbook cross-check for the same region: 68% of scores fall between 85 and 115, so about 16% fall above 115 and about 2.3% above 130, which is the 44 this page returns.

2. An IQ of 145, three standard deviations out

   1. Distance from the average: 145 − 100 = 45 points
   2. In standard deviations: 45 ÷ 15 = 3, so z = 3
   3. Read the normal curve at z = 3: 0.99865 of scores fall at or below it — 99.87%
   4. Above it: 1 − 0.99865 = 0.00135, and 1 ÷ 0.00135 = 740.7 — about 741 people per person

   This is the end of the ruler, and it is where the percentile stops being informative. One more standard deviation would put the percentile at 99.997, which two decimals cannot show at all, while the rarity figure would read about 32000. That is the whole reason the page carries a second reading: in a room of a thousand people you would not expect to meet a score this high, and 'almost everybody is below it' is true but is not the same statement as 'about one in 741'.

3. An IQ of 85, one standard deviation below the average

   1. Distance from the average: 85 − 100 = −15 points
   2. In standard deviations: −15 ÷ 15 = −1, so z = −1 — the sign is the direction and nothing else
   3. Read the normal curve at z = −1: 0.15866 of scores fall at or below it — 15.87%
   4. The share at least that far from 100 is what sits at or below 0.15866, so 1 ÷ 0.15866 = 6.30 — about 6 people per person

   Compare this with 115. The two percentiles, 15.87 and 84.13, look like opposites, but the rarity figure is 6 for both: the same distance from the average on either side is the same rarity, and only the direction differs. This is also why the rarity reading is taken from whichever side of 100 is closer rather than from the top alone — taking it from the top would report 1 ÷ 0.8413 = 1.19 people per person for a score of 85, which reads as though almost nobody shares that distance when in fact one person in six does. The textbook cross-check: 68% of scores fall between 85 and 115, leaving 16% below 85, which is the 15.87% this page returns.

A score you type in is treated here as exact, and it is not. Any IQ figure is one sample from a distribution of possible scores for the same person: the manual for the test you were given prints an interval around it, and that interval is usually several points wide in each direction. Test-retest studies make the same point from the other end. In one follow-up, 344 students were retested after an average of 2.84 years and the reliability of full-scale IQ came out at .82 — and 25% of them moved by ten points or more between the two sittings. Ten points near the middle of the scale is the distance from about the 37th percentile to about the 63rd. A later study of 225 children on a newer edition found the full-scale score stable enough for comparisons between people, but none of the within-person measures stable enough to base a decision on. Two further things date the answer. The norms behind the number come from a sample gathered in a particular place and year, and because scores rise from one generation to the next — the Flynn effect — those norms are periodically replaced; a percentile computed against this year's norms and one computed against those of twenty years ago are not the same claim. And the score's meaning depends on which test it came from and which age group it was normed for, neither of which a bare number carries. So this page reports where a number sits on a scale. It says nothing about what the same person would score on a retest, on another test, or against another year's norms. It is not a diagnostic instrument, it is not a substitute for the report that came with the score, and nothing here should be used to decide anything about a person.

What does an IQ percentile mean?

It is the share of the population scoring at or below your score. An IQ of 115 sits at the 84th percentile, which means about 84% of people score 115 or lower. Two things follow from that phrasing. The word 'or lower' matters — the figure is not the share who scored exactly 115, which would be a much smaller number. And a percentile is a statement about a population, not about a person: it says where the score lands in a distribution, and nothing about what the score is made of or what it predicts.

What IQ is in the top 1 percent?

About 135. The 99th percentile is 2.33 standard deviations above the average, and 100 + 2.33 × 15 = 134.9. For other round landmarks: the top 5% starts at about 125, the top 2% at about 131, the top 0.1% at about 146. You can read the same numbers off the chart further down this page without doing the arithmetic, and the rarity column next to the percentile column is often the faster way to see them — the top 1% and 'one in 100' are the same statement.

Why can't I enter my own average and standard deviation?

Because on an IQ scale those two numbers are the definition of the scale, not inputs. A test is standardised so that the sample it was normed on produces an average of 100 and a standard deviation of 15; change the 15 to 16 and you are no longer reading the score on the scale it was reported on. So this page asks for the score alone and pins the other two. If what you actually have is a measurement with its own average and its own standard deviation — a class, a batch of readings, a different test with a different spread — then the z-score page and the normal distribution page are the ones built for that, and they take both numbers as inputs.

Why doesn't this page tell me whether my score is high?

Because the bands that would answer that are the test publisher's, not this page's. Descriptions like average, high or superior are printed with the test and are tied to the sample it was normed on and the year it was normed in; a page that receives one number does not know which test it came from or which edition's norms apply. There is a second reason that has nothing to do with labels. A score is one measurement with a margin of error around it, and it moves: in a study that retested 344 students after nearly three years, a quarter of them shifted by ten points or more. A badge attached to a single reading would be reporting a verdict the reading cannot support. What this page can do is say where the number sits in the population, which it does.

How accurate is a single IQ score?

Less precise than the number looks. Test manuals print a confidence interval around every score, usually several points wide in each direction, and repeated testing shows the same thing: full-scale IQ has a test-retest reliability around .82 over a couple of years, with about a quarter of people moving ten points or more between sittings. Part of that spread is real change, part is measurement error, and from a single number the two cannot be told apart. The comparison group also ages: norms are gathered from a sample at a point in time and replaced periodically, because scores rise from one generation to the next. So treat the percentile as the position of one reading against one edition of one test, which is exactly what it is.

What does the '1 in N' figure mean?

It is the rarity reading: N is how many people there are per person scoring at least that far from 100 on the side the score falls on. A score of 130 comes out at 44, so roughly one person in 44 scores 130 or above; a score of 85 also comes out at 6, so roughly one person in 6 scores 85 or below. The reason the page carries it alongside the percentile is that the percentile runs out of room at the ends of the range — at 145 it is 99.87 and at 160 it is 100.00, both of which are true and neither of which tells you how rare the score is. The rarity figure keeps working there, which is where the interesting scores live.

• Measures of Intelligence — OpenStax, Psychology 2e, section 7.5 — source for the deviation IQ mean of 100 and standard deviation of 15, for norming and the Flynn effect; the 68% between 85 and 115 and the 2.2% below 70 are in the same section
• Long-term stability of the Wechsler Intelligence Scale for Children—Fourth Edition — Watkins MW, Smith LG — Psychological Assessment, 2013, 25(2):477–483 (344 students, mean retest interval 2.84 years, full-scale reliability .82, and 25% of the students scoring more than 10 points apart; doi 10.1037/a0031653)
• Long-term stability of Wechsler Intelligence Scale for Children—fifth edition scores in a clinical sample — Watkins MW et al. — Applied Neuropsychology: Child, 2022, 11(3):422–428 (225 children, mean retest interval 2.6 years; full-scale scores held steady while individual index scores did not; doi 10.1080/21622965.2021.1875827)