Statistics
Percentile calculator
Enter your data to get the 5th through 99th percentiles, the quartiles and the interquartile range, with a box plot showing where they fall.
Separate numbers with commas, spaces or new lines, or paste a spreadsheet column.
Decimals and negatives are fine; write 10:3 for a value that occurs 3 times.
Your results will appear here: the result first, then the rest of the summary and a chart.
Median (50th percentile)
70.5
The box spans the middle 50% of the data, from Q1 to Q3, with the median inside it. Whiskers reach the furthest values still within 1.5 IQR of the box, and no points fall outside them.
Show the working, step by step
How a percentile is calculated
Sort the data, then find the position (n − 1) × p where p is the percentile as
a decimal. If that position falls between two observations, interpolate linearly between
them.
position = (n − 1) × p Q(p) = x⌊pos⌋ + (pos − ⌊pos⌋) × (x⌈pos⌉ − x⌊pos⌋)
For 12 sorted values, the 75th percentile sits at position 11 × 0.75 = 8.25 — a quarter of the way between the 9th and 10th values counting from zero.
Standard deviations as percentiles
When data is roughly normal, each standard deviation from the mean corresponds to a fixed percentile. This is the table people are usually looking for:
| Standard deviations | Percentile | Meaning |
|---|---|---|
| −3σ | 0.1 | Bottom one in a thousand |
| −2σ | 2.3 | Bottom fortieth |
| −1σ | 15.9 | Bottom sixth |
| −0.674σ | 25 | First quartile |
| 0 | 50 | The mean, and the median |
| +0.674σ | 75 | Third quartile |
| +1σ | 84.1 | Top sixth |
| +1.28σ | 90 | Top tenth |
| +1.645σ | 95 | Top twentieth |
| +2σ | 97.7 | Top fortieth |
| +3σ | 99.9 | Top one in a thousand |
An IQ of 130 is two standard deviations above a mean of 100 with σ = 15, which puts it at the 97.7th percentile. To go the other way — from a value to its percentile — use the z-score calculator, which reports the percentile directly.
Every figure in that table assumes normality. On skewed data they can be badly wrong: for a right-skewed distribution, +1σ might be the 90th percentile rather than the 84th. The percentiles computed from your own data above carry no such assumption, which is why they are the safer answer when you have the raw numbers.
Percentile from a score, the mean and the standard deviation
When all you have is one score plus the group's mean and standard deviation — no raw data — convert the score to a z-score, then read the percentile off the normal distribution.
Example: a law graduate scores 108 on the bar exam, where the average is 132 and the standard deviation is 15.7.
z = (x − μ) ÷ σ = (108 − 132) ÷ 15.7 = −1.529 percentile = Φ(−1.529) = 0.0632 → about the 6th percentile
About 6.3% of test takers scored lower and 93.7% scored higher. The z-score calculator does both steps at once. The answer assumes scores are roughly normally distributed; if they are skewed, the percentile computed from the raw scores above is the one to trust.
Percentile rank versus percentile
They run in opposite directions, and mixing them up is the most common error here. A percentile takes a rank and returns a value: "what score is the 90th percentile?" A percentile rank takes a value and returns a rank: "what percentile is a score of 82?"
For the second question on your own data, rather than a normal curve, the percentile rank calculator counts how many values fall below a score and turns that into its rank.
Related calculators
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Z-score calculator
Converts a value to standard deviations, and to a percentile.
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IQR & outliers
The 25th and 75th percentiles, and what lies beyond them.
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Five-number summary
The quartiles drawn as a box plot.
-
Empirical rule
The percentiles that each standard deviation corresponds to.
Common questions
What is a percentile?
The value below which a given percentage of the data falls. The 90th percentile is the value that 90% of observations sit below.
What is the difference between a percentile and a percentage?
A percentage is a score; a percentile is a rank. Scoring 80% on a test is a percentage. Being in the 80th percentile means you scored higher than 80% of the people who took it. On a hard test those two numbers can be very far apart.
How do I convert a standard deviation to a percentile?
Convert to a z-score first, then read the area below it on a normal curve. One standard deviation above the mean is the 84th percentile, two is the 97.7th, and one below is the 15.9th.
That conversion assumes the data is normally distributed. The percentiles this calculator reports from your actual data make no such assumption.
Why do percentile calculators disagree?
Because there are at least nine accepted definitions of a quantile. This site uses
linear interpolation between order statistics — R's default type 7, and Excel's
PERCENTILE.INC. PERCENTILE.EXC and Minitab use other rules.
The differences shrink as n grows. Full detail here.
Can the median be a value not in my data?
Yes. With an even number of observations the median is the average of the middle two, which need not be an observed value. The same is true of most percentiles.