Percentile Calculator
Paste a list and the percentile calculator finds the value at any percentile and the percentile rank of any value, with quartiles, IQR and outliers. Three methods are shown side by side, because they disagree on small lists.
90th percentile, inclusive method
- 90th percentile Position (n - 1) x p = 9.9 counting from 0, interpolated between the values either side.
- 87.6
- Percentile rank of 79 7 of 12 values are below it and 1 equal it: (7 + 1 / 2) / 12. Counting only values below gives 58.33%; counting values at or below gives 66.67%. Excel's PERCENTRANK.INC gives 63.64%.
- 62.5%
- Five-number summary Minimum, Q1, median, Q3, maximum.
- 48, 69, 75, 81.75, 93
- Interquartile range (IQR) Q3 - Q1 = 81.75 - 69, by the inclusive method. By the median-of-halves method taught in many textbooks and used by the TI-84: Q1 68, Q3 82.5, IQR 14.5.
- 12.75
- Fences Q1 - 1.5 x IQR and Q3 + 1.5 x IQR.
- 49.875 and 100.875
- Outliers Outside the 1.5 x IQR fences. A flag to check each value, not a reason to delete it.
- 48
- Count
- 12
| Percentile | Inclusive (PERCENTILE.INC) | Exclusive (PERCENTILE.EXC) | Nearest rank |
|---|---|---|---|
| 10th | 62.4 | 52.2 | 62 |
| 25th (Q1) | 69 | 67 | 66 |
| 50th (median) | 75 | 75 | 74 |
| 75th (Q3) | 81.75 | 83.25 | 81 |
| 90th | 87.6 | 91.5 | 88 |
- The methods converge as the list grows. On a dozen values they can differ by a whole gap between data points, most of all in the tails, so pick the one your course or spreadsheet uses.
- The box plot generator draws these quartiles, fences and outliers as a picture.
How the Percentile Calculator works
A percentile is the value below which a given share of the data falls. The 90th percentile of a list is a number that about 90% of the values sit below, and the percentile rank of a value is the reverse question: what share of the list falls below it. The median is the 50th percentile, and the quartiles are the 25th and 75th.
Take the default list of twelve exam scores, sorted: 48, 62, 66, 70, 71, 74, 76, 79, 81, 84, 88, 93. The score 79 has 7 values below it and 1 equal to it, so its percentile rank is (7 + 0.5) / 12 = 62.5%. That counts half of the tied values, the textbook convention, and the result shows three other conventions beside it.
The methods disagree. There is no single formula for the value at a percentile, and on small lists the common ones give different answers. For the 90th percentile of these twelve scores, the three methods give three answers.
Inclusive (Excel PERCENTILE.INC, Google Sheets, the default here) counts from 0 to position (12 - 1) x 0.9 = 9.9 and interpolates 90% of the way from 84 to 88: 87.6. Exclusive (Excel PERCENTILE.EXC, Minitab, SPSS) counts from 1 to position (12 + 1) x 0.9 = 11.7, which lands between 88 and 93: 91.5. Nearest rank takes the value at rank ceil(0.9 x 12) = 11: 88. It never interpolates.
None of them is wrong. The inclusive method is the default because it is defined for every percentile from 0 to 100; the exclusive one has no value below rank 1 or above rank n, so on twelve values it stops at the 7.69th and 92.31st.
How to find the IQR. Sort the data, find Q1 and Q3, and subtract. By the inclusive method Q1 sits at position 2.75, three quarters of the way from 66 to 70, so 69, and Q3 at position 8.25, so 81.75; the IQR is 81.75 - 69 = 12.75. Done by hand, most textbooks take the median of each half instead (66 and 70 give 68, 81 and 84 give 82.5) for an IQR of 14.5. Used as an interquartile calculator, the result shows both.
The five-number summary and outliers. As a five-number summary calculator it lists minimum, Q1, median, Q3 and maximum: 48, 69, 75, 81.75, 93. Tukey's fences sit 1.5 IQRs beyond the quartiles, at 49.875 and 100.875, so 48 is flagged as an outlier. The box plot generator draws the same five numbers as a picture.
Percentile Calculator FAQ
How do you calculate a percentile?
Sort the data. For the pth percentile by the inclusive method, find position (n - 1) x p / 100 counting the first value as 0, and interpolate between the values either side. For the 90th percentile of 12 values that is position 9.9: 90% of the way from the 10th value to the 11th.
What is the difference between PERCENTILE.INC and PERCENTILE.EXC?
Where they put the positions. PERCENTILE.INC uses (n - 1) x p from the first value and covers 0 to 100. PERCENTILE.EXC uses (n + 1) x p, which spreads the positions wider, gives more extreme values in the tails, and is undefined for percentiles too close to 0 or 100 for the list size.
How do you find the percentile rank of a score?
Count the values below it, add half the values equal to it, divide by the count and multiply by 100. In the list above, 79 has 7 values below and 1 equal, so (7 + 0.5) / 12 = 62.5%.
How do you find the IQR?
Subtract the first quartile from the third: IQR = Q3 - Q1. Sort the data, find the 25th and 75th percentiles, and take the difference. Different quartile methods can give slightly different IQRs on small data sets, so use the one your class or software uses.