Percentile Calculator

A percentile calculator answers two mirror-image questions: "which value sits at the kth percentile?" and "what percentile is this value?", the percentile rank. This page handles both from raw data, and adds a third route for when you only know the mean and standard deviation of roughly bell-shaped data.

There is no single official formula. In 1996 the statisticians Hyndman and Fan catalogued nine sample percentile definitions used by statistical software, and spreadsheets and textbooks still disagree. The calculator shows the three you are most likely to be asked for side by side, so you can match your course, your spreadsheet or your report.

What a percentile tells you

The kth percentile is a cut point: roughly k percent of the values fall at or below it. The 50th percentile is the median, the 25th and 75th are the quartiles, and the 90th marks the start of the top tenth. Because a percentile depends only on the order of the values, one extreme number cannot drag it around the way it drags an average, which is why growth charts, test reports and response-time dashboards speak in percentiles. A percentile describes position, not quality: the 60th percentile of a strong group can beat the 90th percentile of a weak one.

The three percentile formulas

All three methods begin the same way: sort the n values from smallest to largest, number them 1 to n, and write the percentile as a proportion, p = k ÷ 100. They differ only in how they turn p into a position in the sorted list:

When h is a whole number, both interpolating methods simply return the value at that position. When h has a fractional part, the answer lies that fraction of the way from one value to the next. Nearest rank skips interpolation entirely and rounds the position up, so its answer is always an actual observation.

Worked example: the 90th percentile of twelve exam scores

Take twelve exam scores, already sorted: 48, 52, 55, 61, 64, 67, 70, 73, 77, 82, 88, 95. We want the 90th percentile, so p = 0.9 and n = 12.

Three defensible answers from one data set: 87.4, 88 and 92.9. Twelve values leave big gaps near the top of the list, and PERCENTILE.EXC reaches furthest toward the maximum because its positions run on a scale of n + 1 = 13 slots instead of n − 1 = 11 gaps. Load the "Exam scores (12)" example and set k to 90 to see every step.

The three methods compared at common percentiles

Each method applied to the same twelve scores. The calculator reproduces every figure in its "Common percentiles of your data" card.

The exclusive method cannot reach the 5th or 95th percentile with twelve values, because its valid range is 100 ÷ 13 = 7.69 to 1,200 ÷ 13 = 92.31. Nearest rank reports 67 rather than 68.5 as the median, since it never averages the two middle values of an even-sized set.

How to find a percentile by hand

1. Sort the data from smallest to largest and count the values, n.
2. Convert the percentile to a proportion: the 80th percentile becomes p = 0.8.
3. Compute the position for your method, for example h = (n − 1) × p + 1 for PERCENTILE.INC.
4. If the position is a whole number, the answer is the value in that spot.
5. Otherwise split h into a whole part and a fraction, and move that fraction of the way from the value at the whole-number position to the next value.

For the 80th percentile of the twelve scores, PERCENTILE.INC gives h = 11 × 0.8 + 1 = 9.8, so you go 0.8 of the way from the 9th value (77) to the 10th (82): 77 + 0.8 × 5 = 81.

Percentile rank: the reverse question

A percentile rank calculator starts from a value and reports where it stands. Count the values strictly below it (B) and the values equal to it (E), then apply the midpoint formula:

In the twelve scores, the value 73 has B = 7 values below it (48 through 70) and E = 1 copy of itself, so PR = (7 + 0.5) ÷ 12 × 100 = 62.5. The share strictly below is 58.33% and the share at or below is 66.67%; the midpoint formula splits the difference so a value is not credited for outscoring itself. Excel answers on a 0 to 1 scale: PERCENTRANK.INC returns B ÷ (n − 1) = 7 ÷ 11 = 0.6364, displayed as 0.636, and PERCENTRANK.EXC returns (B + 1) ÷ (n + 1) = 8 ÷ 13 = 0.6154, displayed as 0.615.

Worked example: a percentile from the mean and standard deviation

Sometimes you have no raw data, only a summary: a test with a mean of 72 and a standard deviation of 8, with scores that follow a roughly normal curve. To find the percentile of a score of 84, standardize it and read the normal cumulative distribution:

The reverse direction inverts the normal CDF: the 10th percentile sits 1.2816 standard deviations below the mean, at 61.75, and by symmetry the 90th sits at 82.25. The CDF has no closed form, so the calculator uses a double-precision approximation. The tables below come from the same function: z-score to percentile on the left, percentile to z-score on the right.

The percentiles are unevenly spaced: moving from the 50th to the 60th percentile takes 0.2533 standard deviations, while moving from the 90th to the 99th takes 1.0448. Near the middle of a bell curve a small change in score passes many people; in the tails it takes a large change to pass a few.

Which percentile method should you use?

Match the tool your audience will check against. Excel's PERCENTILE.INC, the older PERCENTILE function it replaced, and R's default quantile type all use linear interpolation, so it is the headline method here. The NIST/SEMATECH e-Handbook of Statistical Methods describes the p(n + 1) rule that PERCENTILE.EXC uses, but clamps to the minimum and maximum at the ends where Excel returns an error. Some courses teach nearest rank, and NIST also lists a textbook variant that averages two values when p × n is a whole number; the calculator mentions that variant whenever it differs. On large data sets all methods converge, so the choice matters most for small samples.

How to use this calculator

Pick a mode with the three tabs. In kth percentile mode, paste your numbers (commas, spaces and line breaks all work), type k or tap a quick button such as P90, and choose which method fills the big result. In Percentile rank mode, enter the value to rank; it does not need to appear in the list. In Mean & SD mode, enter the mean and standard deviation and switch between "value → percentile" and "percentile → value". The sorted table at the bottom labels every value with its own percentile rank. Students use it to check homework against an answer key, analysts to summarize skewed data such as response times, and parents and test takers to read a score report.

Key terms

• kth (or nth) percentile: the value below which about k percent of the data falls.
• Percentile rank: the percentage of values below a given value, usually with ties split in half.
• Position (h): the possibly fractional place in the sorted list where the percentile sits.
• Interpolation: estimating a value between two neighbors in proportion to the fractional part of h.
• Quartiles and deciles: the 25th, 50th and 75th percentiles, and the 10th, 20th and so on.
• z-score: how many standard deviations a value sits from the mean, (x − μ) ÷ σ.
• Normal CDF, Φ(z): the share of a normal distribution below z.

What changes a percentile, and reporting tips

• The method: on twelve values the 90th percentile ranges from 87.4 to 92.9.
• The sample size: small n leaves large gaps between neighbors, so interpolation choices swing the answer.
• Ties: many identical values make several percentiles collapse onto one number.
• Values near the target position: changing the largest value moves P95 on a small set but leaves P25 untouched.
• Tip: name the method next to the number, for example "P90 = 87.4 (PERCENTILE.INC)".
• Tip: round to the precision of the raw data, and say whether a percentile rank counts ties as below, at or halfway.

Limitations

A sample percentile estimates a population percentile, and on small samples it is noisy: one extra observation can move P90 by several points, and no confidence interval is reported. The mean and SD mode is only as good as its normal assumption; for skewed data such as incomes or waiting times, rank the raw values instead. Percentile ranks are ordinal, so equal percentile gaps do not mean equal gaps in the underlying values.

Related calculators

• For Q1, Q3, the interquartile range and an outlier test, use the Quartile Calculator.
• To turn a single value into a z-score and a normal-curve probability, use the Z-Score Calculator.
• For mean, median, mode, range and spread in one pass, use the Statistics Calculator.
• To get the standard deviation that the normal mode needs, use the Standard Deviation Calculator.
• For children's growth percentiles from CDC growth charts, see the Height Percentile Calculator and the Weight Percentile Calculator.
• For z critical values at a chosen confidence level, use the Critical Value Calculator.

Sources

• NIST/SEMATECH e-Handbook of Statistical Methods, section 7.2.6.2, Percentiles: the p(n + 1) rule, the 1 + p(n − 1) rule used by Excel and the textbook method.
• NIST/SEMATECH e-Handbook of Statistical Methods, section 1.3.6.6.1, Normal Distribution: the normal density and its CDF.
• Microsoft Support, PERCENTILE.INC function and PERCENTILE.EXC function: the k ranges and the #NUM! conditions.
• Microsoft Support, PERCENTRANK.INC function and PERCENTRANK.EXC function: the three-digit default and the worked examples.
• Hyndman, R. J., and Fan, Y. (1996). Sample Quantiles in Statistical Packages. The American Statistician, 50(4), 361 to 365: the nine definitions, including types 1, 6 and 7.