Percentile Calculator

What a percentile is

A percentile is a position in an ordered list, not a share of the points on a test. The NAEP glossary kept by the National Center for Education Statistics defines it as a score location below which a specified percentage of the population falls. In the glossary's own example, the 10th percentile of fourth grade reading in 1998 was 167, which means 10 percent of fourth graders scored below 167 and 90 percent scored at or above it.

The same idea runs through federal wage tables. The Bureau of Labor Statistics defines a percentile wage estimate as the value of a wage below which a certain percentage of workers fall, and publishes the 10th, 25th, 50th, 75th and 90th percentile for each occupation. The 50th percentile has a familiar name: the median.

Two directions come out of that definition, and the calculator handles both. Give it a score and it returns the rank: the percentage of values at or below that score. Give it a percentile and it returns the value sitting at that position. The first direction answers "where do I stand", and the second answers "how high is the bar".

Beyond the median, percentiles carry information an average throws away. A class where the 25th percentile is 58 and the 75th is 95 is split into two groups, and no mean captures that. That is the reason percentile columns appear in wage tables, growth charts and score reports instead of a single typical value.

Two directions: rank a score, look up a percentile

Ranking a score takes one division. Count the values at or below it, divide by the total, and multiply by 100.

Take the list 62, 70, 74, 79, 85, 88 and 91. The score 85 has five values at or below it, so the rank is 5 / 7 x 100 = 71.4, the 71.4th percentile. Two values sit above it, and with seven values each one is worth 14.3 percentile points, which is why the number carries one decimal and no more.

Going the other way, from a percentile to a value, needs a position rule. This calculator uses linear interpolation, the same rule spreadsheets call PERCENTILE.INC:

position = 1 + (n - 1) x p, where p is the percentile over 100

A whole-number position lands on a score. A fractional position sits partway between two scores, and the calculator takes that fraction of the gap. For the 90th percentile of the seven scores: position = 1 + 6 x 0.90 = 6.4, which sits 40 percent of the way from the sixth score to the seventh, so 88 + 0.4 x 3 = 89.2.

The quartile calculator uses the same position rule for the 25th, 50th and 75th percentiles, and the z score calculator turns a score into standard deviations from the mean. The nearest rank method is the other convention, and the calculator reports it next to the interpolated value. It rounds the position up to a whole rank, so the 90th percentile of seven scores is the 7th value, 91. Textbooks, software and published tables disagree by a point or two on small lists for exactly this reason, so match the method before you compare digits.

Worked example: a class of 20 scores

Twenty scores, already sorted: 41, 44, 47, 49, 52, 53, 55, 57, 58, 60, 62, 63, 65, 67, 70, 72, 75, 78, 84, 91. With n = 20, each score is worth exactly 5 percentile points, which makes the arithmetic easy to check by hand.

Read the first row slowly, because it is the one people get wrong. The score 72 has sixteen values at or below it: the fifteen scores under 72, plus 72 itself. Sixteen out of twenty is 80 percent, so 72 sits at the 80th percentile of this class. The four scores above it, 75, 78, 84 and 91, are the top 20 percent.

The position rule and the count rule agree here by construction. Position 5.75 for the 25th percentile means the value sits three quarters of the way from the 5th score to the 6th, and both scores bracket it, 52 and 53. Half of the class lives between 52.75 and 70.5, a 17.75 point band, while the full spread runs 50 points from 41 to 91.

Watch what interpolation does at the edges. The 90th percentile, 78.6, is a value that never appears in the data, and that is normal. A percentile of a discrete list is allowed to fall between two observations, the same way an average of 10 and 11 is allowed to be 10.5.

Why one person can be worth 10 percentile points

Percentile points are shares of a group, not fixed units. In a class of 10, one person occupies 10 points. In a class of 20, one person is 5 points. In a group of 100, one person moves the needle by 1 point, and in a national sample of thousands, one person barely registers.

This is why a percentile from a small list should be reported with its sample size. The 71.4th percentile of seven scores sounds precise, yet one more score below 85 gives six of eight values at or below, which is 75, and one more score above 85 gives five of eight, which is 62.5. With n = 7 the rank moves in steps of 14.3 points, so quoting a second decimal implies a stability the data does not have.

Ties complicate the count. Take five scores, 60, 70, 70, 70 and 90, and rank the score 70. Three values match it. Counting everything at or below puts 4 of 5 at or below, a rank of 80, yet 70 is obviously not in the top fifth of this group. The midrank rule fixes that by placing the score at the center of its own block: below 70 there is 1 value and beside it 3, so the rank is (1 + 1.5) / 5 x 100 = 50. The calculator reports both conventions, because the at-or-below count matches how agencies phrase percentiles and the midrank is fairer to tied scores.

Percentiles in public data

Federal agencies publish percentiles constantly, and three examples show the range of uses.

Wages. The percentile wage estimates published by the Bureau of Labor Statistics cover about 830 occupations. Its worked example reads: the hourly 10th percentile is $11.00, the 25th is $15.00, the median is $20.00, the 75th is $24.00 and the 90th is $29.00. Ten percent of workers earn less than $11.00 an hour and 90 percent earn more. Eighty percent of workers earn between the 10th and 90th percentile, which is the band job seekers face.

Health screening. The blood lead reference value that the CDC uses for children is 3.5 micrograms per deciliter, set in 2021 from the 97.5th percentile of blood lead levels among U.S. children ages 1 to 5 in the 2015 to 2018 NHANES survey cycles. The definition is pure percentile language: children at or above the value are the top 2.5 percent. The CDC is careful to add that the value is a screening tool, not a health-based standard, which is a fair description of what any percentile can claim.

Income. The Census Bureau reported median household income of $83,730 for 2024 in its Income in the United States: 2024 report, and the median is the 50th percentile by definition: half of households fall below it and half above. The same report tracks income at the 90th percentile, which rose 4.2 percent from 2023 to 2024, while the 10th and 50th percentiles did not change significantly. That one sentence says more about the shape of 2024 income growth than any average could.

What a percentile cannot tell you

A percentile is a rank, and ranks have limits that averages do not.

It is not a percent correct. A student at the 71st percentile did not answer 71 percent of questions right. The 71 says where the score sits in a group, and the group could have found the test easy or brutal. The two numbers answer different questions and only one of them appears on the score report.

The gaps are not equal. The distance from the 50th to the 60th percentile is usually a small slice of the score scale, while the distance from the 90th to the 99th is a large one, because most people pile up in the middle. On a bell shaped distribution the 84th percentile sits about one standard deviation above the mean, and so does the 16th below it, which no equal-spacing reading of percentiles would suggest.

Percentiles cannot be averaged. The average of the 90th percentile and the 10th percentile is not the 50th, and averaging a student's percentile ranks across subjects produces a number that describes no group. Convert back to scores or use a scale built for averaging before combining, such as the IQ percentile converter for age-scaled scores.

The reference group is the whole meaning. A 90th percentile against a national norm can be a 60th percentile against a selective school's norm, with the same raw score. Every published percentile names its group somewhere in a footnote, and the footnote is not optional.

How to report a percentile

Report a percentile with the same three facts that give it meaning: the group, the size and the method.

For the worked list above, the honest sentence is: "85 sits at the 71.4th percentile of seven scores, counting values at or below, using linear interpolation for lookups." Drop any clause and the number loses its check. A rank without the group is unanswerable, and a rank from seven values should be read as "about the 70th percentile", because one new score moves it 14 points.

Round to match the data. With 20 scores, one decimal is defensible, since each score is worth exactly 5 points. With 7 scores, round to the nearest 5. With a national sample, one decimal is conservative. The digits should never promise more stability than the group size supports.

Name the convention when ties exist. The at-or-below count and the midrank put the same tied score at the 80th and 50th percentile of the same five values, so a table that mixes the two conventions without saying so will look inconsistent when it is not.

Finally, say which question the percentile answers. Ranking answers "where does this score sit". Lookup answers "what score marks this position". The calculator returns both for the same list, plus the median and quartiles, so the number in a report can be traced back to the list it came from in one screen.