What does percentile mean? Definition and examples

What a percentile is

The $k$th percentile is the value at or below which $k$ percent of the observations lie. Set $k$ to any number from 0 to 100 and you get a location in the distribution.

So if the 90th percentile of a college entrance test is 1306 points, then 90 percent of test takers scored 1306 or lower and about 10 percent scored higher. If the 25th percentile of house prices in a city is 310,000, a quarter of the houses cost that or less.

Percentiles describe relative standing: where one value sits inside the whole distribution. That is a different job from center or spread. The mean tells you where the middle is; a percentile tells you where you are.

Three percentiles have their own names. The 25th percentile is the first quartile $Q_1$, the 50th percentile is the median, and the 75th percentile is the third quartile $Q_3$. Everything in how to find quartiles by hand is percentile arithmetic under a different label.

A percentile is a value, not a percentage

This is the single most common confusion, and the grammar of the definition fixes it. In "the 90th percentile is 1306," the number 90 is the percent and the number 1306 is the percentile. The percentile is the value on the right.

A percentile therefore carries the data's units. Percentiles of heights are in centimeters. Percentiles of salaries are in dollars. Percentiles of wait times are in minutes. A percentage never has those units, so if the answer you wrote comes with a percent sign, you have reported the wrong quantity.

The quantity that does come as a percentage has its own name: percentile rank. It runs the definition backwards. You start with a value and ask what percent of the data sits at or below it:

So a percentile takes a percent and returns a value, and a percentile rank takes a value and returns a percent. In everyday speech people run them together and say "I am in the 85th percentile," which really means "my percentile rank is 85." That is fine in conversation and worth keeping straight on an exam. The percentile and percentile rank entries state both directions.

The at-or-below convention

Notice the phrase "at or below" in both definitions. It includes the value itself, and that choice has consequences worth knowing.

The largest value in a finite data set lands at the 100th percentile, since every observation, itself included, is at or below it. Under a strict "below" convention a unique largest value would land lower, at $\frac{n-1}{n} \times 100$ percent. This is why some score reports cap their percentile ranks at 99: they are avoiding a 100th percentile that sounds like perfection.

Repeated values are the other place the convention bites. In 70, 72, 72, 72, 80, the three 72s all share one percentile rank, $\frac{4}{5} \times 100 = 80$, because "at or below 72" counts all three of them plus the 70. Under a "below" convention they would each score 20 instead. Same data, different label, so state which rule you are using when it could matter.

A further wrinkle: software disagrees about how to interpolate a percentile that falls between two data values, and different packages return different answers for the same request. This site uses the median-excluded (TI-84) convention for quartiles. None of this changes the idea, but it can change the reported value by a whole data unit or more. On the twelve exam scores further down this page, the median-excluded convention gives $Q_1 = 65$ and $Q_3 = 83$, while the rule built into Excel and used by default in R and numpy gives 66 and 82. Name the convention whenever your answer has to match someone else's.

Percentile vs percentage score: not the same thing

A percentage score is how much of the test you got right. A percentile rank is how you did compared with everyone else. They are answers to different questions and they can point in opposite directions.

Suppose a student answers 80 percent of the items correctly on a test that most people found easy, and the class median was 84 percent. That 80 percent might correspond to a percentile rank near 40, meaning about 40 percent of test takers did the same or worse. High score, unremarkable standing.

Now flip it. On a brutal exam where the median score was 32 percent, a student who scores 45 percent correct could be above 90 percent of the field, so a percentile rank in the low 90s. Low score, excellent standing.

The rule to carry away: a percentage score depends only on you and the test, while a percentile rank depends on you, the test, and every other person in the comparison group. Change the comparison group and the percentile rank changes even though your answers did not. That is exactly why score reports name the group, such as "national percentile" or "among test takers in your grade."

Finding a percentile in a data set by hand

With a small data set, percentiles are counting. Sort the values, then count positions.

Take 12 exam scores: 55, 61, 63, 67, 70, 72, 74, 78, 81, 85, 88, 94.

To find the percentile rank of a value, count how many observations are at or below it and divide by $n$. For the score 78, there are 8 values at or below it, so the percentile rank is $\frac{8}{12} \times 100 \approx 66.7$, usually reported as about the 67th percentile.

To go the other way, from a percent to a value, use the named percentiles you already know how to compute. Here the median is $\frac{72 + 74}{2} = 73$, so the 50th percentile is 73. With the median-excluded convention, $Q_1 = \frac{63 + 67}{2} = 65$ and $Q_3 = \frac{81 + 85}{2} = 83$, so the 25th percentile is 65 and the 75th percentile is 83.

Those answers agree with the counting rule, which is a good check. Three of the 12 scores are at or below 65, and $\frac{3}{12} \times 100 = 25$. Nine are at or below 83, and $\frac{9}{12} \times 100 = 75$.

Because percentiles are positions, the ones in the interior of the distribution inherit the median's resistance: an extreme value shifts them only by moving observations across them. Percentiles out near the tails are a different matter, since the extreme value can be one of the observations sitting at that position. The 100th percentile is just the maximum, so it moves as far as the largest value does, and in a five-value data set $Q_3$ is the average of the top two values, so one huge value drags it directly. That is the same mechanism behind why the median is not affected by outliers.

Percentiles on a normal distribution

When a variable is modeled by a normal distribution, you do not need a list of values. You convert between percentiles and data values with z-scores.

A z-score measures how many standard deviations a value sits above or below the mean:

Here $\mu$ ("mu") is the mean and $\sigma$ ("sigma") is the standard deviation. The area under the standard normal curve to the left of $z$ is exactly the percentile rank, expressed as a proportion, because area to the left is the same idea as "at or below."

Going from a value to a percentile takes two steps: standardize, then look up the left-tail area in the z-table or with a calculator. Going the other way reverses them: find the z-score with that left-tail area, then convert back with $x = \mu + z\sigma$.

A few standard landmarks are worth memorizing. The mean is the 50th percentile of any normal distribution, since the curve is symmetric. One standard deviation above the mean, $z = 1$, is the 84.13th percentile, and two standard deviations above, $z = 2$, is the 97.72nd percentile. The empirical rule gives the same two landmarks in round numbers, 84 percent and 97.5 percent, because half of whatever falls outside an interval sits in each tail. See the empirical rule for where those come from, and try moving the boundary yourself in the normal curve explorer.

Common mistakes

• Reporting a percent when the question asked for a percentile. The 90th percentile of a test is a score, not "90 percent."
• Reading a percentile as a percentage score. Being at the 70th percentile does not mean answering 70 percent correctly, and the two numbers are often far apart.
• Forgetting to sort. Percentiles are positions in the sorted list, so an unsorted list gives nonsense.
• Treating a percentile as a count of people you beat. Under the at-or-below convention, everyone tied with you is counted too.
• Assuming higher is better. A high percentile is only good if high values of the variable are good. The 95th percentile of cholesterol is not something to celebrate, which is the same context-first reasoning as is a higher standard deviation better.
• Averaging percentiles. Percentile ranks are positions, not measurements, so averaging two of them does not give the percentile rank of anything. If you want a combined standing, average the raw values first and then find the percentile rank of that average.