In statistics, percentiles are used to understand and interpret data. The *n*th percentile of a set of data is the value at which *n* percent of the data is below it. In everyday life, percentiles are used to understand values such as test scores, health indicators, and other measurements. For example, an 18-year-old male who is six and a half feet tall is in the 99th percentile for his height. This means that of all the 18-year-old males, 99 percent have a height that is equal to or less than six and a half feet. An 18-year-old male who is only five and a half feet tall, on the other hand, is in the 16th percentile for his height, meaning only 16 percent of males his age are the same height or shorter.

### Key Facts: Percentiles

• Percentiles are used to understand and interpret data. They indicate the values below which a certain percentage of the data in a data set is found.

• Percentiles can be calculated using the formula n = (P/100) x N, where P = percentile, N = number of values in a data set (sorted from smallest to largest), and n = ordinal rank of a given value.

• Percentiles are frequently used to understand test scores and biometric measurements.

## What Percentile Means

Percentiles should not be confused with percentages. The latter is used to express fractions of a whole, while percentiles are the values below which a certain percentage of the data in a data set is found. In practical terms, there is a significant difference between the two. For example, a student taking a difficult exam might earn a score of 75 percent. This means that he correctly answered every three out of four questions. A student who scores in the 75th percentile, however, has obtained a different result. This percentile means that the student earned a higher score than 75 percent of the other students who took the exam. In other words, the percentage score reflects how well the student did on the exam itself; the percentile score reflects how well he did in comparison to other students.

## Percentile Formula

Percentiles for the values in a given data set can be calculated using the formula:

n = (P/100) x N

where N = number of values in the data set, P = percentile, and n = ordinal rank of a given value (with the values in the data set sorted from smallest to largest). For example, take a class of 20 students that earned the following scores on their most recent test: 75, 77, 78, 78, 80, 81, 81, 82, 83, 84, 84, 84, 85, 87, 87, 88, 88, 88, 89, 90. These scores can be represented as a data set with 20 values: {75, 77, 78, 78, 80, 81, 81, 82, 83, 84, 84, 84, 85, 87, 87, 88, 88, 88, 89, 90}.

We can find the score that marks the 20th percentile by plugging in known values into the formula and solving for *n*:

n = (20/100) x 20

n = 4

The fourth value in the data set is the score 78. This means that 78 marks the 20th percentile; of the students in the class, 20 percent earned a score of 78 or lower.

## Deciles and Common Percentiles

Given a data set that has been ordered in increasing magnitude, the median, first quartile, and third quartile can be used split the data into four pieces. The first quartile is the point at which one-fourth of the data lies below it. The median is located exactly in the middle of the data set, with half of all the data below it. The third quartile is the place where three-fourths of the data lies below it.

The median, first quartile, and third quartile can all be stated in terms of percentiles. Since half of the data is less than the median, and one-half is equal to 50 percent, the median marks the 50th percentile. One-fourth is equal to 25 percent, so the first quartile marks the 25th percentile. The third quartile marks the 75th percentile.

Besides quartiles, a fairly common way to arrange a set of data is by deciles. Each decile includes 10 percent of the data set. This means that the first decile is the 10th percentile, the second decile is the 20th percentile, etc. Deciles provide a way to split a data set into more pieces than quartiles without splitting the set into 100 pieces as with percentiles.

## Applications of Percentiles

Percentile scores have a variety of uses. Anytime that a set of data needs to be broken into digestible chunks, percentiles are helpful. They are often used to interpret test scores—such as SAT scores—so that test-takers can compare their performance to that of other students. For example, a student might earn a score of 90 percent on an exam. That sounds pretty impressive; however, it becomes less so when a score of 90 percent corresponds to the 20th percentile, meaning only 20 percent of the class earned a score of 90 percent or lower.

Another example of percentiles is in children's growth charts. In addition to giving a physical height or weight measurement, pediatricians typically state this information in terms of a percentile score. A percentile is used in order to compare the height or weight of a child to other children of the same age. This allows for an effective means of comparison so that parents can know if their child's growth is typical or unusual.