The *n*th percentile of a set of data is the value at which *n*% of the data is below it. Percentiles generalize the idea of a quartile and allow us to split our data set into many pieces. We will examine percentiles and learn more about their connections to other topics in statistics.

### Quartiles and 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 of 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%, we could call the median the 50th percentile. One-fourth is equal to 25%, and so the first quartile the 25th percentile. Similarly, the third quartile is the same as the 75th percentile.

### An Example of a Percentile

A class of 20 students had 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. The score of 80% has four scores below it. Since 4/20 = 20%, 80 is the 20th percentile of the class. The score of 90 has 19 scores below it.

Since 19/20 = 95%, 90 corresponds to the 95 percentile of the class.

### Percentile vs. Percentage

Be careful with the words percentile and percentage. A percentage score indicates the proportion of a test that someone has completed correctly. A percentile score tells us what percent of other scores are less than the data point we are investigating.

As seen in the above example these numbers are rarely the same.

### Deciles and Percentiles

Besides quartiles, a fairly common way to arrange a set of data is by deciles. A decile has the same root word as decimal and so it makes sense that each decile serves as a demarcation of 10% of a set of data. This means that the first decile is the 10th percentile. The second decile is the 20th percentile. Deciles provide a way to split a data set into more pieces than quartiles without splitting it 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. One common application of percentiles is for use with tests, such as the SAT, to serve as a basis of comparison for those who took the test. In the above example, a score of 80% initially sounds good. However, this does not sound as impressive when we find out that it is the 20th percentile - only 20% of the class scored less than an 80% on the test.

Another example of percentiles being used is in children's growth charts. In addition to a physical height or weight measurement, pediatricians typically state this in terms of a percentile score.

A percentile is used in this situation in order to compare the height or weight of a given child to all children of that age. This allows for an effective means of comparison.