# How to Find the Interquartile Range for a Statistical Sample

To obtain a measure of variation based on the five-number summary of a statistical sample, you can find what’s called the *interquartile range**,** *or* IQR*.

The purpose of the five-number summary is to give descriptive statistics for center, variation, and relative standing all in one shot. The measure of center in the five-number summary is the median, and the first quartile, median, and third quartiles are measures of relative standing.

The *IQR* equals *Q*_{3} – *Q*_{1} (that is, the 75th percentile minus the 25th percentile) and reflects the distance taken up by the innermost 50% of the data. If the *IQR* is small, you know the data are mostly close to the median. If the *IQR* is large, you know the data are more spread out from the median.

For example, suppose you want to find the *IQR* of the following 25 (ordered) exam scores: 43, 54, 56, 61, 62, 66, 68, 69, 69, 70, 71, 72, 77, 78, 79, 85, 87, 88, 89, 93, 95, 96, 98, 99, 99.

Counting from left to right in the data set, the 7th value (the 25th percentile, or *Q*_{1}) is 68. The 19th value on the list (the 75th percentile, or *Q*_{3}) is 89.

The *IQR* for the test scores data set is *Q*_{3} – *Q*_{1}, or 89 – 68 = 21, which is fairly large, seeing as how test scores only go from 0 to 100.

The interquartile range is a much better measure of variation than the regular range (maximum value minus minimum value). That’s because the interquartile range doesn’t take outliers into account; it cuts them out of the data set by only focusing on the distance within the middle 50 percent of the data (that is, between the 25th and 75th percentiles).