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 Q3Q1 (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 Q1) is 68. The 19th value on the list (the 75th percentile, or Q3) is 89.

The IQR for the test scores data set is Q3Q1, 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).

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