We are given the following class scores on a mathematics test, arranged in order from lowest to highest scores:
48, 49, 54, 55, 65, 76, 77, 78, 78, 81, 83, 85, 88,
89, 89, 89, 90, 91, 93, 96, 99To find the quartiles, the first step we need to do is find the median of the data. Since 83 is in the middle of the data, 83 is our median. 83 is also called the second quartile, or Q2, of our data, since two-fourths of the data lie below this value.
Next, we look at the data that lie below 83 (our median, or Q2). The median of only these first 10 test scores is the first quartile, or Q1, of our data. Since 65 and 76 both lie in the middle of the first 10 test scores, we average them to get 70.5 as our Q1. This means that approximately one-fourth of our data lie below 70.5.
Finally, we look at the data that lie above 83 (our median, or Q2). The median of only these last 10 test scores is the third quartile, or Q3, of our data. Since 89 and 90 both lie in the middle of the last 10 test scores, we average them to get 89.5 as our Q3. This means that approximately three-fourths of our data lie below 89.5.
Note that while we only have three quartile values -- Q1, Q2, and Q3 -- these three numbers split our data into four groups of approximately the same number of data. Thus the term quartile.