2.2 Quartiles and Percentiles

In addition to the measures of centre, some other measures can be used to describe a distribution such as quartiles and percentiles. Recall that the median is the middle value that divides the sorted data into two halves with an equal number of observations; that is, 50% of the observations are below the median, and another 50% are above the median. Similarly, quartiles and percentiles of a distribution are defined as follows:

• Quartiles are the three values that divide the sorted data into four parts with an equal number of observations, denoted as [latex]Q_1, Q_2, Q_3[/latex]. Each part contains 25% of the data. Actually, the second quartile [latex]Q_2[/latex] is the median of the entire data set; the first quartile [latex]Q_1[/latex] is the median of the bottom 50% (first half) and the third quartile [latex]Q_3[/latex] is the median of the top 50% (second half). Note that when the number of observations [latex]n[/latex] is odd, we include the median in both the first half and the second half when calculating [latex]Q_1[/latex] and [latex]Q_3[/latex].
• Percentiles are those 99 values that divide the sorted data into 100 parts with an equal number of observations. Each part contains 1% of the data. The first quartile [latex]Q_1[/latex] is the 25th percentile, the second quartile [latex]Q_2[/latex] (median) is the 50th percentile, and the third quartile [latex]Q_3[/latex] is the 75th percentile. In this course, we will not calculate percentiles by hand, except for the important special cases of quartiles. The software can calculate any arbitrary percentiles for us.