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.
Example: Find the Quartiles
Find the quartiles for 3, 1, 9, 7, 5, 11, 21
Steps:
 Sort into 1, 3, 5, 7, 9, 11, 21.
 n = 7 is odd, [latex]Q_2[/latex] = median = 7.
 The bottom half consists of the first three smallest observations and the median, i.e., 1, 3, 5, 7. [latex]Q_1[/latex] is the median of the first half, i.e., [latex]Q_1 = \frac{3+5}{2} = 4[/latex].
 The top half consists of the three largest values and the median, i.e., 7, 9, 11, 21. [latex]Q_3[/latex] is the median of the second half, [latex]Q_3 = \frac{9+11}{2} = 10[/latex].
Therefore, the quartiles are [latex]Q_1 = 4, Q_2=7, Q_3=10[/latex].
Note: since the number of observations [latex]n=7[/latex] which is odd, we include the median [latex]Q_2=7[/latex] in both the first half {1, 3, 5, 7} and the second half {7, 9, 11, 21}.
Exercise: Find the Quartiles
Find the quartiles for the data 3, 1, 9, 7, 5, 11, 21, 19.
Show/Hide Answer
Steps:

 Sort into 1, 3, 5, 7, 9, 11, 19, 21.
 n=8 is even, [latex]Q_2[/latex] = median = [latex]\frac{7+9}{2} = 8[/latex].
 The bottom half is the first four observations in the sorted list 1, 3, 5, 7, and [latex]Q_1[/latex] is the median of the first half, i.e., [latex]Q_1 = \frac{3+5}{2} = 4[/latex].
 The top half is the last four observations 9, 11, 19, 21, and [latex]Q_3 = \frac{11+19}{2} = 15[/latex] is the median of the second half.
Therefore, the quartiles are [latex]Q_1 =4, Q_2=8, Q_3=15[/latex].