Question

in the data set below, what are the lower quartile, the median, and the upper quartile? 32 32 36 36 43 45 45 59 74 78 lower quartile = median = upper quartile =

Explanation:

Step1: Arrange data in order

The data set 32, 32, 36, 36, 43, 45, 45, 59, 74, 78 is already in ascending - order. There are \(n = 10\) data points.

Step2: Find the median

Since \(n = 10\) (an even - numbered data set), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+1)\)th values. \(\frac{n}{2}=5\) and \(\frac{n}{2}+1 = 6\). The 5th value is 43 and the 6th value is 45. Median\(=\frac{43 + 45}{2}=44\).

Step3: Find the lower half and upper half

The lower half of the data set is 32, 32, 36, 36, 43. The upper half of the data set is 45, 59, 74, 78.

Step4: Find the lower quartile

The lower quartile (\(Q_1\)) is the median of the lower half. Since there are 5 data points in the lower half (an odd - numbered data set), the median of the lower half is the 3rd value. So \(Q_1 = 36\).

Step5: Find the upper quartile

The upper quartile (\(Q_3\)) is the median of the upper half. Since there are 5 data points in the upper half (an odd - numbered data set), the median of the upper half is the 3rd value. So \(Q_3=59\).

Answer:

lower quartile = 36
median = 44
upper quartile = 59