Question

in the data set below, what are the lower quartile, the median, and the upper quartile? 20 34 52 52 57 59 59 71 90 97 lower quartile = median = upper quartile =

Explanation:

Step1: Arrange data in order

The data is already in ascending - order: 20, 34, 52, 52, 57, 59, 59, 71, 90, 97.

Step2: Find the median

There are \(n = 10\) data points. 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 57 and the 6th value is 59. Median\(=\frac{57 + 59}{2}=\frac{116}{2}=58\).

Step3: Find the lower half

The lower half of the data is 20, 34, 52, 52, 57.

Step4: Find the lower quartile

There are 5 data - points in the lower half. The lower quartile is the 3rd value. So, lower quartile \(=52\).

Step5: Find the upper half

The upper half of the data is 59, 71, 90, 97.

Step6: Find the upper quartile

There are 4 data - points in the upper half. The upper quartile is the average of the 2nd and 3rd values. \(\frac{71+90}{2}=\frac{161}{2}=80.5\).

Answer:

lower quartile \(=52\)
median \(=58\)
upper quartile \(=80.5\)