Question

in the data set below, what are the lower quartile, the median, and the upper quartile? 16 33 35 52 55 61 75 80 94 99 lower quartile = median = upper quartile =

Explanation:

Step1: Find the median

The data set has \(n = 10\) values. The median is the average of the \(\frac{n}{2}\) - th and \((\frac{n}{2}+1)\) - th values.
\(\frac{n}{2}=\frac{10}{2} = 5\), \(\frac{n}{2}+1=6\)
The \(5\) - th value is \(55\) and the \(6\) - th value is \(61\).
Median \(M=\frac{55 + 61}{2}=\frac{116}{2}=58\)

Step2: Find the lower quartile

The lower half of the data set is \(16,33,35,52,55\). The lower quartile \(Q_1\) is the median of the lower half. Since \(n_1 = 5\) (odd), the median of the lower half is the \(\frac{n_1 + 1}{2}\) - th value. \(\frac{5+1}{2}=3\) - rd value. So \(Q_1 = 35\)

Step3: Find the upper quartile

The upper half of the data set is \(61,75,80,94,99\). The upper quartile \(Q_3\) is the median of the upper half. Since \(n_2=5\) (odd), the median of the upper half is the \(\frac{n_2 + 1}{2}\) - th value. \(\frac{5 + 1}{2}=3\) - rd value. So \(Q_3=80\)

Answer:

lower quartile \(=35\)
median \(=58\)
upper quartile \(=80\)