Question

swati recorded this data set, which contains an outlier.
163, 97, 184, 199, 169, 175
what numbers represent the lower and upper quartiles?
○ lower quartile: 163
upper quartile: 184
○ lower quartile: 163
upper quartile: 172
○ lower quartile: 172
upper quartile: 184
○ lower quartile: 184
upper quartile: 163

Explanation:

Step1: Sort the data set

Sort the data set \(163,97,184,199,169,175\) in ascending order: \(97,163,169,175,184,199\).

Step2: Find the median

Since there are \(n = 6\) data points, the median is the average of the \(\frac{n}{2}\) - th and \((\frac{n}{2}+1)\) - th values. The median \(Q_2=\frac{169 + 175}{2}=172\).

Step3: Find the lower half and upper half

The lower half of the data is \(97,163,169\) and the upper half is \(175,184,199\).

Step4: Find the lower quartile \(Q_1\)

For the lower half (\(n_1=3\) data points), the lower quartile \(Q_1\) is the middle value of the lower half. So \(Q_1 = 163\).

Step5: Find the upper quartile \(Q_3\)

For the upper half (\(n_2 = 3\) data points), the upper quartile \(Q_3\) is the middle value of the upper half. So \(Q_3=184\).

Answer:

lower quartile: \(163\), upper quartile: \(184\) (First option)