Question

1. refer to the data: 2, 2, 4, 4, 5, 5, 6, 7, 9, 15.

a. what is the five - number summary for the data?
b. when the maximum, 15, is removed from the data set, what is the five - number summary?

Explanation:

Step1: Sort the data

The sorted data set is 2, 2, 4, 4, 5, 5, 6, 7, 9, 15.

Step2: Find the minimum

The minimum value is 2.

Step3: Find the first - quartile ($Q_1$)

There are $n = 10$ data points. The position of $Q_1$ is $0.25(n + 1)=0.25\times(10 + 1)=2.75$. So $Q_1=4$.

Step4: Find the median ($Q_2$)

The position of the median is $\frac{n+1}{2}=\frac{10 + 1}{2}=5.5$. So $Q_2=\frac{5 + 5}{2}=5$.

Step5: Find the third - quartile ($Q_3$)

The position of $Q_3$ is $0.75(n + 1)=0.75\times(10+1)=8.25$. So $Q_3 = 7$.

Step6: Find the maximum

The maximum value is 15.

Step7: Remove the maximum and repeat steps

After removing 15, the sorted data set is 2, 2, 4, 4, 5, 5, 6, 7, 9. There are $n = 9$ data points.

• Minimum: 2
• $Q_1$: The position of $Q_1$ is $0.25(n + 1)=0.25\times(9 + 1)=2.5$. So $Q_1 = 4$.
• Median ($Q_2$): The position of the median is $\frac{n + 1}{2}=\frac{9+1}{2}=5$. So $Q_2 = 5$.
• $Q_3$: The position of $Q_3$ is $0.75(n + 1)=0.75\times(9 + 1)=7.5$. So $Q_3=6$.
• Maximum: 9

Answer:

a. Minimum = 2, $Q_1$ = 4, Median = 5, $Q_3$ = 7, Maximum = 15
b. Minimum = 2, $Q_1$ = 4, Median = 5, $Q_3$ = 6, Maximum = 9