Question

the data represents the number of runs allowed by 8 college softball pitchers.
{18, 49, 38, 41, 33, 44, 42, 22}
what is the five - number summary?
minimum = 18
q₁ = 27.5
median = 39.5
q₃ = 43
maximum = 49
minimum = 18
q₁ = 28.5
median = 38.5
q₃ = 42.5
maximum = 49
minimum = 18
q₁ = 33
median = 41
q₃ = 44
maximum = 49
minimum = 18
q₁ = 33
median = 40
q₃ = 41.5
maximum = 49

Explanation:

Step1: Order the data

First, we order the data set \(\{18, 49, 38, 41, 33, 44, 42, 22\}\) from smallest to largest: \(18, 22, 33, 38, 41, 42, 44, 49\).

Step2: Find the minimum and maximum

The minimum value is the smallest number, which is \(18\). The maximum value is the largest number, which is \(49\).

Step3: Find the median (second quartile, \(Q_2\))

Since there are \(n = 8\) (even number of data points), the median is the average of the \(\frac{n}{2}\)-th and \((\frac{n}{2}+1)\)-th values.
\(\frac{n}{2}=\frac{8}{2} = 4\)-th value and \((\frac{n}{2}+1)=5\)-th value.
The 4th value is \(38\) and the 5th value is \(41\).
Median \(=\frac{38 + 41}{2}=\frac{79}{2}=39.5\).

Step4: Find the first quartile (\(Q_1\))

The first quartile is the median of the lower half of the data. The lower half of the data (values below the median) is \(\{18, 22, 33, 38\}\) (since \(n = 8\), lower half has \(4\) values).
For \(n = 4\) (even), the median of the lower half is the average of the 2nd and 3rd values.
2nd value: \(22\), 3rd value: \(33\).
\(Q_1=\frac{22 + 33}{2}=\frac{55}{2}=27.5\).

Step5: Find the third quartile (\(Q_3\))

The third quartile is the median of the upper half of the data. The upper half of the data (values above the median) is \(\{41, 42, 44, 49\}\) (since \(n = 8\), upper half has \(4\) values).
For \(n = 4\) (even), the median of the upper half is the average of the 2nd and 3rd values.
2nd value: \(42\), 3rd value: \(44\).
\(Q_3=\frac{42 + 44}{2}=\frac{86}{2}=43\).

Answer:

Minimum \(= 18\), \(Q_1 = 27.5\), Median \(= 39.5\), \(Q_3 = 43\), Maximum \(= 49\) (corresponding to the first option set)