Question

find the 5 number summary for the data shown below. 53 59 92 21 89 58 34 19 78 72 5 number summary: , , , ,

Explanation:

Step1: Sort the data

First, we list all the data points: \(53,59,92,21,89,58,34,19,78,72\).
Sorting them in ascending order: \(19,21,34,53,58,59,72,78,89,92\).

Step2: Find the minimum

The minimum value is the first - element of the sorted list. So, \(min = 19\).

Step3: Find the first quartile (\(Q_1\))

Since \(n = 10\) (the number of data points), the position of \(Q_1\) is \(\frac{n + 1}{4}=\frac{10+1}{4}=2.75\).
Using the formula \(Q_1=x_{2}+0.75(x_{3}-x_{2})\), where \(x_{2}=21\) and \(x_{3}=34\).
\(Q_1=21 + 0.75\times(34 - 21)=21+0.75\times13=21 + 9.75=30.75\).

Step4: Find the median (\(Q_2\))

The position of the median for \(n = 10\) (even number of data points) is \(\frac{n}{2}=5\) and \(\frac{n}{2}+1 = 6\).
The median \(Q_2=\frac{x_{5}+x_{6}}{2}\), where \(x_{5}=58\) and \(x_{6}=59\). So, \(Q_2=\frac{58 + 59}{2}=\frac{117}{2}=58.5\).

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

The position of \(Q_3\) is \(\frac{3(n + 1)}{4}=\frac{3\times(10 + 1)}{4}=8.25\).
Using the formula \(Q_3=x_{8}+0.25(x_{9}-x_{8})\), where \(x_{8}=78\) and \(x_{9}=89\).
\(Q_3=78+0.25\times(89 - 78)=78 + 0.25\times11=78+2.75 = 80.75\).

Step6: Find the maximum

The maximum value is the last - element of the sorted list. So, \(max=92\).

Answer:

\(19,30.75,58.5,80.75,92\)