Question

find the 5 number summary for the data shown below.
57 18
51 96
97 75
76 61
70 86
5 number summary:
□, □, □, □, □
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Explanation:

Step1: Sort the data

First, we list all the data points: \(57,18,51,96,97,75,76,61,70,86\).
Sorting them in ascending order: \(18,51,57,61,70,75,76,86,96,97\).

Step2: Find the minimum

The minimum value is the first - element of the sorted list. So, \(min = 18\).

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

The formula for the position of \(Q_1\) is \(i=\frac{n + 1}{4}\), where \(n = 10\) (the number of data points).
\(i=\frac{10+1}{4}=2.75\).
Using linear interpolation: \(Q_1=51+(57 - 51)\times0.75=51 + 4.5=55.5\).

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

The formula for the position of the median is \(i=\frac{n + 1}{2}\). For \(n = 10\), \(i=\frac{10 + 1}{2}=5.5\).
Using linear interpolation: \(Q_2=70+(75 - 70)\times0.5=70+2.5 = 72.5\).

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

The formula for the position of \(Q_3\) is \(i=\frac{3(n + 1)}{4}\). For \(n = 10\), \(i=\frac{3\times(10 + 1)}{4}=8.25\).
Using linear interpolation: \(Q_3=86+(96 - 86)\times0.25=86 + 2.5=88.5\).

Step6: Find the maximum

The maximum value is the last - element of the sorted list. So, \(max=97\).

Answer:

\(18\), \(55.5\), \(72.5\), \(88.5\), \(97\)