Question

question #3
find the interquartile range for the following set of data:
72 67 69 78 65 67 66 70
73 73 71 71 70 77 67 67
3.5
9.5
5.5
4.5

question #4
find the standard deviation of the following set of sample data:
14 13 13 17 16 15 18
19 12 17 16 13 17 13
18 16 17
2.71
2.15
3.04
3.3

Explanation:

Step1: Sort the data

Sort the data set \(72,67,69,78,65,67,66,70,73,73,71,71,70,77,67,67\) in ascending order: \(65,66,67,67,67,67,69,70,70,71,71,72,73,73,77,78\)

Step2: Find the median

Since there are \(n = 16\) data points, the median is the average of the \(8^{th}\) and \(9^{th}\) values. The \(8^{th}\) value is \(70\) and the \(9^{th}\) value is \(70\), so the median \(Q_2=70\)

Step3: Find \(Q_1\) and \(Q_3\)

The lower half of the data is \(65,66,67,67,67,67,69,70\). There are \(n_1 = 8\) data points. The median of the lower half (\(Q_1\)) is the average of the \(4^{th}\) and \(5^{th}\) values. The \(4^{th}\) value is \(67\) and the \(5^{th}\) value is \(67\), so \(Q_1 = 67\)
The upper half of the data is \(70,71,71,72,73,73,77,78\). There are \(n_2=8\) data points. The median of the upper half (\(Q_3\)) is the average of the \(4^{th}\) and \(5^{th}\) values. The \(4^{th}\) value is \(72\) and the \(5^{th}\) value is \(73\), so \(Q_3=\frac{72 + 73}{2}=72.5\)

Step4: Calculate the inter - quartile range (IQR)

The formula for \(IQR\) is \(IQR=Q_3 - Q_1\). Substitute \(Q_1 = 67\) and \(Q_3=72.5\) into the formula: \(IQR=72.5-67 = 5.5\)

Answer:

\(5.5\)