Question

here are the scores of 12 students on a geography test.
56, 60, 62, 70, 72, 74, 84, 84, 87, 87, 91, 92
notice that the scores are ordered from least to greatest.
give the five - number summary and the interquartile range for the data set.
five - number summary
minimum:
lower quartile:
median:
upper quartile:
maximum:
interquartile range:

Explanation:

Step1: Find the minimum and maximum

The minimum is the smallest value in the data set, which is \(56\). The maximum is the largest value, which is \(92\).

Step2: Find the median

Since there are \(n = 12\) data points. The median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+ 1)\)th values. \(\frac{12}{2}=6\) and \(\frac{12}{2}+1 = 7\). The \(6\)th value is \(74\) and the \(7\)th value is \(84\). Median \(M=\frac{74 + 84}{2}=\frac{158}{2}=81\).

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

The lower half of the data set is \(56,60,62,70,72,74\). Since \(n_1=6\) (number of data points in the lower half), the lower quartile \(Q_1\) is the average of the \(\frac{n_1}{2}\)th and \((\frac{n_1}{2}+1)\)th values. \(\frac{6}{2}=3\) and \(\frac{6}{2}+1 = 4\). The \(3\)rd value is \(62\) and the \(4\)th value is \(70\). \(Q_1=\frac{62+70}{2}=\frac{132}{2}=66\).

Step4: Find the upper quartile (\(Q_3\))

The upper half of the data set is \(84,87,87,91,92\). Since \(n_2 = 6\) (number of data points in the upper half), the upper quartile \(Q_3\) is the average of the \(\frac{n_2}{2}\)th and \((\frac{n_2}{2}+1)\)th values. \(\frac{6}{2}=3\) and \(\frac{6}{2}+1=4\). The \(3\)rd value is \(87\) and the \(4\)th value is \(91\). \(Q_3=\frac{87 + 91}{2}=\frac{178}{2}=89\).

Step5: Calculate the inter - quartile range (IQR)

The inter - quartile range \(IQR=Q_3 - Q_1\). \(IQR=89-66 = 23\).

Answer:

Minimum: \(56\)
Lower quartile: \(66\)
Median: \(81\)
Upper quartile: \(89\)
Maximum: \(92\)
Interquartile range: \(23\)