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 first number in the ordered list, and the maximum is the last number.
Minimum: \(56\)
Maximum: \(92\)

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. \(\frac{12}{2}=6\), \(\frac{12}{2}+1 = 7\). The \(6\)th value is \(74\) and the \(7\)th value is \(84\).
Median: \(\frac{74 + 84}{2}=\frac{158}{2}=79\)

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

The lower half of the data is \(56,60,62,70,72,74\). Since \(n_1=6\) (number of data points in the lower half), \(Q_1\) is the average of the \(\frac{n_1}{2}\)th and \((\frac{n_1}{2}+1)\)th values. \(\frac{6}{2}=3\), \(\frac{6}{2}+1 = 4\). 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 is \(84,87,87,91,92\). Since \(n_2 = 6\) (number of data points in the upper half), \(Q_3\) is the average of the \(\frac{n_2}{2}\)th and \((\frac{n_2}{2}+1)\)th values. \(\frac{6}{2}=3\), \(\frac{6}{2}+1=4\). 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)

\(IQR=Q_3 - Q_1\)
\(IQR=89-66 = 23\)

Answer:

Minimum: \(56\)
Lower quartile: \(66\)
Median: \(79\)
Upper quartile: \(89\)
Maximum: \(92\)
Interquartile range: \(23\)