Question

the following are the ages of 12 music teachers in a school district.
26, 28, 32, 34, 35, 36, 38, 38, 45, 45, 57, 60
notice that the ages 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, and the maximum is the largest value.
Minimum: \(26\)
Maximum: \(60\)

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.
\(\frac{n}{2}=6\), \(\frac{n}{2}+1 = 7\)
The \(6\)th value is \(36\) and the \(7\)th value is \(38\)
Median: \(\frac{36 + 38}{2}=\frac{74}{2}=37\)

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

The lower half of the data is \(26,28,32,34,35,36\). Since \(n_1=6\) (number of data points in the lower half), \(Q_1\) is the average of the \(3\)rd and \(4\)th values.
The \(3\)rd value is \(32\) and the \(4\)th value is \(34\)
\(Q_1=\frac{32+34}{2}=\frac{66}{2} = 33\)

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

The upper half of the data is \(38,45,45,57,60\). Since \(n_2 = 6\) (number of data points in the upper half), \(Q_3\) is the average of the \(3\)rd and \(4\)th values.
The \(3\)rd value is \(45\) and the \(4\)th value is \(57\)
\(Q_3=\frac{45 + 57}{2}=\frac{102}{2}=51\)

Step5: Calculate the inter - quartile range (IQR)

\(IQR=Q_3 - Q_1\)
\(IQR=51-33 = 18\)

Answer:

Minimum: \(26\)
Lower quartile: \(33\)
Median: \(37\)
Upper quartile: \(51\)
Maximum: \(60\)
Interquartile range: \(18\)