Question

consider the following hypothetical scores: 21, 25, 37, 38, 39, 42, 44, 45. what is the value of the interquartile range?

Explanation:

Step1: Find the median

The data set has \(n = 8\) values. The median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+ 1)\)th values. \(\frac{n}{2}=4\), \(\frac{n}{2}+1 = 5\). The median \(Q_2=\frac{38 + 39}{2}=38.5\)

Step2: Split the data into lower and upper halves

Lower - half: \(21,25,37,38\); Upper - half: \(39,42,44,45\)

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

For the lower - half (\(n_1=4\)), \(Q_1=\frac{25+37}{2}=31\)
For the upper - half (\(n_2 = 4\)), \(Q_3=\frac{42 + 44}{2}=43\)

Step4: Calculate the inter - quartile range (IQR)

\(IQR=Q_3 - Q_1\)
\(IQR=43-31\)

Answer:

\(12\)