Chapter 13.2, Problem 21E

To determine

To find: The measure of central tendency that best describes the test scores.

Answer to Problem 21E

The median best describes the test scores, based on the explanation given below.

Explanation of Solution

Given:

The scores of students are 99, 64, 80, 77, 59, 72, 87, 79, 92, 88, 90, 42, 20, 89, 42, 100, 98, 84, 78 and 91.

Calculation:

The measures of central tendencies and their values are listed below:

1) The mean, $\overline{x}$ , is calculated using the formula,

  $\overline{x}=\frac{\text{sum of all observations}}{\text{number of observations}}$

Substitute the required values and determine the mean as shown below:

  $\begin{array}{c}\overline{x}=\frac{\left(\begin{array}{l}99+64+80+77+59+72+87+79\\ +92+88+90+42+20+89+42\\ +100+98+84+78+91\end{array}\right)}{20}\\ =76.55\end{array}$

2) The median, M, is calculated using the formula,

  $\begin{array}{c}M=\frac{\left(\frac{n}{2}\right)\text{th observation}+\left(\frac{n}{2}+1\right)\text{th observation}}{2}\\ =\frac{\left(\frac{20}{2}\right)\text{th observation}+\left(\frac{20}{2}+1\right)\text{th observation}}{2}\\ =\frac{\left(10\right)\text{th observation}+\left(11\right)\text{th observation}}{2}\end{array}$

First, arrange the data in ascending order- 20, 42, 42, 59, 64, 72, 77, 78, 79, 80, 84, 87, 88, 89, 90, 91, 92, 98, 99, 100.

So, the value of the median is,

  $\begin{array}{c}M=\frac{\left(10\right)\text{th observation}+\left(11\right)\text{th observation}}{2}\\ =\frac{80+84}{2}\\ =82\end{array}$

3) The mode, $M_O$ , is the most frequent occurring number in the data which is 42 here.

So, based on the above calculations, it is observed that the median 82 best describes the test scores, it being the central value in the data set and hence closer to most of the data points.