Chapter B.2, Problem 15E

To determine

the best measure of central tendency.

Answer to Problem 15E

The mean of the data best describes the central tendency test score.

Explanation of Solution

Given information :

The given set of data whose mean median and mode are to be found are:

  $99,64,80,77,59,72,87,79,92,88,90,42,20,89,42,100,98,84,78,91$

Formula used :

  $Mean=\frac{{\displaystyle \sum x}}{N}$

  $Median=\frac{{\left(\frac{n}{2}\right)}^{th}term+{\left(\frac{n}{2}+1\right)}^{th}term}{2}$

  $Mode=3median-2mean$

Calculation:

The given numbers are $99,64,80,77,59,72,87,79,92,88,90,42,20,89,42,100,98,84,78,91$ .

Substitute $10$ for N in the formula to find mean.

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

Now arrange the given data in ascending order

  $20,42,42,59,64,72,77,78,79,80,84,87,88,89,90,91,92,98,99,100$

Calculate median of the numbers:

  $Median=\frac{{\left(\frac{n}{2}\right)}^{th}term+{\left(\frac{n}{2}+1\right)}^{th}term}{2}$

Substitute $20$ for $n$ .

  $\begin{array}{c}Median=\frac{{\left(\frac{20}{2}\right)}^{th}term+{\left(\frac{20}{2}+1\right)}^{th}term}{2}\\ =\frac{{10}^{th}term+{11}^{th}term}{2}\end{array}$

Substitute $80$ for ${10}^{th}term$ and $84$ for ${11}^{th}term$ .

  $\begin{array}{c}Median=\frac{80+84}{2}\\ =82\end{array}$

Calculate Mode of the data:

  $Mode=3median-2mean$

Substitute $76.55$ for mean and $82$ for median.

  $\begin{array}{c}Mode=3\times 82-2\times 76.55\\ =92.9\end{array}$

Hence, the mean of the data best describes the central tendency test score.