Chapter B.2, Problem 10E

To determine

To calculate: the mean median mode of a given set of data

Answer to Problem 10E

The mean of the data is $10.1$ , the median is $8.5$ and the mode is $5.3$ .

Explanation of Solution

Given information :

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

  $2,10,19,3,7,5,3,10,31,11$

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 $2,10,19,3,7,5,3,10,31,11$ .

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

  $\begin{array}{c}Mean=\frac{2+10+19+3+7+5+3+10+31+11}{10}\\ =10.1\end{array}$

Now arrange the given data in ascending order

  $2,3,3,5,7,10,10,11,19,31$

Calculate median of the numbers:

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

Substitute $10$ for $n$ .

  $\begin{array}{c}Median=\frac{{\left(\frac{10}{2}\right)}^{th}term+{\left(\frac{10}{2}+1\right)}^{th}term}{2}\\ =\frac{5^{th}term+6^{th}term}{2}\end{array}$

Substitute $7$ for $5^{th}term$ and $10$ for $6^{th}term$ .

  $\begin{array}{c}Median=\frac{7+10}{2}\\ =\frac{17}{2}\\ =8.5\end{array}$

Calculate Mode of the data:

  $Mode=3median-2mean$

Substitute $10.1$ for mean and $8.5$ for median.

  $\begin{array}{c}Mode=3\times 8.5-2\times 10.1\\ =5.3\end{array}$

Hence, the mean of the data is $10.1$ , the median is $8.5$ and the mode is $5.3$ .