Chapter 13, Problem 9RE

To determine

which measure of central tendency is the most representative of the data shown in the frequency distribution.

Answer to Problem 9RE

Median or mean is the best measure of central tendency.

Explanation of Solution

Given information:

  $\begin{array}{cccccccccc}\text{Number}& 1& 2& 3& 4& 5& 6& 7& 8& 9\\ \text{Frequency}& 3& 2& 5& 6& 2& 1& 0& 0& 1\end{array}$

Formula used:

For n number of observations

  $\text{Mean}=\frac{1}{n}{\displaystyle \sum _{i=1}^n\left(f_i\right)\left(x_i\right)}$ , where $f_i=$ Frequency of the i^thterm and $x_i$ is the number of i^thterm.

Median for even number of ordered observations can be calculated as

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

Mode is the data value which has maximum number of frequency.

Calculation:

Here total number of observations is 20

So, mean cal be calculated as

  $\begin{array}{c}\text{Mean}=\frac{1}{20}{\displaystyle \sum _{i=1}^{20}\left(f_i\right)\left(x_i\right)}\\ =\frac{1}{20}\left(\left(1\right)\left(3\right)+\left(2\right)\left(2\right)+\left(3\right)\left(5\right)+\left(4\right)\left(5\right)+\left(5\right)\left(2\right)+\left(6\right)\left(1\right)+\left(7\right)\left(0\right)+\left(8\right)\left(0\right)+\left(9\right)\left(1\right)\right)\\ =3.55\end{array}$

Median can be calculated as

  $\begin{array}{c}\text{Median}=\frac{1}{2}\left({\left(\frac{20}{2}\right)}^{\text{th}}\text{term}+{\left(\frac{20}{2}+1\right)}^{\text{th}}\text{term}\right)\\ =\frac{1}{2}\left({10}^{\text{th}}\text{term}+{11}^{\text{th}}\text{term}\right)\\ =\frac{1}{2}\left(3+4\right)\\ =3.5\end{array}$

From the given data we can see that the frequency of 4 is maximum. That is, 6

Hence mode is 4

So,

Mean = 3.55

Median = 3.5

Mode = 4

Here median or mean described the data set in the best way as they are close to most of the data.

Conclusion:

For the given data median or mean is the best measure of central tendency.