Question

If each of the four numbers 821,783,834,and855in the example is divided by 2, how will the mean, standard deviation, and coefficient of variation be affected?

Short answer

When the given numbers are divided by two, the mean and standard deviations are divided by two. The coefficient of the variation will remain unchanged.

Step-by-step solution

Formula for Mean, SD, and Coefficient of Variance

Mean$\overline{x}=\frac{{\displaystyle \sum _i}{x}_i}{n}$.

SD $s=\sqrt{\frac{{\displaystyle \sum _i}{\left(x_i-\overline{x}\right)}^2}{n-1}}$.

Coefficient of Variance$=100\times \frac{s}{\overline{x}}$.

Given Data

The four numbers are 821,783,834, and 855

The given numbers are divided by 2.

The mean of the given numbers before dividing by 2 is 823.2.

The standard deviation of the given numbers before dividing by 2 is 30.3

The coefficient of variance of the given numbers before dividing by 2 is 3.7%

Calculate Mean

The four numbers after dividing by 2 are 410.5, 391.5, 391.5, 417, and 427.5.

Mean:

$$\begin{gathered} \overline{x}=\frac{{\displaystyle \sum _i}{x}_i}{n} \\ =\frac{410.5+391.5+417+427.5}{4} \\ =\frac{1646.5}{4} \\ =411.625. \end{gathered}$$

The obtained mean value is half of the mean value calculated from the numbers before dividing by 2 $\left(i.e.,823.2/2=411.6\right)$

Calculate the Standard Deviation

Standard deviation:

$$\begin{gathered} s=\sqrt{\frac{{\displaystyle \sum _i}{\left(x_i-\overline{x}\right)}^2}{n-1}} \\ =\sqrt{\frac{{\left(411.6-410.5\right)}^2+{\left(411.6-391.5\right)}^2+{\left(411.6-417\right)}^2+\left(411.6-427.5\right)}{\left(4-1\right)}} \\ =\sqrt{229.06} \\ =15.15. \end{gathered}$$

The obtained standard deviation value is half the standard deviation value calculated from the numbers before dividing by two$\left(i.e.,30.3/2=15.15\right).$

Calculate the Coefficient of Variance

Coefficient of variation$=100\times \frac{s}{\overline{x}}$

$$\begin{gathered} =100\times \frac{15.15}{411.6} \\ =3.68\% \\ =3.67\% \end{gathered}$$

The obtained coefficient of variance is the same as the coefficient of variance calculated from the numbers before dividing by 2.

Therefore, when the given numbers are divided by 2, the mean and standard deviation will be divided by 2.The coefficient of variation remains unchanged.