Chapter B.2, Problem 48E

i. To find the Mean, Variance and Standard Deviation of the given data.

To determine

Compare the Central Tendencies for the given data.

Answer to Problem 48E

• Mean_(a)=4,
• Mean_(b)=12;
• Variance_(a)=4,
• Variance_(b)=36;
• S.D._(a)=2,
• S.D._(b)=6

Explanation of Solution

Given:

• 1,2,3,4,5,6,7
• (1,2,3,4,5,6,7) multiplied by 3

Calculation:

(a) Arrange the data in ascending order

1

2

3

4

5

6

7

  $\begin{array}{l}\text{Mean = }\frac{\text{Sum of Observations}}{\text{Number of Observations}}\\ {\text{Mean}}_{(\mathrm{a})}=\frac{1+2+3+4+5+6+7}{7}\\ {\text{Mean}}_{(\mathrm{a})}=\frac{28}{7}=\text{ 4}\end{array}$

  $\begin{array}{l}{\text{Variance}}_{(\mathrm{a})}{\text{(σ}}^{\text{2}}\text{) = Average of the squared difference from mean}\\ {\text{Variance}}_{(\mathrm{a})}=\frac{{(1-4)}^2+{(2-4)}^2+{(3-4)}^2+{(4-4)}^2+{(5-4)}^2+{(6-4)}^2+{(7-4)}^2}{7}\\ {\text{Variance}}_{(\mathrm{a})}\text{ = }\frac{9+4+1+0+1+4+9}{7}=\frac{28}{7}=4\\ \\ {\text{Standard Deviation}}_{(\mathrm{a})}(\mathrm{\sigma })=\sqrt{\text{Variance}}\\ \text{Thus,}\\ {\mathrm{\sigma }}_{(\mathrm{a})}=\sqrt{4}=2\end{array}$

(b) Arrange the data in ascending order

3

6

9

12

15

18

21

  $\begin{array}{l}{\text{Mean}}_{(\mathrm{b})}=\frac{3+6+9+12+15+18+21}{7}\\ {\text{Mean}}_{(\mathrm{b})}=\frac{84}{7}=\text{ 12}\end{array}$

  $\begin{array}{l}{\text{Variance}}_{(\mathrm{b})}{\text{(σ}}^{\text{2}}\text{) = Average of the squared difference from mean}\\ {\text{Variance}}_{(\mathrm{b})}=\frac{{(3-12)}^2+{(6-12)}^2+{(9-12)}^2+{(12-12)}^2+{(15-12)}^2+{(18-12)}^2+{(21-12)}^2}{7}\\ {\text{Variance}}_{\text{(b)}}\text{ = }\frac{81+36+9+0+9+36+81}{7}=\frac{252}{7}=36\\ \\ {\text{Standard Deviation}}_{(\mathrm{b})}(\mathrm{\sigma })=\sqrt{\text{Variance}}\\ \text{Thus,}\\ {\mathrm{\sigma }}_{(\mathrm{b})}=\sqrt{36}=6\end{array}$

Conclusion:

When multiplied by 3 the Mean is multiplied by 3, Variance is multiplied by the square of 3 and Standard Deviation is multiplied by 3.

ii. Effect of constant k on Central Tendency

To determine

Change in central tendency.

Answer to Problem 48E

Mean is multiplied by the same constant (k), Variance is multiplied by the square of the constant i.e. (k²) and Standard Deviation is multiplied by constant (k).

Explanation of Solution

Given:

• 1,2,3,4,5,6,7
• (1,2,3,4,5,6,7) multiplied by 3

Calculation:

Arrange the data in ascending order

3

6

9

12

15

18

21

  $\begin{array}{l}{\text{Mean}}_{(\mathrm{b})}=\frac{3+6+9+12+15+18+21}{7}\\ {\text{Mean}}_{(\mathrm{b})}=\frac{84}{7}=\text{ 12}\end{array}$

  $\begin{array}{l}{\text{Variance}}_{(\mathrm{b})}{\text{(σ}}^{\text{2}}\text{) = Average of the squared difference from mean}\\ {\text{Variance}}_{(\mathrm{b})}=\frac{{(3-12)}^2+{(6-12)}^2+{(9-12)}^2+{(12-12)}^2+{(15-12)}^2+{(18-12)}^2+{(21-12)}^2}{7}\\ {\text{Variance}}_{\text{(b)}}\text{ = }\frac{81+36+9+0+9+36+81}{7}=\frac{252}{7}=36\\ \\ {\text{Standard Deviation}}_{(\mathrm{b})}(\mathrm{\sigma })=\sqrt{\text{Variance}}\\ \text{Thus,}\\ {\mathrm{\sigma }}_{(\mathrm{b})}=\sqrt{36}=6\end{array}$

Conclusion:

Mean is Multiplied by the same constant (k), Variance is multiplied by the square of the constant i.e. (k²) and Standard Deviation is multiplied by constant (k).