Question

5. The frequency distribution shows the salaries in million for specific year of top 25ceo’s in United States. Find the mean, variance and standard deviation. 

Class Boundaries Frequency 

15.5-20.5 13 

20.5-25.5   6 

25.5-30.5   4 

30.5-35.5 1 

35.5-40.5 1 

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c} Classes & Frequency & Midpoint & M_i\cdot f_i \\ & f_i & M_i \\ \hline 15.5-20.5 & 13 & 18 & 234 \\ \hdashline 20.5-25.5 & 6 & 23 & 138 \\ \hdashline 25.5-30.5 & 4 & 28 & 112 \\ \hdashline 30.5-35.5 & 1 & 33 & 33 \\ \hdashline 35.5-40.5 & 1 & 38 & 38 \\ \hdashline \end{array}$$

$$\bar{x}=\dfrac{1}{N}\sum_iM_i\cdot f_i$$

$$=\dfrac{234+138+112+33+38}{13+6+4+1+1}=22.2$$

$$s^2=\dfrac{1}{N-1}\sum _i(M_i-\bar{x})^2\cdot f_i$$

$$=\dfrac{1}{25-1}((18-22.2)^2(13)+(23-22.2)^2(6)$$

$$+(28-22.2)^2(4)+(33-22.2)^2(1)$$

$$+(38-22.2)^2(1))=\dfrac{734}{24}\approx30.583333$$

$$s=\sqrt{s^2}=\sqrt{\dfrac{734}{24}}\approx5.530220$$