Answer to Question #350323 in Statistics and Probability for sergu

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

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

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

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

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

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

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

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

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