Answer to Question #201021 in Statistics and Probability for Rameen

Question

1. Calculate the first four moments about the mean from the following frequency distribution. 

      Also calculate the Moment Ratios b1 and b2.                         

Groups

2------4

4-------6

6------8

8-----10

10----12

12----14

14----16

16----18

18----20

Frequency

18

24

47

80

102

66

40

21

15

                              

Answer 1

Given data is:

Fist four moments about mean:

$\mu_1'=\dfrac{1}{n}\sum x_if_i=\dfrac{4445}{413}=10.762$

$\mu_2'=\dfrac{1}{n}\sum x_i^2f_i=\dfrac{48110}{413}=116.489$

$\mu_3'=\dfrac{1}{n}\sum x_i^3f_i=\dfrac{699749}{413}=1694.30$

$\mu_4'=\dfrac{1}{n}\sum x_i^4f_i=\dfrac{9775376}{413}=23669.19$

Now,

$\mu_1=\mu_1'=10.762\\\ \\\mu_2=\mu_2'-\mu_1'=105.727\\\ \\\mu_3=\mu_3'-3\mu_2'\mu_1'+2\mu_1'^3=1694.30-3760.96+2492.92=426.26$

$\mu_4=\mu_4'-4\mu_3'\mu_1'+6\mu_2'\mu_1'^2-3\mu_1'^4=23669.19-72936.2264+80950.98-3739.38\\\mu_4=27944.563$

Now,

$b_1=\dfrac{\mu_3^2}{\mu_2^3}=\dfrac{(426.26)^2}{(105.727)^3}=\dfrac{181697.58}{1181837.39}=0.1537$

$b_2=\dfrac{\mu_4}{\mu_2^2}=\dfrac{27944.563}{11178.19853}=2.499$