Question

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

The first central moment m₁ is always zero.

The second central moment is:

$m_2=\frac{1}{N}\sum f_i(x_i-\overline{x})^2$

where f_i is frequency,

$x_i=\{3,5,7,9,11,13,15,17,19\}$

The third central moment is

$m_3=\frac{1}{N}\sum f_i(x_i-\overline{x})^3$

The fourth central moment is

$m_3=\frac{1}{N}\sum f_i(x_i-\overline{x})^4$

Using online calculator vrcacademy.com, we get:

Number of observations:

$N=413$

Mean of X values:

$\overline{x}=10.76$

$m_2=13.76$

$m_3=3.17$

$m_4=526.43$

Moment Ratios:

skewness:

$b_1=\frac{1}{N\sigma^3}\sum (x_i-\overline{x})^3$

kurtosis

$b_2=\frac{1}{N\sigma^4}\sum (x_i-\overline{x})^4$

where $\sigma$ is standard deviation.

Using online calculator www.socscistatistics.com, we get:

$b_1=0$

$b_2=-1.2$