Question

Calculate the 1st four moment about the mean
for the following data :
x 1 2 3 4 5 6 7 8 9
y 1 6 13 25 30 22 9 5 2

Answer 1

Solution:

Mean$=\mu=\dfrac{\Sigma f_i x_i}{\Sigma f_i}=\dfrac{555}{91}=6.0989\approx6.1$

Now, 1st four moments about the mean:

First order moment$=\dfrac{\Sigma f_i (x_i-\mu)}{\Sigma f_i}=\dfrac{-134.3}{91}=-1.476$

Second order moment$=\dfrac{\Sigma f_i (x_i-\mu)^2}{\Sigma f_i}=\dfrac{440.73}{91}=4.843$

Third order moment$=\dfrac{\Sigma f_i (x_i-\mu)^3}{\Sigma f_i}=\dfrac{-1115.303}{91}=-12.256$

Fourth order moment$=\dfrac{\Sigma f_i (x_i-\mu)^4}{\Sigma f_i}=\dfrac{4315.2033}{91}=47.42$