Question

For the given data find Karl Pearson's coefficient of skewness
CI:130-134 135-139 140-144 145-149
F: 3 12 21 28
CI:150-154 155-159 160-164
F: 19 12 5

Answer 1

We'll find Karl Pearson's mode coefficient of skewness.

From the table one can see that the maximal frequency corresponds to the 4-th interval with midpoint 147. Therefore, mode M(X)=147.

Calculate the expectation of the random variable:

$E(X)=(132\cdot 3 +137\cdot 12 +142\cdot 21 +147\cdot 28 +152\cdot 19 +157\cdot 12 +162\cdot 5)/100=147.2$

Calculate the variance of the random variable:

$V(X)=(132-147.2)^2\cdot 3 +(137-147.2)^2\cdot 12 +(142-147.2)^2\cdot 21 +(147-147.2)^2\cdot 28 +(152-147.2)^2\cdot 19 +(157-147.2)^2\cdot 12 +(162-147.2)^2\cdot 5)/100=51.96$

Calculate the standard deviation:

$\sigma=\sqrt{V(X)}=\sqrt{51.96}=7.21$

Karl Pearson's mode coefficient of skewness:

$\frac{E(X)-M(X)}{\sigma} = \frac{147.2-147}{7.21}=2.77$