Question

Given the data:

 

6

8

K

2

4

K

10

3K

 

 The mean of the above data is 9.375 where K is a constant. Find

a)    The value of K.

b)    The coefficient of skewness and interpret.

Answer 1

$E(X)=( 6+8+K+2+4+K+10+3K)/8=(30+5K)/8$

$K=(8E(X)-30)/5=(8\cdot 9.375-30)/5=9$

$E(X^2)=(6^2+8^2+9^2+2^2+4^2+9^2+10^2+27^2)/8=138.875$

$Var(X)=E(X^2)-E(X)^2=138.875-9.375^2=50.98$

$\sigma(X)=\sqrt{Var(X)}=\sqrt{50.98}=7.14$

Now we calculate Z-scores of each value of the random variable X:

$\tilde X=(X-E(X))/\sigma(X)$

$(6-9.375)/7.14=-0.4727$

$(8-9.375)/7.14=-0.1926$

$(9-9.375)/7.14=-0.0525$

$(2-9.375)/7.14=-1.0329$

$(4-9.375)/7.14=-0.7528$

$(9-9.375)/7.14=-0.0525$

$(10-9.375)/7.14=0.0875$

$(27-9.375)/7.14=2.4684$

The coefficient of skewness is by definition

$E(\tilde X^3)=(-0.4727^3-0.1926^3-0.0525^3-1.0329^3-0.7528^3-0.0525^3+0.0875^3+2.4684^3)/8=13.4$

One can see that $E(\tilde X^3)>>0$, this means that some values of X (X=27) are located far to the right of the average value. The value $X=27$ one should consider as extremal (unusual) big.