Question

In a data distribution, the first quartile, the median and the means are 30.8, 48.5 and 42.0 respectively. If the coefficient skewness is −0.38

a) What is the approximate value of the third quartile (Q3 ), correct to 2 decimal places.
b)What is the approximate value of the variance, correct to the nearest whole number

Answer 1

a) The third quartile Q₃ = 56.45

b) Variance
`\mathbf{ \sigma^2 =2633.31}`

Explanation:

Given that :

`Q_1` = 30.8

Median
`Q_2` = 48.5

Mean = 42

a) The mean is less than median; thus the expression showing the coefficient of skewness is given by the formula :

`SK = (Q_3+Q_1-2Q_2)/(Q_3-Q_1)`

`-0.38 = (Q_3+30.8-2(48.5))/(Q_3-30.8)`

`-0.38Q_3 + 11.704 = Q_3 +30.8 - 97`

`1.38Q_3 = 77.904`

Divide both sides by 1.38

`Q_3 = 56.45`

b) The objective here is to determine the approximate value of the variance;

Using the relation

`SK_p = (Mean- (3*Median-2 *Mean) )/(\sigma)`

`-0.38= (42- (3 *48.5-2*42) )/(\sigma)`

`-0.38= ((-19.5) )/(\sigma)`

`-0.38* \sigma = {(-19.5) }{}`

`\sigma =\frac {(-19.5) }{-0.38 }`

`\sigma =51.32`

Variance =
`\sigma^2 =51.32^2`

`\mathbf{ \sigma^2 =2633.31}`