An analysis of the monthly allowances received by a sample of five engineers from a telco company has been conducted recently. The mean and median of the allowances is RM7000. A unimodal among the observations is RM12,000. Find the standard deviation if the allowances paid to each engineer were in full thousands

Solution:

Mean=7000

Median=7000

Mode (one)=12000

n=5

Assume two unknown values are x and y.

Arranging in ascending order,

x, y, 7000, 12000, 12000

Mean = 7000

$\dfrac{x+y+7000+12000+12000}5=7000 \\ \Rightarrow x+y=4000\ ...(i)$

Also, $x\ne y$ as there is unimode...(ii)

Also, x, y are in thousands...(iii)

From (i), (ii), (iii)

$x=1000,y=3000$

Now, standard deviation$=\sqrt{\dfrac1{n-1}\Sigma(x_i-\bar x)^2}$

$=\sqrt{\dfrac1{5-1}[(1000-7000)^2+(3000-7000)^2+(7000-7000)^2+(12000-7000)^2+(12000-7000)^2}$

$=5049.752469$