Question

Average of daily wage of 50 workers of a factory was Rs.200 with a standard deviation of Rs.40. Each worker is given a raise of Rs.20. What are the new average daily wage and standard deviation? Have the wages becomes more or less uniform?

Answer 1

$N=50\overline{X}=200\sigma =40\text{Average wage}=\frac{\text{Total wages}}{\text{Number of workers}}200=\frac{Totalwages}{50}$

So, total wages = 200 $\times$ 50 = Rs 10,000

Now, increase in wage rate = Rs.20

Total raise = 50 $\times$ 20 = Rs 1,000

Total wage after raise = 10,000+1,000 = Rs 11,000

$\text{New Average wage}=\frac{\text{New Total wages}}{\text{Number of workers}}=\frac{11,000}{50}=Rs220$

Thus, mean increases by the amount of increase in wage of each worker as the absolute increase was equal for all.

Standard deviation will remain the same that is Rs. 40 as it is independent of origin and hence an addition of equal amount in all the value will not cause any change in the standard deviation.

Uniformity of wages can be seen by coefficient of variation.

Previously, the coefficient of variation was:
$CV=\frac{\sigma }{x}\times 100=\frac{40}{200}\times 100=20$

The new coefficient of variation after wage increases is given by :
$CV=\frac{\sigma }{x}\times 100=\frac{40}{220}\times 100=18.18$

This shows that wages become more uniform now as the new CV is lower.