Question

The mean and the standard deviation of 19 observations are given to be 6 and 2 respectively. If every observation is multiplied by 3, the mean and the variance of the new series will be respectively.
1. 18 and 2√3 
2. 2 and 4
3. 18 and 36
4. 6 and 12

Answer 1

Correct Answer - Option 3 : 18 and 36

Concept:

Mean (x̅)= \(\rm \frac{\sum x}{n}\)

Variance = \(\rm \frac1n\sum(x_i-\bar x)^2\)

Standard deviation =  \(\sqrt{\rm \frac1n\sum(x_i-\bar x)^2}\)

∴ Variance = (Standard deviation)²

Multiplying every value by a constant: If we multiply each observation by constant, then a new mean and standard deviation can be obtained by multiplying by that constant.

 

Calculation:

Here, mean = 6, standard deviation = 2

Now, every observation is multiplied by 3

So, new mean = 3 × 6 = 18

And new standard deviation = 3 ×  2 = 6

∴ Variance = (6)² = 36

Hence, option (3) is correct.

 

New mean = constant × (old mean)

New standard deviation = constant × (old standard deviation)

New Variance =  (constant)2 × (old Variance)