# The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations

**Solution:**

Let the observations be x_{1}, x_{2}, x_{3}, x_{4}, x_{5} and x_{6}.

It is given that the mean is 8 and standard deviation is 4.

If each observation is multiplied by 3 and the resulting observations are y_{i}, then

y_{i} = 3x_{i} i.e., x_{i} = 1 y_{i}, for i = 1 to 6

Therefore, new mean,

y = (y_{1} + y_{2} + y_{3} + y_{4} + y_{5} + y_{6})/6

= 3( x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6})/6

= 3 × 8 ....from (1)

= 24

(σ) = 1/6 ∑^{6}_{i = 1} (x_{1 }- x)²

(4²) = 1/6 ∑^{6}_{i = 1} (x_{1 }- x)²

∑^{6}_{i = 1} (x - x)² = 96 ....(2)

From (1) and (2) , it can be observed that,

y = 3x and x = 1/3 y

Substituting the values of x_{1} and x in (2), we obtain

∑^{6}_{i = 1} (1/3 y_{i} - 1/3 y)² = 96

∑^{6}_{i = 1} (y_{i} - y)² = 864

Therefore, variance of new observations is (1/6 x 864) = 144

Hence, the standard deviation of new observations is 144 = 12

NCERT Solutions Class 11 Maths Chapter 15 Exercise ME Question 3

## The mean and standard deviation of six observations are 8 and 4, respectively. If each observation is multiplied by 3, find the new mean and new standard deviation of the resulting observations.

**Summary:**

Therefore, the new mean and new standard deviation are 24 and 12