# The Variance of 20 Observations is 5. If Each Observation is Multiplied by 2, Find the Variance of the Resulting Observations. - Mathematics

The variance of 20 observations is 5. If each observation is multiplied by 2, find the variance of the resulting observations.

#### Solution

Let $x_1 , x_2 , x_3 , . . . , x_{20}$  be  the 20 given observations.

$\text{ Variance} (X) = 5$

$\text{ Variance } (X) = {\frac{1}{20}} \times \sum \left( {x_i - X} \right)^2 = 5 (\text{ Here , is the mean of the given observations } . )$

Let u1,u2,,u3, ..., u20 be the new observations, such that

$u_i = 2 x_i (\text{ for } i = 1, 2, 3, . . . , 20) . . . (1)$

$\text{ Mean } = \bar{U} = \frac{\sum^{20}_{i = 1} u_i}{n}$

$= \frac{\sum^{20}_{i = 1} 2 x_i}{20} \left[ \text{ substituting} u_i \text{ from eq (1) and taking n as } 20 \right]$

$= 2 \times \frac{\sum^{20}_{i = 1}{ x_i} }{20}$

$= 2 \bar{X}$

$u_i - \bar{U} = 2 x_i - 2 \bar{X} (\text{ for } i = 1, 2, . . . , 20)$

$= 2\left( x_i - \bar{X} \right)$

$\left( u_i - \bar{U} \right)^2 = \left( 2\left( x_i - \bar{X} \right) \right)^2 \left(\text{ squaring both the sides } \right)$

$= 4 \left( x_i - \bar{X} \right)^2$

$\therefore \sum^{20}_{i = 1} \left( u_i - \bar{U} \right)^2 = \sum 4^{20}_{i = 1} \left( x_i - \bar{X} \right)^2$

$\frac{\sum^{20}_{i = 1} \left( u_i - \bar{U} \right)^2}{20} = \frac{\sum 4^{20}_{i = 1} \left( x_i - \bar{X} \right)^2}{20}$

$= 4 \frac{\sum^{20}_{i = 1} \left( x_i - \bar{X} \right)^2}{20}$

$\text{ Variance } (U) = 4 \times \text{ Variance }(X)$

$= 4 \times 5$

$= 20$

Thus, variance of the new observations is 20.

Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 32 Statistics
Exercise 32.4 | Q 2 | Page 28