Find the mean and variance for the first 10 multiples of 3.

Question

Let $X=\{x\in\mathbb{N}:1\le x\le19\}$ and for some $a,b\in\mathbb{R}$, $Y=\{ax+b:x\in X\}$. If the mean and variance of the elements of $Y$ are $30$ and $750$ respectively, then the sum of all possible values of $b$ is

Answer 1

The first 10 multiples of 3 are

3, 6, 9, 12, 15, 18, 21, 24, 27, 30

Here, number of observations, n = 10

Mean, $\bar{x}=\frac{\sum_{i=1}^{10}x_i}{10}=\frac{165}{10}=16.5$

The following table is obtained.

$x_i$	$(x_i-\bar{x})$	$(x_i-\bar{x})^2$
3	-13.5	182.25
6	-10.5	110.25
9	-7.5	56.25
12	-4.5	20.25
15	-1.5	2.25
18	1.5	2.25
21	4.5	20.25
24	7.5	56.25
27	10.5	110.25
30	13.5	182.25
		742.5

Variance(σ²) = $\frac{1}{n}\sum_{i=1}^{10}(x_i-\bar{x})^2=\frac{1}{10}×742.5=74.25$

Find the mean and variance for the first 10 multiples of 3.

Solution and Explanation

Top Questions on Variance and Standard Deviation

Questions Asked in CBSE Class XI exam

\(x_i\)	\((x_i-\bar{x})\)	\((x_i-\bar{x})^2\)
3	-13.5	182.25
6	-10.5	110.25
9	-7.5	56.25
12	-4.5	20.25
15	-1.5	2.25
18	1.5	2.25
21	4.5	20.25
24	7.5	56.25
27	10.5	110.25
30	13.5	182.25
		742.5