First slide

Introduction to statistics

Question

Consider the numbers 1 ,2,3, 4, 5, 6,7 ,8, 9 and 10. If 1 is added to each number, the variance of the numbers so obtained is

Moderate

A

6.5
B

2.87
C

8.25
D

3.87

Solution

$Given numbers are 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 .$
If 1 is added to each number, then observations will be 2, 3, 4, 5, 6,7 ,8, 9, 10 and 11.
$\begin{matrix} ∴ Σ x_{i} = 2 + 3 + 4 + \dots + 11 \\ = \frac{10}{2} [2 \times 2 + 9 \times 1] = 5 [4 + 9] = 65 \\ and Σ x_{i}^{2} = 2^{2} + 3^{2} + 4^{2} + 5^{2} + \dots + 11^{2} \\ = (1^{2} + 2^{2} + 3^{2} + \dots + 11^{2}) - (1^{2}) \\ = \frac{11 \times 12 \times 23}{6} - 1 = 505 \end{matrix}$
$\begin{matrix} ∴ V = s^{2} = \frac{Σ x_{i}^{2}}{n} - {(\frac{Σ x_{i}}{n})}^{2} = \frac{505}{10} - {(\frac{65}{10})}^{2} \\ = 50.5 - (6.5)^{2} \\ = 50.5 - 42.25 \\ = 8.25 \end{matrix}$

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