Question

The variance of the scores 2, 4, 6, 8, 10
A.2
B.4
C.6
D.8

Answer 1

In statistics, the measurement of the spread out between numbers in the data sets is known as variance and is denoted by ${\sigma ^2}$. By formula it is defined as average of the squared deviation of a random variable from its mean i.e.
$\Rightarrow {\sigma ^2} = \dfrac{{\sum\limits_{i = 1}^n {{{({X_i} - \bar X)}^2}} }}{n}$
Where, ${X_i}$ is the value of ${ i}^{th}$ element, $\bar X$is mean and n is the number of terms.

Complete step-by-step answer:
To calculate the variance first of all, calculate the mean of the numbers i.e. $\overline X$ is given by
$\Rightarrow \overline X = \dfrac{{\sum\limits_{i = 1}^n {{X_i}} }}{n}$
Here in this question there are 5 numbers of terms and hence n is 5. And the numbers are 2, 4, 6, 8, 10. So, mean can be calculated as:
$\Rightarrow \overline X = \dfrac{{2 + 4 + 6 + 8 + 10}}{5}$
By adding all the numbers, we get.
$\Rightarrow \overline X = \dfrac{{30}}{5}$
By dividing 30 with 5 we get,
$\Rightarrow \overline X = 6$
The difference of each number with the mean is given by:
$\Rightarrow {X_1} - \bar X = 2 - 6 = - 4$
$\Rightarrow {X_2} - \bar X = 4 - 6 = - 2$
$\Rightarrow {X_3} - \bar X = 6 - 6 = 0$
$\Rightarrow {X_4} - \bar X = 8 - 6 = 2$
$\Rightarrow {X_5} - \bar X = 10 - 6 = 4$
Now, squaring each term and adding i.e. we will find squared difference of mean
$\Rightarrow {( - 4)^2} + {( - 2)^2} + {0^2} + {2^2} + {4^2} \\\ \Rightarrow 16 + 4 + 4 + 16 = 40 \\\$
To find the variance divide it by n i.e. number of terms. So,
$\Rightarrow {\sigma ^2} = \dfrac{{40}}{5}$
$\Rightarrow {\sigma ^2} = 8$
Hence, option D is correct.

Note: Students make mistakes while squaring the negative number, they may take negative signs after squaring which is wrong. As square is always positive either it is negative or positive.
If the variance is zero, it indicates that the value of all numbers within a set are identical. For the comparison of relative performance of each asset in a portfolio variance is used. Here variance is used in investigation.
The variance statistic can be helpful in determining the risk while purchasing a specific security by investors.