Question

The variance of \[10,10,10,10,10\] is
${\text{(A) 10}}$
${\text{(B) }}\sqrt {10} $
${\text{(C) 0}}$
${\text{(D) 5}}$

Answer 1

Hint: We will calculate the variance from the given distribution by using the formula of variance. Finally we get the required answer.

Formula used: $s = \dfrac{{\sum {{{({x_i} - \overline x )}^2}} }}{{n - 1}}$
$\overline x = \dfrac{{\sum {{x_i}} }}{n}$
Here $s$ is the variance
${x_i}$ is the value of one observation,
$\overline x $ is the mean of the observation and $n$ is the number of terms in the distribution.

Complete step-by-step solution:
It is given that the question stated as the value \[10,10,10,10,10\]
From the above distribution we can see that there are total $5$terms in the distribution therefore,
$n = 5$
Now to calculate the mean $\overline x $ we have to use the formula of mean which is:
$\overline x = \dfrac{{\sum {{x_i}} }}{n}$
On substituting the values, we get:
$\overline x = \dfrac{{10 + 10 + 10 + 10 + 10}}{5}$
On adding the values, we get:
$\overline x = \dfrac{{50}}{5}$
$\overline x = 10$.
Therefore, the mean $\overline x $ is $10$.
Now we will find $({x_i} - \overline x )$ for each of the value, using the table below we get:

${x_i}$	$({x_i} - \overline x )$
$10$	$0$
$10$	$0$
$10$	$0$
$10$	$0$
$10$	$0$

Since the value of $({x_i} - \overline x )$ is $0$ for all the values of $x$,
The value of $s = \dfrac{{\sum {{{({x_i} - \overline x )}^2}} }}{{n - 1}}$ will be $0$.
Therefore, the variance is $0$.

Therefore, the correct answer is option ${\text{(C)}}$.

Note: The variance in statistics is a measurement which tells the distance of a number from the mean in the distribution. It tells how much a value in the distribution will differ from the mean.
The variance is an important measurement in the distribution because it tells us how spread out the data is in a distribution; it means that if the variance is low the data is more similar and if it is high the data is a lot different from each other.
There is also another measurement in statistics which is called the standard deviation which is the square root value of the variance. It is represented using the symbol $\sigma $. Another notation of the variance other than $s$ is ${\sigma ^2}$ since it is the square value of the standard deviation.