Question

If the mean of the data : $7,8,9,7,8,7,\lambda ,8$ is $8$, then the variance of this data is
A) $\dfrac{7}{8}$
B) $1$
C) $\dfrac{9}{8}$
D) $2$

Answer 1

For approaching such kind of questions we need to know what mean and the variance actually is mean is the average of the given set of the numbers which is the sum of all the numbers given divided by the total count of the numbers and the term variance is the statistical measurement of the spread between numbers in the data set that is the variance measures how far each number in the set is from the mean and thus from every number in the set here we would use the concept of the mean to find the missing term and later on find the variance that is asked in the question.

Complete step-by-step answer:
Here we are given a set of the numbers that is $7,8,9,7,8,7,\lambda ,8$ and we are also given mean of this set of the data which is $8$ and we are asked to find the variance of the data
For which we have to find the missing term $\lambda$ of the given data which can be done by using the given value to us that is mean of the data is $8$
Mean of the data is the average of the data which is equal to the sum of the whole numbers in the sample divided by the count of number present in the sample size
Hence the formula for the mean is –
\mu = \dfrac{{{x_1} + {x_2} + {x_3}\\_\\_\\_\\_\\_\\_{x_n}}}{n}
Where $n =$ count of the terms present in the given space
And {x_1} + {x_2} + {x_3}\\_\\_\\_\\_\\_\\_{x_n}is expressed as the total sum of the terms present in the sample space
Now here $\mu = 8$(which is the mean of the data that is given to us in the question)
And $n = 8$(total count of the elements is$8$)

$$\Rightarrow \mu = \dfrac{{7 + 8 + 9 + 7 + 8 + 7 + \lambda + 8}}{8} \\\ \Rightarrow 8 = \dfrac{{7 + 8 + 9 + 7 + 8 + 7 + \lambda + 8}}{8} \\\ \Rightarrow 64 = 7 + 8 + 9 + 7 + 8 + 7 + \lambda + 8 \\\ \Rightarrow 64 = 54 + \lambda \\\$$

$\Rightarrow \lambda = 10$
Now we have to find the variance that is asked in the question –
Variance is the statistical measurement of the spread between numbers in the data set that is the variance measures how far each number in the set is from the mean and thus from every number in the set and its formula is –
$\sigma = \sqrt {\dfrac{{\sum\limits_{i = 1}^n {{{\left( {{x_i} - \overline x } \right)}^2}} }}{n}}$
Where $\overline x =$ the mean of the data in the consideration
$\sigma =$ The variance of the data in the consideration
${x_i} =$ Each member of the data
$n =$ Total count of the data in the consideration
So the variance of the given data $7,8,9,7,8,7,10,8$is –

$$\Rightarrow \sigma = \sqrt {\dfrac{{{{\left( {7 - 8} \right)}^2} + {{\left( {7 - 8} \right)}^2} + {{\left( {10 - 8} \right)}^2} + {{\left( {7 - 8} \right)}^2} + {{\left( {9 - 8} \right)}^2} + {{\left( {8 - 8} \right)}^2} + {{\left( {8 - 8} \right)}^2} + {{\left( {8 - 8} \right)}^2}}}{8}} \\\ \Rightarrow \sigma = \sqrt {\dfrac{{1 + 1 + 1 + 4 + 1 + 0 + 0 + 0}}{8}} \\\ \Rightarrow \sigma = \sqrt {\dfrac{8}{8}} \\\ \Rightarrow \sigma = 1 \\\$$

So the variance of the data comes out to be $1$
So the option which is similar to the $1$ is option B
The correct answer is option B $1$.

Note: While approaching such kind of the question one should know the basic formulas of mean and the variance and how to apply them also care should be taken while copying the terms from the question and so while solving and calculating too the concentration should be fullest as a little mistake in the calculation can lead the wrong solution.