Variance Examples - Word Problems
Mean is the average of a given set of data. Variance is the measure of variability of a given set of data. The formula for variance where $\mu $ is the mean of N number of data given is,
$V$ = $\frac{\sum (X -\mu )^2}{N}$

Now, suppose that a data of 100 elements is being given. How to find the variance of such data. For that sample variance is taken. A sample of the population is taken and the variance of the population is predicted by the sample variance. The sample variance is found by the formula,
$SV$ = $\frac{\sum (X -\mu )^2}{n-1}$, where $\mu $ is the mean and n is the number of data in the sample.

The standard deviation of a data is the square root of the sample.


Word Problems

Problem 1:

The heights (in cm) of students of a class is given to be 163, 158, 167, 174, 148. Find the variance.

Solution:


To find the variance, we need to find the mean of the given data and total members in the data set.

Total number of elements, N = 5

Mean, $\mu$ = $\frac{163+158+167+174+148}{5}$ = $162$

The formula for variance is, $V$ = $\frac{\sum (X -\mu )^2}{N}$

Now putting the values in the formula we get,

$V$ = $\frac{(162-163)^2+(158-163)^2+(167-163)^2+(174-163)^2+(148-163)^2}{5}$

   = $\frac{(-1)^2+(-5)^2+(4)^2+(11)^2+(-15)^2}{5}$

   = $\frac{388}{5}$ = $77.6$

Hence, the variance is found to be $77.6$.
Problem 2:

The marks of 10 students of a class is given to be 0, 4, 9, 12, 25, 2, 21, 7, 11, 12. What is the variance of the data set?

Solution:

Total number of elements, N = 10

Mean, $\mu$ = $\frac{0+4+9+12+25+2+21+7+11+12}{10}$ = $10.3$

The formula for variance is, $V$ = $\frac{\sum (X -\mu )^2}{N}$

Now putting the values in the formula we get,

$V$ = $\frac{(0-10.3)^2+(4-10.3)^2+(9-10.3)^2+(12-10.3)^2+(25-10.3)^2+(2-10.3)^2+(21-10.3)^2+(7-10.3)^2+(11-10.3)^2+(12-10.3)^2}{10}$

   = $\frac{601.3}{10}$ = $60.13$

Hence, the variance is found to be $60.13$.
Problem 3:

There are 500 staffs in a company. The hours used in leisure per week by 5 of the employees is 5, 2, 8, 10, 7. Find the sample variance.

Solution:


The formula for sample variance is given as,

$SV$ = $\frac{\sum (X -\mu )^2}{n-1}$, where $\mu $ is the mean and n is the number of data in the sample.

Mean of the sample, $\mu$ = $\frac{5 + 2 + 8 + 10 + 7}{5}$ = $6.4$

Putting the values in the formula we get,

$SV$ = $\frac{(5-6.4)^2+(2-6.4)^2+(8-6.4)^2+(10-6.4)^2+(7-6.4)^2}{5}$

        = $\frac{37.2}{5}$ = $7.44$
Problem 4:

If the standard deviation of a sample is given to be 62.5 then find its variance.

Solution:


Variance is square of the standard deviation.

Standard deviation = $62.5$

Variance = $\sqrt{sd}$ = $7.90569$