Question

Find the mean and variance for the following frequency distribution.

Classes	0-10	10-20	20-30	30-40	40-50
Frequency	5	8	15	16	6

Answer 1

Hint: We solve this problem by first considering the formula for mean and variance of a grouped frequency distribution, $Mean=\dfrac{\sum{{{x}_{i}}{{f}_{i}}}}{\sum{{{f}_{i}}}}$ and $Variance=\left( \dfrac{\sum{x_{i}^{2}{{f}_{i}}}}{\sum{{{f}_{i}}}} \right)-{{\left( Mean \right)}^{2}}$. Then we take the midpoint of each class as ${{x}_{i}}$. Then we make a table with ${{x}_{i}}$, ${{f}_{i}}{{x}_{i}}$ and $x_{i}^{2}{{f}_{i}}$ values of each class. Then we add them and find the values of $\sum{{{f}_{i}}}$, $\sum{{{f}_{i}}{{x}_{i}}}$ and $\sum{x_{i}^{2}{{f}_{i}}}$. Then we substitute the values in the formula for mean and variance and find those values respectively.

Complete step by step answer:
Here we need to find the mean and variance of the given frequency distribution.
Here we can see that the given frequency distribution is grouped.
First let us consider the formulas for mean and variance of the grouped frequency distribution.
$\begin{align}
  & Mean=\dfrac{\sum{{{x}_{i}}{{f}_{i}}}}{\sum{{{f}_{i}}}} \\
 & Variance=\left( \dfrac{\sum{x_{i}^{2}{{f}_{i}}}}{\sum{{{f}_{i}}}} \right)-{{\left( Mean \right)}^{2}} \\
\end{align}$
As the data is grouped data, let us take the midpoint of each class to represent each class.
So, let us draw a table and find the values of classes accordingly.

Classes	${{f}_{i}}$	${{x}_{i}}$	${{f}_{i}}{{x}_{i}}$	$x_{i}^{2}{{f}_{i}}$
0-10	5	5	25	125
10-20	8	15	120	1800
20-30	15	25	375	9375
30-40	16	35	560	19600
40-50	6	45	270	12150
	$\sum{{{f}_{i}}=}50$		$\sum{{{f}_{i}}{{x}_{i}}=}1350$	$\sum{x_{i}^{2}{{f}_{i}}=}43050$

Now let us consider the formula for the mean.
$Mean=\dfrac{\sum{{{x}_{i}}{{f}_{i}}}}{\sum{{{f}_{i}}}}$
So, substituting the values obtained in the above table we can find the value of mean as,
$\begin{align}
  & \Rightarrow Mean=\dfrac{1350}{50} \\
 & \Rightarrow Mean=27 \\
\end{align}$
So, the mean of the given frequency distribution is 27.
Now let us consider the formula for the variance.
$Variance=\left( \dfrac{\sum{x_{i}^{2}{{f}_{i}}}}{\sum{{{f}_{i}}}} \right)-{{\left( Mean \right)}^{2}}$
So, substituting the values obtained in the above table we can find the value of variance as,
$\begin{align}
  & \Rightarrow Variance=\left( \dfrac{43050}{50} \right)-{{\left( 27 \right)}^{2}} \\
 & \Rightarrow Variance=861-729 \\
 & \Rightarrow Variance=132 \\
\end{align}$
So, variance of the given frequency distribution is 132.
So, we get that $Mean=27$ and $Variance=132$.

Hence the answer is 27 and 132.

Note: We can also solve this question by using a different formula for variance.
$Variance=\dfrac{\sum{{{f}_{i}}{{\left( {{x}_{i}}-Mean \right)}^{2}}}}{\sum{{{f}_{i}}}}$
If we use this formula, we need to make some extra calculations and take more time to solve. So, it is easier to solve if we use the formula $Variance=\left( \dfrac{\sum{x_{i}^{2}{{f}_{i}}}}{\sum{{{f}_{i}}}} \right)-{{\left( Mean \right)}^{2}}$ to find the variance.