Question

The number of villages with respect to their population as per India census 2011 is given below. Find the average population of each village.

Population (in thousands)	125	30	20	15	8
Villages	20	153	23	53	67

Answer 1

Hint: We solve this formula by extending the table by adding one column. We assume the population as the class which is represented as \[{{x}_{i}}\] and the number of villages as the frequency of corresponding class represented as \[{{f}_{i}}\]. Then we create a column that contains the product of class and frequency that is\[{{f}_{i}}{{x}_{i}}\] then we use the formula of average as
\[\Rightarrow \bar{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}\]

Complete step-by-step solution:
Let us assume the population as the class which is represented as \[{{x}_{i}}\] and the number of villages as the frequency of corresponding class represented as \[{{f}_{i}}\].
Now let us extend the table by creating a column of the product of class and frequency that is \[{{f}_{i}}{{x}_{i}}\]

Population(in thousands) (\[{{x}_{i}}\])	Villages (\[{{f}_{i}}\])	\[{{f}_{i}}{{x}_{i}}\]
125	20	2500
30	153	4590
20	23	460
15	53	795
8	67	536

We know that the formula of average is given as
\[\Rightarrow \bar{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}\]
By substituting the required values in the above formula we get
\[\begin{align}
  & \Rightarrow \bar{x}=\dfrac{2500+4590+460+795+536}{20+153+23+53+67} \\
 & \Rightarrow \bar{x}=\dfrac{8881}{316} \\
 & \Rightarrow \bar{x}=28.10\simeq 28 \\
\end{align}\]
Therefore, the average population of each village according to the 2011 census in India is 28 thousand.

Note: Students will do mistake in solving the problem that is, they calculate the mean by using the formula
\[\Rightarrow \bar{x}=\dfrac{\sum{{{x}_{i}}}}{\sum{{{f}_{i}}}}\]
This formula is wrong for this type of problem because in the average formula the numerator is the total population. But in the above formula, the numerator doesn’t lead to the total population. But if the formula is taken as
\[\Rightarrow \bar{x}=\dfrac{\sum{{{f}_{i}}{{x}_{i}}}}{\sum{{{f}_{i}}}}\]
Here, the numerator gives the total population. So, this is the correct formula to use.