Statistics | Exercise 13.1

Question 1

A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.

Which method did you use for finding the mean, and why?

Solution

Understand the Question

To find the mean number of plants per house from grouped data, we find the class mark ( $x_i$ ) for each interval, where $x_i = \dfrac{\text{Upper limit} + \text{Lower limit}}{2}$ .
Compute the product $f_i x_i$ for each interval.
Use the Direct Method formula for grouped data: $\bar{x} = \dfrac{\sum f_i x_i}{\sum f_i}$
The Direct Method is best suited when the numerical values of frequencies ( $f_i$ ) and class marks ( $x_i$ ) are small.

Step 1 · Calculate Class Marks and Products

Compute the class mark $x_i = \dfrac{\text{Upper limit} + \text{Lower limit}}{2}$ and the product $f_i x_i$ for each class interval:

\begin{array}{|c|c|c|c|} \hline \text{Number of plants} & \text{Number of houses } (f_i) & \text{Class mark } (x_i) & f_i x_i \\ \hline 0 - 2 & 1 & 1 & 1 \\ \hline 2 - 4 & 2 & 3 & 6 \\ \hline 4 - 6 & 1 & 5 & 5 \\ \hline 6 - 8 & 5 & 7 & 35 \\ \hline 8 - 10 & 6 & 9 & 54 \\ \hline 10 - 12 & 2 & 11 & 22 \\ \hline 12 - 14 & 3 & 13 & 39 \\ \hline \text{Total} & \sum f_i = 20 & & \sum f_i x_i = 162 \\ \hline \end{array}

Sum of frequencies: $\sum f_i = 1 + 2 + 1 + 5 + 6 + 2 + 3 = 20$

Sum of $f_i x_i$ : $\sum f_i x_i = 1 + 6 + 5 + 35 + 54 + 22 + 39 = 162$

Step 2 · Calculate the Mean

Using the Direct Method formula:

\begin{aligned} \bar{x} &= \dfrac{\sum f_i x_i}{\sum f_i} \\[0.6em] &= \dfrac{162}{20} \\[0.6em] &= 8.1 \end{aligned}

The Direct Method is used because the numerical values of class marks ( $x_i$ ) and frequencies ( $f_i$ ) are small, which makes calculation straightforward.

Answer

Mean number of plants per house is $8.1$ .

The Direct Method was used because the values of $f_i$ and $x_i$ are small.

Common Mistakes

Dividing by Class Count: Dividing $\sum f_i x_i$ by the number of classes ( $7$ ) instead of the total frequency ( $\sum f_i = 20$ ).
Class Mark Formula Error: Computing class marks by subtracting limits instead of taking their average: $x_i = \dfrac{\text{Lower limit} + \text{Upper limit}}{2}$ .

Question 1

More questions in Exercise 13.1