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. Number of plants 0 - 2 2 - 4 4 - 6 6 - 8 8 - 10 10 - 12 12 - 14 Number of houses 1 2 1 5 6 2 3 Which method did you use for finding the mean, and why?
Step-by-Step Solution
Key Concept: For grouped data, the mean is obtained by using the class‑midpoint (assumed mean) as a representative value for each class. The formula is \(\displaystyle \bar{x}=\frac{\sum f\,x}{\sum f}\), where \(f\) is the frequency of the class and \(x\) is the class‑midpoint.
1. Identify class intervals and frequencies
\[\begin{array}{c|c}
\text{Class (plants)} & \text{Number of houses (}f\text{)}\\ \hline
0-2 & 1\\
2-4 & 2\\
4-6 & 1\\
6-8 & 5\\
8-10 & 6\\
10-12 & 2\\
12-14 & 3\\
\end{array}\]
Total \(\sum f = 20\).
2. Find the class‑midpoint (assumed value) for each interval
\[\begin{array}{c|c}
\text{Class} & \text{Midpoint }(x)\\ \hline
0-2 & \frac{0+2}{2}=1\\
2-4 & \frac{2+4}{2}=3\\
4-6 & \frac{4+6}{2}=5\\
6-8 & \frac{6+8}{2}=7\\
8-10 & \frac{8+10}{2}=9\\
10-12 & \frac{10+12}{2}=11\\
12-14 & \frac{12+14}{2}=13\\
\end{array}\]
3. Compute \(f\times x\) for each class
\[\begin{array}{c|c|c}
\text{Midpoint }(x) & f & f\times x\\ \hline
1 & 1 & 1\\
3 & 2 & 6\\
5 & 1 & 5\\
7 & 5 & 35\\
9 & 6 & 54\\
11 & 2 & 22\\
13 & 3 & 39\\
\end{array}\]
Sum \(\sum f x = 1+6+5+35+54+22+39 = 162\).
4. Apply the formula for mean
\[\bar{x}=\frac{\sum f x}{\sum f}=\frac{162}{20}=8.1\]
5. Interpretation
The average (mean) number of plants per house in the surveyed locality is 8.1 plants.
6. Method used and justification
- *Method*: Assumed‑mean (grouped data) method using class midpoints.
- *Why*: The original data are given in class intervals, not as individual observations. By taking the midpoint of each interval as a representative value, we can compute an approximate mean for the grouped data, which is the standard technique prescribed in the NCERT textbook for such problems.
Correct Answer: 8.1 plants per house