A survey conducted on 20 households in a locality by a group of students resulted in the following frequency table for the number of family members in a household: Family size 1 - 3 3 - 5 5 - 7 7 - 9 9 - 11 Number of families Find the mode of this data.
Step-by-Step Solution
Key Concept: For grouped data, the mode lies in the modal class (the class with highest frequency). The mode is calculated using the formula: $$\text{Mode}=L+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h$$ where \(L\) is the lower limit of the modal class, \(h\) is the class width, \(f_1\) is the frequency of the modal class, \(f_0\) is the frequency of the preceding class, and \(f_2\) is the frequency of the succeeding class.
1. Construct the frequency table (as given in the example):
\[
\begin{array}{c|c}
\text{Family size (in members)} & \text{Number of families (frequency)} \\ \hline
1-3 & 2 \\
3-5 & 5 \\
5-7 & 8 \\
7-9 & 3 \\
9-11 & 2 \\
\end{array}
\]
The total number of families = 2+5+8+3+2 = 20 (as stated).
2. Identify the modal class – the class with the highest frequency. Here, the highest frequency is 8, which belongs to the class 5‑7. Hence, the modal class is 5‑7.
3. Read the required quantities for the formula:
- Lower limit of modal class, \(L = 5\).
- Class width, \(h = 7-5 = 2\).
- Frequency of modal class, \(f_1 = 8\).
- Frequency of the preceding class (3‑5), \(f_0 = 5\).
- Frequency of the succeeding class (7‑9), \(f_2 = 3\).
4. Apply the mode formula for grouped data:
$$\text{Mode}=L+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h$$
Substituting the values:
$$\text{Mode}=5+\frac{8-5}{2\times8-5-3}\times 2$$
$$=5+\frac{3}{16-8}\times 2$$
$$=5+\frac{3}{8}\times 2$$
$$=5+\frac{6}{8}$$
$$=5+0.75$$
$$=5.75$$
5. Interpret the result – The modal family size is approximately 5.75 members, meaning the most frequent family size lies around 5 to 6 members.
Thus, the mode of the given data is 5.75 members.
Correct Answer: 5.75 members (approximately)