The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate. Literacy rate (in %) 45 - 55 55 - 65 65 - 75 75 - 85 85 - 95 Number of cities 3 10 11 8 3
Step-by-Step Solution
Key Concept: For grouped data, the mean is obtained by using the class‑mark (mid‑point) of each interval as a representative value. Compute Σ(f·x) where f is the frequency and x is the class‑mark, then divide by the total number of observations.
1. Identify the class intervals and their frequencies
- 45–55 : f = 3
- 55–65 : f = 10
- 65–75 : f = 11
- 75–85 : f = 8
- 85–95 : f = 3
Total number of cities, \(N = 3+10+11+8+3 = 35\).
2. Find the class‑mark (mid‑point) of each interval
\[\text{Class‑mark} = \frac{\text{lower limit}+\text{upper limit}}{2}\]
- 45–55 : \(x = \frac{45+55}{2}=50\)
- 55–65 : \(x = \frac{55+65}{2}=60\)
- 65–75 : \(x = \frac{65+75}{2}=70\)
- 75–85 : \(x = \frac{75+85}{2}=80\)
- 85–95 : \(x = \frac{85+95}{2}=90\)
3. Calculate \(f\times x\) for each class
\[
\begin{aligned}
50\times3 &= 150\\
60\times10 &= 600\\
70\times11 &= 770\\
80\times8 &= 640\\
90\times3 &= 270
\end{aligned}
\]
Sum of \(f\times x\) = \(150+600+770+640+270 = 2430\).
4. Compute the mean
\[
\bar{x} = \frac{\sum f x}{N} = \frac{2430}{35} \approx 69.4286\%\]
Rounded to two decimal places, the mean literacy rate is \(69.43\%\).
5. Answer: The mean literacy rate of the 35 cities is approximately 69.43 %.
Correct Answer: Approximately 69.43 % (mean literacy rate)