Consider the following distribution of daily wages of 50 workers of a factory. Daily wages (in `) 500 - 520 520 -540 540 - 560 560 - 580 580 -600 Number of workers 12 14 8 6 10 Find the mean daily wages of the workers of the factory by using an appropriate method.
Step-by-Step Solution
Key Concept: For grouped data, the mean is obtained by taking the class‑midpoint (assumed value) of each interval, multiplying it by the frequency, summing these products and dividing by the total number of observations: $$\bar{x}=\frac{\sum f_i x_i}{\sum f_i}$$ where $f_i$ is the frequency and $x_i$ is the class‑midpoint.
1. Form the class‑midpoints (also called class‑marks).
\[\begin{array}{c|c}
\text{Class interval (₹)} & \text{Midpoint } x_i \\ \hline
500-520 & \frac{500+520}{2}=510 \\
520-540 & \frac{520+540}{2}=530 \\
540-560 & \frac{540+560}{2}=550 \\
560-580 & \frac{560+580}{2}=570 \\
580-600 & \frac{580+600}{2}=590 \\
\end{array}\]
2. Multiply each midpoint by its frequency $f_i$.
\[\begin{array}{c|c|c}
x_i & f_i & f_i x_i \\ \hline
510 & 12 & 12\times510 = 6120 \\
530 & 14 & 14\times530 = 7420 \\
550 & 8 & 8\times550 = 4400 \\
570 & 6 & 6\times570 = 3420 \\
590 & 10 & 10\times590 = 5900 \\ \hline
\text{Total} & \sum f_i = 50 & \sum f_i x_i = 27260 \\
\end{array}\]
3. Apply the formula for the mean.
$$\bar{x}=\frac{\sum f_i x_i}{\sum f_i}=\frac{27260}{50}=545.2$$
4. Interpretation: The average (mean) daily wage of the 50 workers is ₹ 545.2.
Correct Answer: ₹ 545.2 (approximately)