To calculate the arithmetic average (mean) of the given data, we should follow these steps:
1. Determine the Midpoints of Each Interval:
- For each interval of weekly wages, calculate the midpoint (average of the lower and upper limits of the interval).
2. Calculate the Total Sum of Weekly Wages:
- Multiply each midpoint by the number of workers in that interval to find the total wage contribution of each interval.
- Sum these contributions to get the total sum of weekly wages for all workers.
3. Find the Total Number of Workers:
- Sum the number of workers across all intervals.
4. Compute the Arithmetic Average:
- Divide the total sum of weekly wages by the total number of workers.
### Step-by-Step Solution
1. Calculate the Midpoints:
[tex]\[
\text{Midpoints} = \left\{ \frac{(100 + 105)}{2}, \frac{(105 + 110)}{2}, \frac{(110 + 115)}{2}, \frac{(115 + 120)}{2}, \frac{(120 + 125)}{2}, \frac{(125 + 130)}{2}, \frac{(130 + 135)}{2}, \frac{(135 + 140)}{2}, \frac{(140 + 145)}{2}, \frac{(145 + 150)}{2}, \frac{(150 + 155)}{2}, \frac{(155 + 160)}{2} \right\}
\][/tex]
Calculating these midpoints:
[tex]\[
\text{Midpoints} = \left\{ 102.5, 107.5, 112.5, 117.5, 122.5, 127.5, 132.5, 137.5, 142.5, 147.5, 152.5, 157.5 \right\}
\][/tex]
2. Calculate the Total Sum of Weekly Wages:
Multiply each midpoint by the corresponding number of workers and sum these products:
[tex]\[
\begin{align*}
& 102.5 \times 200 \\
& + 107.5 \times 210 \\
& + 112.5 \times 230 \\
& + 117.5 \times 320 \\
& + 122.5 \times 350 \\
& + 127.5 \times 320 \\
& + 132.5 \times 410 \\
& + 137.5 \times 320 \\
& + 142.5 \times 280 \\
& + 147.5 \times 210 \\
& + 152.5 \times 160 \\
& + 157.5 \times 90 \\
\end{align*}
\][/tex]
The resulting total sum of weekly wages is:
[tex]\[
\text{Total Sum of Weekly Wages} = 398000.0
\][/tex]
3. Find the Total Number of Workers:
Sum the number of workers across all intervals:
[tex]\[
\text{Total Number of Workers} = 200 + 210 + 230 + 320 + 350 + 320 + 410 + 320 + 280 + 210 + 160 + 90
\][/tex]
This sums to:
[tex]\[
\text{Total Number of Workers} = 3100
\][/tex]
4. Compute the Arithmetic Average:
Divide the total sum of weekly wages by the total number of workers:
[tex]\[
\text{Arithmetic Average} = \frac{\text{Total Sum of Weekly Wages}}{\text{Total Number of Workers}} = \frac{398000.0}{3100}
\][/tex]
Simplifying this gives:
[tex]\[
\text{Arithmetic Average} \approx 128.38709677419354 \text{ Rs}
\][/tex]
Therefore, the arithmetic average of the weekly wages of the workers is approximately Rs. 128.39.
