The following table gives the weekly wages of workers in a factory. Weekly wages(Rs) 50-55 55-60 60-65 65-70 70-75 75-80 80-85 85-90 No. of workers 5 20 10 10 9 6 12 18 Calculate the mean, by using Short Cut Method.

Answer

To calculate the mean of the weekly wages of workers in a factory using the Short Cut Method, we will follow these steps:

### Step 1: Create a Table
We start with the given data:

| Weekly Wages (Rs) | No. of Workers (Frequency) |
|--------------------|----------------------------|
| 50 - 55 | 5 |
| 55 - 60 | 20 |
| 60 - 65 | 10 |
| 65 - 70 | 10 |
| 70 - 75 | 9 |
| 75 - 80 | 6 |
| 80 - 85 | 12 |
| 85 - 90 | 18 |

### Step 2: Calculate the Midpoint (X) for Each Class Interval
The midpoint for each class interval is calculated as follows:

- For 50 - 55: \(X = \frac{50 + 55}{2} = 52.5\)
- For 55 - 60: \(X = \frac{55 + 60}{2} = 57.5\)
- For 60 - 65: \(X = \frac{60 + 65}{2} = 62.5\)
- For 65 - 70: \(X = \frac{65 + 70}{2} = 67.5\)
- For 70 - 75: \(X = \frac{70 + 75}{2} = 72.5\)
- For 75 - 80: \(X = \frac{75 + 80}{2} = 77.5\)
- For 80 - 85: \(X = \frac{80 + 85}{2} = 82.5\)
- For 85 - 90: \(X = \frac{85 + 90}{2} = 87.5\)

### Step 3: Create a New Table with Midpoints
Now we create a new table with the midpoints:

| Weekly Wages (Rs) | Midpoint (X) | No. of Workers (Frequency) |
|--------------------|--------------|----------------------------|
| 50 - 55 | 52.5 | 5 |
| 55 - 60 | 57.5 | 20 |
| 60 - 65 | 62.5 | 10 |
| 65 - 70 | 67.5 | 10 |
| 70 - 75 | 72.5 | 9 |
| 75 - 80 | 77.5 | 6 |
| 80 - 85 | 82.5 | 12 |
| 85 - 90 | 87.5 | 18 |

### Step 4: Assume a Mean (A)
We will assume the mean (A) to be 72.5 (the midpoint of the 70-75 class).

### Step 5: Calculate D (X - A)
Now we calculate \(D = X - A\):

| Midpoint (X) | D (X - A) |
|--------------|-----------|
| 52.5 | -20 |
| 57.5 | -15 |
| 62.5 | -10 |
| 67.5 | -5 |
| 72.5 | 0 |
| 77.5 | 5 |
| 82.5 | 10 |
| 87.5 | 15 |

### Step 6: Calculate FD (Frequency × D)
Next, we calculate \(FD\):

| No. of Workers (Frequency) | D (X - A) | FD (Frequency × D) |
|----------------------------|-----------|---------------------|
| 5 | -20 | -100 |
| 20 | -15 | -300 |
| 10 | -10 | -100 |
| 10 | -5 | -50 |
| 9 | 0 | 0 |
| 6 | 5 | 30 |
| 12 | 10 | 120 |
| 18 | 15 | 270 |

### Step 7: Sum the FD and Frequency Columns
Now we sum the \(FD\) and frequency columns:

- \(\sum FD = -100 - 300 - 100 - 50 + 0 + 30 + 120 + 270 = -280\)
- \(\sum Frequency = 5 + 20 + 10 + 10 + 9 + 6 + 12 + 18 = 80\)

### Step 8: Calculate the Mean
Using the shortcut formula for mean:

\[ \text{Mean} = A + \frac{\sum FD}{\sum Frequency} \]

Substituting the values:

\[ \text{Mean} = 72.5 + \frac{-280}{80} \]
\[ \text{Mean} = 72.5 - 3.5 = 69 \]

### Final Answer
The mean weekly wage of the workers is Rs 69.

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