For a group of 50 male workers, the mean and standard deviations of their daily wages are ₹ 63 and ₹ 9 respectively for a group of 40 female workers, these are ₹ 54 and ₹ 6 respectively. Find the mean and standard deviation for the combined group.

Answer

To find the combined mean and standard deviation for the group of male and female workers, we will follow these steps:

### Step 1: Calculate the Combined Mean

The formula for the combined mean \( \bar{X} \) of two groups is given by:

\[ \bar{X} = \frac{n_m \cdot \bar{X}_m + n_f \cdot \bar{X}_f}{n_m + n_f} \]

Where:
- \( n_m \) = number of male workers = 50
- \( \bar{X}_m \) = mean wage of male workers = ₹ 63
- \( n_f \) = number of female workers = 40
- \( \bar{X}_f \) = mean wage of female workers = ₹ 54

Substituting the values:

\[ \bar{X} = \frac{50 \cdot 63 + 40 \cdot 54}{50 + 40} \]

Calculating the numerator:

\[ 50 \cdot 63 = 3150 \]
\[ 40 \cdot 54 = 2160 \]
\[ 3150 + 2160 = 5310 \]

Now, calculating the denominator:

\[ 50 + 40 = 90 \]

Now, substituting back into the formula:

\[ \bar{X} = \frac{5310}{90} = 59 \]

### Step 2: Calculate the Combined Standard Deviation

The formula for the combined standard deviation \( \sigma \) is:

\[ \sigma = \sqrt{\frac{n_m \cdot \sigma_m^2 + n_f \cdot \sigma_f^2 + n_m \cdot (d_m^2) + n_f \cdot (d_f^2)}{n_m + n_f}} \]

Where:
- \( \sigma_m \) = standard deviation of male workers = ₹ 9
- \( \sigma_f \) = standard deviation of female workers = ₹ 6
- \( d_m \) = deviation of male mean from combined mean = \( |\bar{X} - \bar{X}_m| = |59 - 63| = 4 \)
- \( d_f \) = deviation of female mean from combined mean = \( |\bar{X} - \bar{X}_f| = |59 - 54| = 5 \)

Calculating \( \sigma_m^2 \) and \( \sigma_f^2 \):

\[ \sigma_m^2 = 9^2 = 81 \]
\[ \sigma_f^2 = 6^2 = 36 \]

Now substituting into the formula:

\[ \sigma = \sqrt{\frac{50 \cdot 81 + 40 \cdot 36 + 50 \cdot 4^2 + 40 \cdot 5^2}{90}} \]

Calculating each term:

\[ 50 \cdot 81 = 4050 \]
\[ 40 \cdot 36 = 1440 \]
\[ 50 \cdot 16 = 800 \quad \text{(since } 4^2 = 16\text{)} \]
\[ 40 \cdot 25 = 1000 \quad \text{(since } 5^2 = 25\text{)} \]

Now summing these:

\[ 4050 + 1440 + 800 + 1000 = 7290 \]

Now substituting back into the formula:

\[ \sigma = \sqrt{\frac{7290}{90}} = \sqrt{81} = 9 \]

### Final Results

- Combined Mean = ₹ 59
- Combined Standard Deviation = ₹ 9