A simple of 25 variates has mean 40 and standard deviation 5 and a second sample of 35 variates has mean 45 and the standard deviation 2. Find the mean and standard deviation of the two samples of variates , taken together.

Answer

To find the combined mean and standard deviation of two samples, we can follow these steps:

### Step 1: Calculate the Combined Mean
Given:
- Sample A:
- Number of variates (n1) = 25
- Mean (x̄1) = 40
- Sample B:
- Number of variates (n2) = 35
- Mean (x̄2) = 45

The formula for the combined mean (x̄) is:
\[ \bar{x} = \frac{(x̄_1 \cdot n_1) + (x̄_2 \cdot n_2)}{n_1 + n_2} \]

Substituting the values:
\[ \bar{x} = \frac{(40 \cdot 25) + (45 \cdot 35)}{25 + 35} \]
Calculating the numerator:
\[ = \frac{1000 + 1575}{60} = \frac{2575}{60} \approx 42.92 \]

### Step 2: Calculate the Combined Standard Deviation
Given:
- Standard deviation of Sample A (σ1) = 5
- Standard deviation of Sample B (σ2) = 2

The formula for the combined standard deviation (σ) is:
\[ \sigma = \sqrt{\frac{(σ_1^2 \cdot n_1) + (σ_2^2 \cdot n_2) + (d_1^2 \cdot n_1) + (d_2^2 \cdot n_2)}{n_1 + n_2}} \]
Where:
- \(d_1 = |\bar{x} - x̄_1|\)
- \(d_2 = |\bar{x} - x̄_2|\)

Calculating \(d_1\) and \(d_2\):
\[ d_1 = |42.92 - 40| = 2.92 \]
\[ d_2 = |42.92 - 45| = 2.08 \]

Now substituting the values into the standard deviation formula:
\[ \sigma = \sqrt{\frac{(5^2 \cdot 25) + (2^2 \cdot 35) + (2.92^2 \cdot 25) + (2.08^2 \cdot 35)}{60}} \]
Calculating each term:
- \(5^2 \cdot 25 = 25 \cdot 25 = 625\)
- \(2^2 \cdot 35 = 4 \cdot 35 = 140\)
- \(2.92^2 \cdot 25 = 8.5284 \cdot 25 = 213.21\)
- \(2.08^2 \cdot 35 = 4.3264 \cdot 35 = 151.65\)

Now summing these values:
\[ = 625 + 140 + 213.21 + 151.65 = 1130.86 \]

Now substituting back into the standard deviation formula:
\[ \sigma = \sqrt{\frac{1130.86}{60}} \approx \sqrt{18.84767} \approx 4.34 \]

### Final Results:
- Combined Mean ≈ 42.92
- Combined Standard Deviation ≈ 4.34