Means and standard deviations of the scores of an intelligence test of two classes of different sizes of 25 and 75 are `M_(1) = 80` marks and `M_(2) = 85` marks and `SD_(1) = 15` marks and `SD_(2) = 20` marks
Calculate the combined mean and the standard deviation of the two classes.

Answer

To solve the problem of finding the combined mean and standard deviation of the scores from two classes, we will follow these steps:

### Step 1: Calculate the Combined Mean
The formula for the combined mean \( M_{12} \) of two groups is given by:

\[ M_{12} = \frac{n_1 \cdot M_1 + n_2 \cdot M_2}{n_1 + n_2} \]

Where:
- \( n_1 = 25 \) (size of the first class)
- \( n_2 = 75 \) (size of the second class)
- \( M_1 = 80 \) (mean of the first class)
- \( M_2 = 85 \) (mean of the second class)

Substituting the values:

\[ M_{12} = \frac{25 \cdot 80 + 75 \cdot 85}{25 + 75} \]

Calculating the numerator:

\[ = \frac{2000 + 6375}{100} = \frac{8375}{100} = 83.75 \]

Thus, the combined mean \( M_{12} \) is **83.75 marks**.

### Step 2: Calculate the Deviations
Next, we need to calculate the deviations \( d_1 \) and \( d_2 \):

\[ d_1 = M_{12} - M_1 = 83.75 - 80 = 3.75 \]
\[ d_2 = M_{12} - M_2 = 83.75 - 85 = -1.25 \]

### Step 3: Calculate the Combined Standard Deviation
The formula for the combined standard deviation \( SD_{12} \) is given by:

\[ SD_{12} = \sqrt{\frac{n_1 \cdot SD_1^2 + n_1 \cdot d_1^2 + n_2 \cdot SD_2^2 + n_2 \cdot d_2^2}{n_1 + n_2}} \]

Where:
- \( SD_1 = 15 \) (standard deviation of the first class)
- \( SD_2 = 20 \) (standard deviation of the second class)

Substituting the values:

\[ SD_{12} = \sqrt{\frac{25 \cdot 15^2 + 25 \cdot (3.75)^2 + 75 \cdot 20^2 + 75 \cdot (-1.25)^2}{100}} \]

Calculating each term:

1. \( 15^2 = 225 \)
2. \( (3.75)^2 = 14.0625 \)
3. \( 20^2 = 400 \)
4. \( (-1.25)^2 = 1.5625 \)

Now substituting these values into the equation:

\[ = \sqrt{\frac{25 \cdot 225 + 25 \cdot 14.0625 + 75 \cdot 400 + 75 \cdot 1.5625}{100}} \]

Calculating the products:

1. \( 25 \cdot 225 = 5625 \)
2. \( 25 \cdot 14.0625 = 351.5625 \)
3. \( 75 \cdot 400 = 30000 \)
4. \( 75 \cdot 1.5625 = 117.1875 \)

Now summing these:

\[ = \sqrt{\frac{5625 + 351.5625 + 30000 + 117.1875}{100}} = \sqrt{\frac{35893.75}{100}} = \sqrt{358.9375} \]

Calculating the square root:

\[ SD_{12} \approx 18.94 \]

Thus, the combined standard deviation \( SD_{12} \) is approximately **18.94 marks**.

### Final Answers:
- Combined Mean \( M_{12} = 83.75 \) marks
- Combined Standard Deviation \( SD_{12} \approx 18.94 \) marks