The standard deviations of two sets containing 10 and 20 members are 2 and 3 respectively measured from their common mean 5. Find the S.D. for the whole set of 30 members.

Answer

To find the standard deviation for the combined set of 30 members, we will follow these steps:

### Step 1: Understand the given information
We have two sets of data:
- Set 1: 10 members with a standard deviation (SD) of 2
- Set 2: 20 members with a standard deviation (SD) of 3
- Both sets have a common mean of 5.

### Step 2: Use the formula for standard deviation
The formula for standard deviation \( S \) is given by:
\[ S = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2} \]
Where:
- \( \sum x^2 \) = sum of the squares of the data points
- \( n \) = number of data points
- \( \bar{x} \) = mean of the data points

### Step 3: Calculate \( \sum x^2 \) for Set 1
For the first set:
- Given \( S_1 = 2 \) and \( n_1 = 10 \)
Using the formula:
\[ 2 = \sqrt{\frac{\sum x_1^2}{10} - 5^2} \]
Squaring both sides:
\[ 4 = \frac{\sum x_1^2}{10} - 25 \]
Rearranging gives:
\[ \frac{\sum x_1^2}{10} = 29 \]
Multiplying by 10:
\[ \sum x_1^2 = 290 \]

### Step 4: Calculate \( \sum x^2 \) for Set 2
For the second set:
- Given \( S_2 = 3 \) and \( n_2 = 20 \)
Using the formula:
\[ 3 = \sqrt{\frac{\sum x_2^2}{20} - 5^2} \]
Squaring both sides:
\[ 9 = \frac{\sum x_2^2}{20} - 25 \]
Rearranging gives:
\[ \frac{\sum x_2^2}{20} = 34 \]
Multiplying by 20:
\[ \sum x_2^2 = 680 \]

### Step 5: Calculate the combined \( \sum x^2 \)
Now, we can combine the sums of squares:
\[ \sum x^2 = \sum x_1^2 + \sum x_2^2 = 290 + 680 = 970 \]

### Step 6: Calculate the combined mean
The total number of members is:
\[ n = n_1 + n_2 = 10 + 20 = 30 \]
The common mean is \( \bar{x} = 5 \).

### Step 7: Calculate the standard deviation for the combined set
Now we can use the formula for standard deviation for the combined set:
\[ S = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2} \]
Substituting the values:
\[ S = \sqrt{\frac{970}{30} - 5^2} \]
Calculating:
\[ S = \sqrt{\frac{970}{30} - 25} \]
Calculating \( \frac{970}{30} \):
\[ \frac{970}{30} = 32.33 \]
Then:
\[ S = \sqrt{32.33 - 25} = \sqrt{7.33} \]

### Step 8: Final calculation
Calculating \( \sqrt{7.33} \):
\[ S \approx 2.71 \]

Thus, the standard deviation for the whole set of 30 members is approximately **2.71**.

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