The mean of 100 observations is 50 and their standard deviation is 5. the sum of squares all the observations is
(i) 50000
(ii) 250000
(iii) 252500
(iv) 255000

Options

A. 50000
B. 250000
C. 252500
D. 255000

Answer

To find the sum of squares of all the observations, we can use the relationship between the standard deviation, mean, and the sum of squares. Here are the steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- Mean (\( \bar{x} \)) = 50
- Standard Deviation (\( \sigma \)) = 5
- Number of observations (\( n \)) = 100

2. **Use the formula for standard deviation**:
The formula for the standard deviation is:
\[ \sigma^2 = \frac{\sum x^2}{n} - \bar{x}^2 \]
Here, \( \sigma^2 \) is the variance.

3. **Substitute the known values into the formula**:
\[ 5^2 = \frac{\sum x^2}{100} - 50^2 \]
This simplifies to:
\[ 25 = \frac{\sum x^2}{100} - 2500 \]

4. **Rearrange the equation to solve for \( \sum x^2 \)**:
\[ 25 + 2500 = \frac{\sum x^2}{100} \]
\[ 2525 = \frac{\sum x^2}{100} \]

5. **Multiply both sides by 100 to find \( \sum x^2 \)**:
\[ \sum x^2 = 2525 \times 100 \]
\[ \sum x^2 = 252500 \]

6. **Conclusion**:
The sum of squares of all the observations is \( 252500 \).

### Final Answer:
The correct option is (iii) 252500.