If the mean of 100 observations is 50 and their standard deviations is 5,than the sum of all squares of all the observations is

Options

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

Explanation

To find the sum of squares of all observations given the mean and standard deviation, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Information:**
- Mean (\( \bar{x} \)) = 50
- Number of observations (\( n \)) = 100
- Standard deviation (\( \sigma \)) = 5

2. **Understand the Relationship Between Variance and Standard Deviation:**
- The variance (\( \sigma^2 \)) is the square of the standard deviation.
- Therefore, \( \sigma^2 = 5^2 = 25 \).

3. **Use the Formula for Variance:**
- The formula for variance in terms of the sum of squares is:
\[ \sigma^2 = \frac{\sum_{i=1}^{n} x_i^2}{n} - \bar{x}^2 \]
- Rearranging this gives:
\[ \sum_{i=1}^{n} x_i^2 = n \cdot \sigma^2 + n \cdot \bar{x}^2 \]

4. **Substitute the Values:**
- Substitute \( n = 100 \), \( \sigma^2 = 25 \), and \( \bar{x} = 50 \):
\[ \sum_{i=1}^{100} x_i^2 = 100 \cdot 25 + 100 \cdot (50^2) \]

5. **Calculate \( 50^2 \):**
- Calculate \( 50^2 = 2500 \).

6. **Complete the Calculation:**
- Now substitute back into the equation:
\[ \sum_{i=1}^{100} x_i^2 = 100 \cdot 25 + 100 \cdot 2500 \]
- This simplifies to:
\[ \sum_{i=1}^{100} x_i^2 = 2500 + 250000 \]
- Therefore:
\[ \sum_{i=1}^{100} x_i^2 = 252500 \]

### Final Answer:
The sum of all squares of all the observations is \( 252500 \).

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