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

### Step 1: Understand the given information
We have:
- Mean (\(\bar{X}\)) = 50
- Standard deviation (\(\sigma\)) = 5
- Number of observations (n) = 100

### Step 2: Use the formula for variance
The variance (\(\sigma^2\)) is given by the formula:
\[ \sigma^2 = \frac{\sum X_i^2}{n} - \left(\frac{\sum X_i}{n}\right)^2 \]
where:
- \(\sum X_i\) is the sum of all observations.
- \(\sum X_i^2\) is the sum of the squares of all observations.

### Step 3: Substitute the known values
We know:
- \(\sigma = 5\) so \(\sigma^2 = 5^2 = 25\)
- \(\frac{\sum X_i}{n} = 50\) so \(\left(\frac{\sum X_i}{n}\right)^2 = 50^2 = 2500\)

Now substituting these values into the variance formula:
\[ 25 = \frac{\sum X_i^2}{100} - 2500 \]

### Step 4: Rearrange the equation to find \(\sum X_i^2\)
First, add \(2500\) to both sides:
\[ 25 + 2500 = \frac{\sum X_i^2}{100} \]
\[ 2525 = \frac{\sum X_i^2}{100} \]

Now, multiply both sides by \(100\) to isolate \(\sum X_i^2\):
\[ \sum X_i^2 = 2525 \times 100 \]
\[ \sum X_i^2 = 252500 \]

### Conclusion
The sum of the squares of all observations is \(252500\).