If mean and standard deviation of 100 items are 50 and 4 respectively the find the sum of all the item and the sum of the squares of item.

Explanation

To solve the problem, we need to find the sum of all the items and the sum of the squares of the items given the mean and standard deviation.

### Step 1: Find the sum of all items
The mean (average) is given by the formula:

\[ \text{Mean} = \frac{\text{Sum of all items}}{n} \]

Where \( n \) is the number of items. We are given:
- Mean = 50
- \( n = 100 \)

Using the formula, we can rearrange it to find the sum of all items:

\[ \text{Sum of all items} = \text{Mean} \times n \]

Substituting the values:

\[ \text{Sum of all items} = 50 \times 100 = 5000 \]

### Step 2: Find the sum of the squares of the items
We know that the standard deviation (SD) is given by the formula:

\[ \text{SD} = \sqrt{\frac{\sum x_i^2}{n} - \left(\frac{\sum x_i}{n}\right)^2} \]

Where:
- \( \sum x_i^2 \) is the sum of the squares of the items.
- \( \sum x_i \) is the sum of all items, which we found to be 5000.
- \( n = 100 \)

We are given:
- Standard Deviation = 4

First, we calculate the variance (which is the square of the standard deviation):

\[ \text{Variance} = \text{SD}^2 = 4^2 = 16 \]

Now, substituting into the variance formula:

\[ 16 = \frac{\sum x_i^2}{100} - \left(\frac{5000}{100}\right)^2 \]

Calculating \( \left(\frac{5000}{100}\right)^2 \):

\[ \left(\frac{5000}{100}\right)^2 = 50^2 = 2500 \]

Now substituting this back into the variance equation:

\[ 16 = \frac{\sum x_i^2}{100} - 2500 \]

Rearranging to find \( \sum x_i^2 \):

\[ \frac{\sum x_i^2}{100} = 16 + 2500 \]

Calculating the right side:

\[ \frac{\sum x_i^2}{100} = 2516 \]

Now multiplying both sides by 100 to find \( \sum x_i^2 \):

\[ \sum x_i^2 = 2516 \times 100 = 251600 \]

### Final Results
- The sum of all items is **5000**.
- The sum of the squares of the items is **251600**.