Mean of 40 terms is 25 and S.D. is 4, then find the sum of the squares of all terms

Answer

To find the sum of the squares of all terms given that the mean of 40 terms is 25 and the standard deviation is 4, we can follow these steps:

### Step 1: Calculate the sum of the terms
The mean (x̄) of a set of numbers is given by the formula:

\[ \bar{x} = \frac{\sum x_i}{n} \]

Where:
- \(\bar{x}\) = mean
- \(\sum x_i\) = sum of all terms
- \(n\) = number of terms

Given:
- \(\bar{x} = 25\)
- \(n = 40\)

We can rearrange the formula to find \(\sum x_i\):

\[ \sum x_i = \bar{x} \times n = 25 \times 40 = 1000 \]

### Step 2: Use the standard deviation formula
The formula for standard deviation (SD) is:

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

Given:
- \(SD = 4\)

We can substitute the known values into the formula:

\[ 4 = \sqrt{\frac{\sum x_i^2}{40} - \left(\frac{1000}{40}\right)^2} \]

### Step 3: Simplify the equation
First, calculate \(\frac{1000}{40}\):

\[ \frac{1000}{40} = 25 \]

Now substitute this back into the equation:

\[ 4 = \sqrt{\frac{\sum x_i^2}{40} - 25^2} \]

This simplifies to:

\[ 4 = \sqrt{\frac{\sum x_i^2}{40} - 625} \]

### Step 4: Square both sides
To eliminate the square root, square both sides of the equation:

\[ 16 = \frac{\sum x_i^2}{40} - 625 \]

### Step 5: Solve for \(\sum x_i^2\)
Rearranging gives:

\[ \frac{\sum x_i^2}{40} = 16 + 625 \]

Calculating the right side:

\[ \frac{\sum x_i^2}{40} = 641 \]

Now, multiply both sides by 40 to find \(\sum x_i^2\):

\[ \sum x_i^2 = 641 \times 40 = 25640 \]

### Final Answer
The sum of the squares of all terms is:

\[ \sum x_i^2 = 25640 \]

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