Question

The mean and standard deviation of 100 items are 50 and 4, respectively then the sum of all squares of the items is

• A. 250000
• B. 251600
• C. 256100
• D. 265100

Hint:

A useful rearranged form of the variance formula to memorize for direct calculation is: $\sum x_i^2 = N \cdot (\sigma^2 + \mu^2)$. This allows you to plug numbers straight in: $100 \times (16 + 2500) = 251600$.

Answer 1

The Correct Option is B

To solve this problem, we need to calculate the sum of all squares of the items given their mean and standard deviation. Given:

• Mean (\(\bar{x}\)) = 50
• Standard Deviation (\(s\)) = 4
• Number of items (\(n\)) = 100

The formula for the sum of squares of deviations from the mean is given by:

\(S = n \times \left[(\bar{x}^2 + \sigma^2)\right]\)

Where \(S\) is the sum of all squares of the items.

Substitute the given values into the formula:

• \(n = 100\)
• \(\bar{x} = 50\)
• \(\sigma = 4\) (Standard deviation is given as \(s\))

Now compute:

\(S = 100 \times \left[(50^2 + 4^2)\right]\)

Calculate each part:

• \(\bar{x}^2 = 50^2 = 2500\)
• \(\sigma^2 = 4^2 = 16\)
• Adding these gives \(2500 + 16 = 2516\)

Now, compute the total sum:

\(S = 100 \times 2516 = 251600\)

Therefore, the sum of all squares of the items is 251600.

The correct answer is therefore: 251600.