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

Step 1: Understanding the Concept:
This problem relates the fundamental statistical measures: mean, standard deviation (or variance), and the sum of the squares of individual data points.

Step 2: Key Formula or Approach:
Use the computational formula for variance, which links all these quantities: $\text{Variance } (\sigma^2) = \frac{\sum x_i^2}{N} - (\text{Mean } \mu)^2$ Where: $\sigma$ = standard deviation $\sum x_i^2$ = sum of squares of all items (this is what we need to find) $N$ = total number of items $\mu$ = mean of the items Rearrange this formula to solve for $\sum x_i^2$.

Step 3: Detailed Explanation:
From the problem statement, we have the following values: Total number of items, $N = 100$ Mean, $\mu = 50$ Standard deviation, $\sigma = 4$ First, calculate the variance, which is the square of the standard deviation: $\text{Variance } (\sigma^2) = 4^2 = 16$ Now, substitute the known values into the variance formula: \[ \sigma^2 = \frac{\sum x_i^2}{N} - \mu^2 \] \[ 16 = \frac{\sum x_i^2}{100} - (50)^2 \] Calculate the square of the mean: \[ 16 = \frac{\sum x_i^2}{100} - 2500 \] Now, solve for the unknown term $\frac{\sum x_i^2}{100}$: Add 2500 to both sides: \[ 16 + 2500 = \frac{\sum x_i^2}{100} \] \[ 2516 = \frac{\sum x_i^2}{100} \] Finally, to find the sum of all squares of the items ($\sum x_i^2$), multiply both sides by 100: \[ \sum x_i^2 = 2516 \times 100 \] \[ \sum x_i^2 = 251600 \]

Step 4: Final Answer:
The sum of all squares of the items is 251600.