Question

If the mean and standard deviation of 10 observations are 24 and 4 respectively, then the sum of the squares of all observations is

• A. 5920
• B. 5820
• C. 5720
• D. 5640
• E. 5660

Hint:

Logic Tip: The sum of squares can always be found quickly by multiplying the number of observations by the sum of the variance and the mean squared: $\sum x^2 = n(\sigma^2 + \mu^2)$. Using this, $10 \times (16 + 576) = 5920$.

Answer 1

The Correct Option is A

Concept:
The variance ($\sigma^2$) of a dataset is the square of the standard deviation ($\sigma$). The computational formula for variance is: $$\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$$ where $\sum x_i^2$ is the sum of the squares of the observations, $n$ is the number of observations, and $\bar{x}$ is the mean.
Step 1: Identify the given values.
Number of observations, $n = 10$ Mean, $\bar{x} = 24$ Standard deviation, $\sigma = 4$
Step 2: Calculate the variance.
Variance is the standard deviation squared: $$\sigma^2 = 4^2 = 16$$
Step 3: Apply the variance formula to find the sum of squares.
Substitute the known values into the computational formula: $$16 = \frac{\sum x_i^2}{10} - (24)^2$$ Calculate the square of the mean: $$16 = \frac{\sum x_i^2}{10} - 576$$
Step 4: Isolate and solve for the sum of squares.
Add 576 to both sides: $$16 + 576 = \frac{\sum x_i^2}{10}$$ $$592 = \frac{\sum x_i^2}{10}$$ Multiply by 10 to find the final value: $$\sum x_i^2 = 5920$$