Question

by studying the standard deviation formula, given below, explain why the standard deviation of a set of data will always be greater than or equal to 0.

\\s = \sqrt{\frac{\sum (x-\bar{x})^2}{n-1}}\\

choose the correct answer below.

a. the mean, \\(\bar{x}\\), will always be greater than or equal to 0.
b. the values of \\((x-\bar{x})^2\\) will always be greater than or equal to 0.
c. the values of \\(x\\) will always be greater than or equal to 0.
d. the values of \\(x-\bar{x}\\) will always be greater than or equal to 0.

Explanation:

Analyze the standard deviation formula

Using the Standard Deviation and Sample Standard Deviation knowledge points

$$ s = \sqrt{\frac{\sum(x-\bar{x})^2}{n-1}} $$

Evaluate the terms inside the square root

Using the Standard Deviation and Sample Standard Deviation knowledge points

$$
LATEXBLOCK0
$$

Determine the correct multiple-choice option

Using the Standard Deviation knowledge point
The principal square root of a non-negative number is always greater than or equal to 0. The term \((x-\bar{x})^2\) is a squared real number, which ensures it is always non-negative.

Answer:

• A. The mean, \(\bar{x}\), will always be greater than or equal to 0.
• B. The values of \((x-\bar{x})^2\) will always be greater than or equal to 0. (Correct answer)
• C. The values of \(x\) will always be greater than or equal to 0.
• D. The values of \(x-\bar{x}\) will always be greater than or equal to 0.