Question

which formula is used to calculate the standard deviation of sample data?
$s = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n - 1}$
$\sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \dots + (x_n - \mu)^2}{n}}$
$s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n - 1}}$
$a^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \dots + (x_n - \mu)^2}{n}$

Explanation:

Step1: Recall sample standard deviation formula

Sample standard deviation (\(s\)) uses the sample mean (\(\bar{x}\)) and divides by \(n - 1\) (degrees of freedom for sample). The formula involves taking the square root of the sum of squared deviations from the sample mean, divided by \(n - 1\).

Step2: Analyze each option

• First option: Missing square root, so it's sample variance (not standard deviation).
• Second option: Uses population mean (\(\mu\)) and population size \(N\), so it's population standard deviation (\(\sigma\)), not sample.
• Third option: Uses sample mean (\(\bar{x}\)), divides by \(n - 1\), and has square root—matches sample standard deviation formula.
• Fourth option: Uses population mean and \(N\), and is variance (\(a^2\) likely \(\sigma^2\)), not sample standard deviation.

Answer:

\( s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \dots + (x_n - \bar{x})^2}{n - 1}} \) (the third formula)