Question

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

Explanation:

Step1: Recall standard - deviation concepts

The standard deviation of a sample is denoted by $s$.

Step2: Identify sample and population formulas

The formula for the variance of a population is $\sigma^{2}=\frac{(x_1-\mu)^2+(x_2-\mu)^2+\cdots+(x_N-\mu)^2}{N}$, and its standard - deviation is $\sigma=\sqrt{\frac{(x_1-\mu)^2+(x_2-\mu)^2+\cdots+(x_N-\mu)^2}{N}}$, where $\mu$ is the population mean and $N$ is the population size. For a sample, the mean is $\bar{x}$ and the formula for the sample standard deviation uses $n - 1$ in the denominator to correct for bias, and is $s=\sqrt{\frac{(x_1-\bar{x})^2+(x_2 - \bar{x})^2+\cdots+(x_n-\bar{x})^2}{n - 1}}$. The fourth option is the formula for sample variance, not sample standard deviation. So the correct formula for the standard deviation of sample data is the first option.

Answer:

The first option: $s = \sqrt{\frac{(x_1-\bar{x})^2+(x_2 - \bar{x})^2+\cdots+(x_n-\bar{x})^2}{n - 1}}$