In introductory statistics, which formula should you use to compute the sample variance from data values $x_1$, $x_2$, ..., $x_n$ with sample mean $\overline{x}$?

• A. $s=\frac{\sum (x_i-\overline{x})}{n-1}$
• B. $s^2=\frac{\sum {(x_i-\overline{x})}^2}{n}$
• C. $s^2=\frac{\sum {(x_i-\overline{x})}^2}{n-1}$
• D. $s^2=\sqrt{\frac{\sum {(x_i-\overline{x})}^2}{n-1}}$

Answer: (C)

Verified step by step guidance

1. Identify that the sample variance is a measure of how spread out the data values are around the sample mean. It is calculated by averaging the squared deviations from the mean, but with a correction in the denominator to account for sample size.
2. Recall the formula for the sample mean: \(\overline{x} = \frac{1}{n} \sum_{i=1}^n x_i\), where \(x_i\) are the data values and \(n\) is the sample size.
3. Calculate the squared deviations for each data point: \((x_i - \overline{x})^2\) for \(i = 1, 2, ..., n\).
4. Sum all the squared deviations: \(\sum_{i=1}^n (x_i - \overline{x})^2\).
5. Divide this sum by \(n - 1\) (not \(n\)) to get the sample variance: \(s^2 = \frac{\sum_{i=1}^n (x_i - \overline{x})^2}{n - 1}\). This denominator adjustment is known as Bessel's correction and provides an unbiased estimate of the population variance.