Sample Variance Formulas

B

I and III are equivalent and correct; III is the computational formula for sample variance.

Let's go through each option and explain the formulas for sample variance:

I. [1/(n-1)] * sum (as i goes from 1 to n) of (X_i - Xbar)^2.

This formula represents the sample variance using Bessel's correction, which is commonly used when calculating the sample variance. In this formula, n represents the sample size, X_i represents each individual data point in the sample, and Xbar represents the sample mean. The formula calculates the squared difference between each data point and the sample mean, sums them up, and then scales the result by dividing it by (n-1). Dividing by (n-1) instead of n corrects for the bias in the sample variance estimation and provides an unbiased estimate of the population variance.

II. [1/n] * sum (as i goes from 1 to n) of (X_i - Xbar)^2.

This formula represents an incorrect calculation for the sample variance. It uses the same elements as the previous formula but scales the squared differences by dividing them by n instead of (n-1). This formula does not use Bessel's correction, which means it does not account for the bias in the sample variance estimation. Therefore, this formula does not provide an unbiased estimate of the population variance.

III. [1/(n-1)] * {sum (as i goes from 1 to n) of (X_i)^2 - (1/n) * [sum (as i goes from 1 to n) of (X_i)]^2}.

This formula also represents the sample variance using Bessel's correction. It involves two parts: the sum of squared data points and the square of the sum of data points. The squared sum of data points divided by n corrects for the bias due to squaring the data points, and the result is subtracted from the sum of squared data points. Finally, the entire expression is scaled by dividing by (n-1). This formula also provides an unbiased estimate of the population variance.

From the explanations above, it is clear that the correct answer is:

A. I only.

Only the first formula correctly calculates the sample variance using Bessel's correction, providing an unbiased estimate of the population variance. The second formula lacks the correction, while the third formula includes the correction but does not calculate the variance correctly.