Why Does Sample Variance Divide by n or n - 1?

Dividing by $n$ gives the variance of the observed data treated as the complete population. Dividing by $n-1$ gives the unbiased estimator of the variance of a larger population from which the sample was drawn.

Why Are the Denominators Different?

For data values $x_1,x_2,\ldots,x_n$ with mean $\bar{x}$, the sum of squared deviations is

$$\sum_{i=1}^{n}(x_i-\bar{x})^2.$$

Once $\bar{x}$ has been calculated from the sample, only $n-1$ deviations can vary independently because

$$\sum_{i=1}^{n}(x_i-\bar{x})=0.$$

Therefore, the calculation has $n-1$ degrees of freedom. Dividing by $n-1$, called Bessel's correction, compensates for the tendency of a sample to underestimate the population variance.

The formulas are related by

$$s_{n-1}^2=\frac{n}{n-1}s_n^2.$$

The distinction is about what the data represent, not simply the size of the data set. A list of 1,000 measurements can still be a sample if it was selected from a larger population. Conversely, five recorded outcomes can be the whole population if those five are all the cases being studied. The denominator therefore follows the statistical purpose, not whether the list seems large or small.

For the sample $2,4,6$, $\bar{x}=4$ and the sum of squared deviations is $8$. Thus,

$$s_n^2=\frac{8}{3},\qquad s_{n-1}^2=\frac{8}{2}=4.$$

The second value is larger because $n-1<n$; its role is to correct bias across repeated samples, not to claim that this one sample contains more variation.

A common misconception is that dividing by $n-1$ makes the variance “more accurate” in every situation. It is appropriate only when using a sample to estimate a larger population's variance.

IB Exam Technique

This is AHL 4.14: unbiased estimators. On a GDC, $\sigma_x$ usually corresponds to division by $n$, while $S_x$ corresponds to division by $n-1$. Read the context carefully: use when the listed data are treated as the population, but use when estimating from a sample. Keep full GDC precision and round only the final result.