Final answer:
The variance of a sample is divided by n-1 because it provides an unbiased estimate of the population variance, known as Bessel's correction, and accounts for the sample's degrees of freedom.
Step-by-step explanation:
When we calculate the variance of a sample, we divide by n-1 instead of n because this approach provides an unbiased estimator of the population variance. If we divide by n, the variance tends to be underestimated because a sample tends not to be as variable as the entire population. This adjustment, which is known as Bessel's correction, accounts for the degrees of freedom in the sample data. In effect, by dividing by n-1, we're correcting for the fact that a sample is just an estimate of the whole population and hence doesn't capture all the variability.
The degrees of freedom, typically denoted as df, equals n-1 where n is the sample size. When calculating statistics like the sample variance or the sample standard deviation, the degrees of freedom come into play to compensate for the fact that we're using sample data to estimate parameters of the entire population. This concept is critical in hypothesis testing and constructing confidence intervals.
The rationale behind using n-1 also applies when calculating the sample standard deviation and in the context of the F ratio, which is used to compare variances between two samples. Similarly, when dealing with the sampling distribution of the mean, especially as it pertains to the central limit theorem, the sample size affects the variance of the distribution of sample means.