The adjusted sample variance is a measure of the dispersion of a sample around its mean.
It is obtained by:
summing the squared deviations from the mean;
dividing the result thus obtained by the number of observations minus one.
Below we provide a precise definition, we illustrate its calculation with an example, and we introduce some of its properties.
It is also often called unbiased sample variance because, under certain conditions, it is an unbiased estimator of the population variance.
It is defined as follows.
Definition Given a sample of observations, their adjusted sample variance iswhere is their sample mean:
The adjective "adjusted" refers to the fact that the sum of squared deviations is divided by rather than by .
Suppose that we have observed the following sample of six observations:
The sample mean is
The adjusted sample variance is
Another common way to compute the sample variance is which is called unadjusted or biased sample variance.
The main difference is that the sum of squared deviations:
is divided by in the unadjusted variance;
is divided by in the adjusted variance.
We can write the adjusted variance in terms of the unadjusted one:
The ratio is called a degrees of freedom adjustment. It is also sometimes called Bessel's correction.
Suppose that the observations are all drawn from probability distributions having the same mean and the same variance .
It is possible to prove (see Variance estimation) that, if are independent, then the adjusted sample variance is an unbiased estimator of .
The degrees of freedom adjustment is not a free lunch: it eliminates the bias, but it usually increases the variance of the sample variance.
This can be proved rigorously when are drawn from a normal distribution (see Variance estimation).
The lecture entitled Variance estimation presents more details about the adjusted sample variance and its properties (e.g., its consistency and lack of bias).
For more details about the biased version, you can instead see the glossary entry on the Unadjusted sample variance.
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Please cite as:
Taboga, Marco (2021). "Adjusted sample variance", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/adjusted-sample-variance.
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