What is variance
Variance measures how spread out a set of numbers is around its mean. To calculate it, find each value's deviation from the mean (the difference between the value and the mean), square each deviation to remove negative signs, and average those squares. The larger the variance, the more spread out the data is; the smaller it is, the closer the data sits to the mean.
Population variance and sample variance
When the data represents the entire population of interest, the sum of squared deviations is divided by n (the number of values), giving the population variance. When the data is only a sample drawn from a larger population, the division is by n − 1 instead of n, giving the sample variance. This correction, called Bessel's correction, exists because a sample tends to underestimate the true population variance, and dividing by a smaller number compensates for that tendency.
What is pooled variance
When two different groups are assumed to share the same population variance, their sample variances can be combined into a single, more precise estimate called the pooled variance. The formula weights each variance by its degrees of freedom (n − 1): s²pooled = [(n₁−1)s²₁ + (n₂−1)s²₂] / (n₁+n₂−2). This calculation is common in statistical tests that compare two groups, such as Student's t-test with equal variances.
Practical example
In an experiment, group A has 10 observations with sample variance 4, and group B has 15 observations with sample variance 6. Assuming both populations share the same variance, the pooled variance is approximately 5.22, a single estimate that makes use of data from both groups.