Explain the concept of variance in statistics.

Official answer

For a random variable with finite second moment, variance is Var(X)=E[(X−E[X])²]. It measures squared spread about the mean and is nonnegative. Standard deviation is its square root and has the original measurement units.

For the complete population {1,2,3}, the mean is 2 and the average squared deviation is (1+0+1)/3=2/3. If these values are an iid sample and we estimate an unknown population variance, the usual unbiased sample estimator divides by n−1, giving s²=2/2=1. The two denominators answer different questions.

Adding a constant does not change variance; multiplying values by c multiplies variance by c². Variance is sensitive to extreme observations and may not be finite for some distributions.

In experiment planning, outcome variability affects the precision and power available from a given sample. Lower variability can improve precision, but neither a chosen sample size nor a variance estimate guarantees statistical significance. Examine whether spread reflects real heterogeneity, measurement issues or a mixture of populations before interpreting it.