Variance

A dispersion measure representing the average squared deviation of individual data points from the mean of a dataset.

Source: Casella G, Berger RL. Statistical Inference. 2nd ed. Duxbury; 2002.

Variance measures how spread out a dataset is by averaging the squared distances of the values from their mean. Squaring the deviations makes every one positive and gives extra weight to points far from the centre, so a large variance means the data scatter widely and a small one that they cluster. It is the foundation of many statistical methods, though its squared units make it less directly interpretable than its square root, the standard deviation. The average squared spread around the mean is what it captures. Kirkwood and Sterne (2003) describe this measure.

Source: Kirkwood & Sterne 2003

Variance is calculated by taking the difference between each value and the mean, squaring these differences, and averaging them; for a sample, the sum of squared deviations is often divided by the number of observations minus one to give an unbiased estimate of the population variance. So variance is calculated as the average of the squared deviations from the mean, with the sample variance using a divisor of the sample size minus one for unbiasedness, and because it squares the deviations, it uses all the data and is sensitive to values far from the mean, which is why it is expressed in squared units and the standard deviation is often preferred for interpretation.

Source: Casella & Berger 2002

Variance measures the variability or spread of a dataset around its mean: a small variance means the values cluster closely around the mean, while a large variance means they are widely dispersed. Being in squared units, it is less directly interpretable than the standard deviation. So variance measures how much the values differ from the mean on average, in squared terms, quantifying dispersion, and although its squared units make it harder to interpret directly than the standard deviation, it is central to statistical theory, since many methods, including analysis of variance and the decomposition of prediction error, are built upon it.

Source: Casella & Berger 2002

Variance relates to the standard deviation as its square: the variance is the average squared deviation from the mean, in squared units, and the standard deviation is its square root, in the original units of the data. So variance and standard deviation measure the same variability, differing in scale, with the standard deviation usually preferred for reporting because it is on the data's scale and directly comparable to the mean, while variance is often used in calculations and theory, since squaring makes it mathematically convenient, and the two are directly convertible through the square root.

Source: Casella & Berger 2002

Variance is important in statistics because it quantifies variability, which underlies the precision of estimates, the design of studies, and many analytical methods; for example, analysis of variance partitions variance to compare groups, and the mean squared error decomposes into bias and variance. So variance matters as a foundational measure of dispersion and a building block of statistical theory, since much of inference and modelling concerns explaining or accounting for variance, which is why understanding and estimating variance is central, from summarising data and gauging precision to comparing groups and evaluating estimators and predictions.

Source: Casella & Berger 2002