Variance

Variance is a statistical measure of the dispersion or spread of a set of values around their mean. It is calculated as the average of the squared deviations of each value from the mean, using the sum of squared differences divided by the number of observations (for a population variance) or by the number of observations minus one (for a sample variance, which uses Bessel's correction to provide an unbiased estimate). Variance quantifies how much individual data points differ from the average, with larger variances indicating greater variability and smaller variances indicating that values are clustered closely around the mean. The positive square root of the variance is the standard deviation, which is expressed in the same units as the original data and is often more intuitive to interpret.

Variance is central to many aspects of statistical analysis. It determines the precision of estimates, the power of statistical tests, the width of confidence intervals, and the ability to detect differences between groups. In ANOVA (analysis of variance), the total variance in the data is partitioned into components attributable to different sources (such as between group and within group variance), forming the basis for testing whether group means differ significantly. In regression analysis, the coefficient of determination (R squared) represents the proportion of the total variance in the dependent variable that is explained by the independent variables. Understanding variance is also essential for sample size calculation, because the required sample size increases as the variance of the outcome measure increases. In measurement theory, total observed variance is decomposed into true score variance and error variance, with reliability defined as the ratio of true score variance to total variance.

For a sample of n observations with values x1 through xn and a sample mean of x bar, the sample variance (s squared) is calculated as the sum of (xi minus x bar) squared for all observations, divided by (n minus 1). The division by (n minus 1) rather than n corrects for the downward bias that occurs because the sample mean is estimated from the same data. The variance is always nonnegative, and equals zero only when all values are identical. In the context of ANOVA, the F statistic is the ratio of the between group variance (mean square between) to the within group variance (mean square within). A large F statistic indicates that the between group variance exceeds what would be expected from random variation alone, providing evidence for group differences. In multivariate statistics, the variance covariance matrix extends the concept to multiple variables simultaneously, capturing both the variance of each variable and the covariances between pairs of variables.

Report and interpret variance (or its square root, the standard deviation) in any descriptive statistical summary, as a measure of dispersion complements a measure of central tendency. Use variance components in ANOVA to test hypotheses about group differences. Use variance in sample size calculations by plugging in the expected variance of the outcome to determine the number of participants needed for adequate power. Use the variance covariance matrix in multivariate analyses, structural equation modeling, and principal component analysis. When comparing the variability of two distributions or testing whether variances are equal (an assumption of some parametric tests), use Levene's test or its median based Brown Forsythe version, which are generally preferred to the F test for equality of variances because that test is highly sensitive to departures from normality.

A common mistake is reporting only the mean without the variance or standard deviation, which gives an incomplete picture of the data. Confusing variance with standard deviation in sample size formulas or when interpreting results leads to incorrect calculations; the variance is the squared standard deviation, and formulas may require either one. Assuming equal variances across groups without checking this assumption (homogeneity of variance) can distort the error rates and confidence intervals of ANOVA and t tests, especially when group sizes differ. Another error is interpreting a large variance as indicating a problem with the data when in fact the population being studied is inherently variable. Using the population variance formula (dividing by n) for a sample estimate produces a biased underestimate. Finally, in meta analysis, confusing between study variance (tau squared) with within study variance (sampling error) can lead to incorrect interpretation of heterogeneity.

A researcher measures systolic blood pressure in three groups: patients receiving drug A (n = 30, mean = 128, SD = 14), drug B (n = 30, mean = 132, SD = 16), and placebo (n = 30, mean = 140, SD = 15). The sample variances are 196, 256, and 225, respectively. The one way ANOVA partitions the total sum of squares into between group and within group parts, giving the between group variance (mean square between = 1120) and the within group variance (mean square within = 226). The F statistic is 1120 / 226 = 4.96 with 2 and 87 degrees of freedom (p = 0.009), indicating that the group means differ significantly. Post hoc pairwise comparisons reveal that only the difference between drug A and placebo is significant after Tukey adjustment.