Measures of Variability

Variance and Standard Deviation

Among the most common measures of variability is the variance, σ2, which is a function of the differences between each data point and the average (mean) of the data. Larger variances indicate more variability (see Figure 1, which shows the difference in variability for two groups with identical means when σ2 = 1 and σ2 = 9). The variance is always greater than or equal to 0: If all the values are identical, σ2 = 0. While the variance is affected if each measurement in the data is multiplied by the same number (change in the scale), it is not changed if the same number is added to each measurement (shift in location). Another useful property of the variance is that the variance of a sum of uncorrelated measures is equal to the sum of their variances.

The variance of a population may be estimated from a sample of size n using the unbiased estimator

, where X1, …, Xn is a random sample and is the sample mean. The biased version, σ2 which replaces the denominator, n — 1, with n, is less commonly used. As an example, Table 1 gives the birth weights of 12 male and 12 female infants in kilograms. The mean birth weight for males in this sample is = 3.48, and the sample variance is s2 = .26. For females, the sample mean is = 3.30, and the sample variance is s2 = .15.

The standard deviation, σ or SD, is the square root of the variance and is often used with the average to describe the distribution of a measure. It is greater than or equal to 0 and is measured in the same units as the measure of interest, which is useful for interpretation. The sample standard deviation, s, the square root of the sample variance, s2, is used in many formulas for confidence intervals and hypothesis testing. In Table 1, s = .51 [Page 735]for males and 39 for females. When estimating the combined (pooled) standard deviation from more than one group or sample, the following formula is often used:

Table 1 Birth weight (kg) of full-term male and female infants
Males		Females
4.1	3.4	3.8	3.1
3.5	3.2	2.8	3.6
3.3	3.0	3.8	3.8
4.0	3.2	3.4	3.0
3.7	2.7	3.4	2.7
4.5	3.2	3.0	3.2