4.6 : Variance

The variance is a statistic estimating the variability of the dataset values from the mean. It is numerically equal to the square of the standard deviation of a dataset.

The variance is a valuable statistical tool used in the analysis of variance, estimation of risk, or volatility in financial markets.

The sample variance is denoted as the square of the sample standard deviation s, while the population variance is expressed as the square of the population standard deviation sigma.

Imagine if one were to estimate sample variance of weight of polar bears in different Arctic regions. On dividing the population into random samples and calculating the sample variances, one observes that the values center around the constant population variance value. Thus, the sample variance is an impartial estimator of the population variance.

The major disadvantage of variance is that its units vastly differ from the dataset units. For instance, the units of variance of rainfall in a year will be millimeters squared, which is unhelpful. Therefore, in most analyses, standard deviation is preferred to variance.

The deviations show how spread out the data are about the mean. A positive deviation occurs when the data value exceeds the mean, whereas a negative deviation occurs when the data value is less than the mean. If the deviations are added, the sum is always zero. So one cannot simply add the deviations to get the data spread. By squaring the deviations, the numbers are made positive; thus, their sum will also be positive.

The standard deviation measures the spread in the same units as the data. The variance is defined as the square of the standard deviation. Thus, its units differ from that of the original data. The sample variance is represented by Equation1 , while the population variance is represented by Equation2 .

For variance, the calculation uses a division by n – 1 instead of n because the data is a sample. This change is due to the sample variance being an estimate of the population variance. Based on the theoretical mathematics behind these calculations, dividing by (n – 1) gives a better estimate of the population variance.

This text is adapted from Openstax, Introductory Statistics, Section 2.7 Measure of the Spread of the Data.

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Variance Standard Deviation Deviations Mean Data Spread Sample Variance Population Variance Positive Deviation Negative Deviation Calculations Statistical Measurement Openstax

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