# measures of variation

**measures of variation** Quantities that express the amount of variation in a random variable (compare measures of location). Variation is sometimes described as *spread* or *dispersion* to distinguish it from systematic trends or differences. Measures of variation are either properties of a probability distribution or sample estimates of them.

The *range* of a sample is the difference between the largest and smallest value. The *interquartile range* is potentially more useful. If the sample is ranked in ascending order of magnitude two values of *x* may be found, the first of which is exceeded by 75% of the sample, the second by 25%; their difference is the interquartile range. An analogous definition applies to a probability distribution.

The *variance* is the expectation (or mean) of the square of the difference between a random variable and its mean; it is of fundamental importance in statistical analysis. The variance of a continuous distribution with mean μ is

and is denoted by σ^{2}. The variance of a discrete distribution is

and is also denoted by σ^{2}. The sample variance of a sample of *n* observations with mean *x̄* is

and is denoted by *s*^{2}. The value (*n* – 1) corrects for bias.

The *standard deviation* is the square root of the variance, denoted by σ (for a distribution) or *s* (for a sample). The standard deviation has the same units of measurement as the mean, and for a normal distribution about 5% of the distribution lies beyond about two standard deviations each side of the mean. The standard deviation of the distribution of an estimated quantity is termed the *standard error*.

The *mean deviation* is the mean of the absolute deviations of the random variable from the mean.

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**measures of variation**