## Quick Reference

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 ∫(*x* − μ)^{2}*f*(*x*) d*x* and is denoted by σ^{2}. The variance of a discrete distribution is ∑ (*x* − μ)^{2}*p*(*x*) and is also denoted by σ^{2}. The sample variance of a sample of *n* observations with mean *x̄* is ∑ (*x** _{i}* −

*x̄*)

^{2}/ (

*n*− 1) and is denoted by

*s*

^{2}. The value (

*n*− 1) corrects for bias.

∫(*x* − μ)^{2}*f*(*x*) d*x*

∑ (*x* − μ)^{2}*p*(*x*)

∑ (*x** _{i}* −

*x̄*)

^{2}/ (

*n*− 1)

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.

*Subjects:*
Computing.

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