Business Mathematics and Basic Statistics · Ch 7 — Measures of Dispersion

Variance and Standard Deviation — Direct Method

Mean deviation uses absolute values to stop positive and negative deviations from cancelling out. An alternative, more mathematically convenient way to avoid the cancellation is to square every deviation instead of taking its absolute value — squaring, like the absolute value, always produces a non-negative number, but squares are far easier to work with algebraically (which is why almost all of statistics beyond this chapter is built on squared deviations rather than absolute ones).

The average of the squared deviations from the mean is called the variance; its square root, which brings the measure back to the same units as the original data, is called the standard deviation — by far the most widely used measure of dispersion in statistics.

Note

Variance and Standard Deviation — Individual Observations

$\sigma^2 = \dfrac{\sum (x_i-\bar{x})^2}{n}, \qquad \sigma = \sqrt{\sigma^2} = \sqrt{\dfrac{\sum(x_i-\bar{x})^2}{n}}$

Note

Variance and Standard Deviation — Frequency Distribution (Direct Method)

$\sigma^2 = \dfrac{\sum f(x-\bar{x})^2}{N}, \qquad \sigma = \sqrt{\dfrac{\sum f(x-\bar{x})^2}{N}}$

An algebraically equivalent, often faster route (expand the square and simplify) gives the same result without first tabulating every deviation:

$\sigma^2 = \dfrac{\sum fx^2}{N} - \left(\dfrac{\sum fx}{N}\right)^2 = \dfrac{\sum fx^2}{N} - \bar{x}^2$ …

Definition 1Variance

The average of the squared deviations of every observation from the mean; $\sigma^2 = \sum(x-\bar{x})^2/n$ for individual data, or $\sum f(x-\bar{x})^2/N$ f …

Definition 2Standard Deviation

The (positive) square root of the variance, $\sigma = \sqrt{\sigma^2}$ — brought back to the same unit as the original data, unlike variance, …

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