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Business Mathematics and Statistics · Ch 9 — Measures of Dispersion

Standard Deviation and Variance — Direct Method

5

Standard Deviation and Variance — Direct Method

Mean Deviation uses absolute values to stop positive and negative deviations from cancelling out. An alternative — mathematically far more convenient, and the one on which almost all further statistics is built — is to square every deviation instead. The average of the squared deviations from the mean is the Variance; its positive square root, which restores the original unit of measurement, is the Standard Deviation (SD) — the single most widely used measure of dispersion in statistics.

Note

Variance and SD — Individual Observations

σ2=∑(xi−xˉ)2n,σ=∑(xi−xˉ)2n\sigma^2 = \dfrac{\sum (x_i - \bar{x})^2}{n}, \qquad \sigma = \sqrt{\dfrac{\sum(x_i-\bar{x})^2}{n}}

Note

Variance and SD — Discrete/Continuous Series (Direct Method)

σ2=∑f(x−xˉ)2N,σ=∑f(x−xˉ)2N\sigma^2 = \dfrac{\sum f(x-\bar{x})^2}{N}, \qquad \sigma = \sqrt{\dfrac{\sum f(x-\bar{x})^2}{N}}

An algebraically equivalent, usually faster route (expand the square) skips tabulating every deviation directly:

σ2=∑fx2N−(∑fxN)2=∑fx2N−xˉ2\sigma^2 = \dfrac{\sum fx^2}{N} - \left(\dfrac{\sum fx}{N}\right)^2 = \dfrac{\sum fx^2}{N} - \bar{x}^2 …

Definition 1Variance ($\sigma^2$)

The average of the squared deviations of every observation from the mean; σ2=∑(x−xˉ)2/n\sigma^2=\sum(x-\bar{x})^2/n for individual data, ∑f(x−xˉ)2/N\sum f(x-\bar{x})^2/N fo …

Definition 2Standard Deviation ($\sigma$)

The positive square root of the variance, σ=σ2\sigma=\sqrt{\sigma^2} — restores the same unit as the original data, unlike variance, wh …

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