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Statistics · Ch 4 — Measures of Dispersion

Standard Deviation and Variance

6

Standard Deviation and Variance

Standard Deviation (S.D., denoted σ\sigma) is the most widely used and most reliable measure of dispersion. It is the positive square root of the variance (σ2\sigma^2), the mean of the squared deviations of all observations from the arithmetic mean.

Individual series (actual/direct mean method):

σ=∑(xi−x‾)2nσ2=∑(xi−x‾)2n\sigma = \sqrt{\dfrac{\sum (x_i-\overline{x})^2}{n}} \qquad \sigma^2 = \dfrac{\sum (x_i-\overline{x})^2}{n}

Discrete / continuous series (direct method):

σ=∑fi(xi−x‾)2N\sigma = \sqrt{\dfrac{\sum f_i (x_i-\overline{x})^2}{N}}

Short-cut (assumed mean) / step-deviation method — convenient when values or class mid-points are large, using an assumed mean AA and, for grouped data of equal class width hh, the step-deviation di=xi−Ahd_i = \dfrac{x_i-A}{h}:

σ=h∑fidi2N−(∑fidiN)2\sigma = h\sqrt{\dfrac{\sum f_i d_i^2}{N} - \left(\dfrac{\sum f_i d_i}{N}\right)^2}

(For an individual series, drop hh and use di=xi−Ad_i = x_i - A directly.) Both the direct method and the step-deviation method must always give the same value of σ\sigma for the same data — computing a numerical both ways is the standard way to check the arithmetic.

Properties of Standard Deviation:

  • It is based on every observation and is capable of further algebraic treatment (variances of independent series can be combined), which is why it is preferred over Range, Q.D., and M.D. in advanced statistical work.
  • It is the least affected by sampling fluctuations among the common measures of dispersion. …
Definition 1Variance and Standard Deviation

σ2=∑f(x−x‾)2N\sigma^2=\dfrac{\sum f(x-\overline{x})^2}{N}; σ=σ2\sigma=\sqrt{\sigma^2}, computable by the direct method or the equivalent step-deviation method $\sigma=h\sqrt{\dfrac{\sum fd^2}{N}-\le …

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