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Business Mathematics and Statistics · Ch 9 — Measures of Dispersion

Standard Deviation — Assumed-Mean and Step-Deviation Methods

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Standard Deviation — Assumed-Mean and Step-Deviation Methods

When the actual mean xˉ\bar{x} is not a convenient round number, computing ∑fx2\sum fx^2 or ∑f(x−xˉ)2\sum f(x-\bar{x})^2 directly can involve large, awkward decimals. The assumed-mean method sidesteps this by measuring every deviation from a conveniently chosen assumed mean AA (any value close to the middle of the data, usually a class mark) instead of the actual mean, using d=x−Ad = x - A:

Note

Standard Deviation — Assumed-Mean Method

σ=∑fd2N−(∑fdN)2,d=x−A\sigma = \sqrt{\dfrac{\sum fd^2}{N} - \left(\dfrac{\sum fd}{N}\right)^2}, \qquad d = x - A

When, in addition, every class shares a common width hh, the step-deviation method goes one step further, coding u=x−Ahu = \dfrac{x-A}{h} so the arithmetic stays on small integers throughout, converting back to the real scale only in the final step:

Note

Standard Deviation — Step-Deviation Method

σ=h×∑fu2N−(∑fuN)2,u=x−Ah\sigma = h \times \sqrt{\dfrac{\sum fu^2}{N} - \left(\dfrac{\sum fu}{N}\right)^2}, \qquad u = \dfrac{x-A}{h}

and correspondingly σ2=h2[∑fu2N−(∑fuN)2]\sigma^2 = h^2\left[\dfrac{\sum fu^2}{N} - \left(\dfrac{\sum fu}{N}\right)^2\right] — note the multiplying factor is hh for SD but h2h^2 for variance, a detail worth remembering carefully. …

Definition 1Assumed Mean ($A$)

A conveniently chosen value (close to the middle of the data) used in place of the actual mean to keep the arithmetic of a variance/SD calculation simple; the final answer is unaf …

Definition 2Step-Deviation ($u$)

The coded value u=(x−A)/hu=(x-A)/h, used when every class shares a common width hh, to keep the SD/variance arithmeti …

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