Standard Deviation — Step-Deviation Method

Exactly as with the mean (measures-of-central-tendency chapter), computing $\sum fx^2$ directly can involve large, awkward numbers when the class marks themselves are large. The step-deviation method sidesteps this by working with small coded values $u = \dfrac{x-A}{h}$ instead — where $A$ is a conveniently chosen assumed mean and $h$ is the (common) class width — and only converting back to the real scale at the very end.

Notice the multiplying factor is $h$ for the standard deviation but $h^2$ for the variance — a detail worth remembering, since forgetting to square $h$ when converting the variance (rather than the standard deviation) back to the real scale is a common slip. The step-deviation method always gives exactly the same standard deviation as the direct method on the same data; Worked Example 9 below re-solves Worked Example 8's own table by this shortcut and confirms the two agree. …