Shortcut (Step-Deviation) Method for Grouped Data

When a continuous frequency distribution has midpoints that are inconveniently large or do not share a common factor, computing $\sum f_ix_i$ and $\sum f_ix_i^2$ directly by the methods of the previous two sections can involve unwieldy arithmetic. The step-deviation method avoids this by first shifting and scaling every midpoint into a small, simple integer before doing any further arithmetic — the underlying variance and standard deviation come out identical, but the numbers being multiplied and summed along the way are far smaller.

Change of origin and scale. Choose any convenient midpoint $A$ (the assumed mean, usually the midpoint of a middle class) and let $h$ be the common class width. Define the step-deviation

for every class. Since every $x_i = A+h\,u_i$ is obtained from $u_i$ by a change of origin (shifting by $A$) followed by a change of scale (stretching by $h$), the mean and variance of the $x_i$'s can be recovered from the mean and variance of the much simpler $u_i$'s.

Effect on the mean. $\bar x = A + h\cdot\bar u$, where $\bar u = \dfrac1N\sum f_iu_i$ — that is,

(A shift by a constant $A$ shifts the mean by exactly $A$; a scaling by $h$ scales the mean by exactly $h$.)

Effect on the variance. A shift of origin (subtracting the constant $A$) does not change the variance at all, since variance measures spread around the mean, and shifting every value (including the mean itself) by the same constant leaves every deviation $x_i-\bar x$ completely unchanged. A scaling by $h$, however, scales every deviation by $h$, and therefore scales the variance (built from squared deviations) by $h^2$:

using the shortcut identity of the previous section applied to the $u_i$ values. The standard deviation is then $\sigma_x = h\cdot\sigma_u = h\sqrt{\dfrac{\sum f_iu_i^2}{N}-\left(\dfrac{\sum f_iu_i}{N}\right)^2}$ — note that, unlike the variance, the standard deviation scales by $h$ itself, not $h^2$, since it is a square root.

Steps to compute, using the step-deviation method.

1. Choose an assumed mean $A$ (a midpoint near the centre of the data) and note the common class width $h$.
2. Compute $u_i=\dfrac{x_i-A}{h}$ for every class — these will be small integers such as $-2,-1,0,1,2,\ldots$
3. Form the columns $f_iu_i$ and $f_iu_i^2$, and sum each.
4. Recover $\bar x = A+h\cdot\dfrac{\sum f_iu_i}{N}$ and $\sigma^2 = h^2\left[\dfrac{\sum f_iu_i^2}{N}-\left(\dfrac{\sum f_iu_i}{N}\right)^2\right]$. …