Summary

This chapter developed four measures of how spread out a data set is, in increasing order of sophistication:

• Range $=X_{\max}-X_{\min}$ (ungrouped/discrete); for continuous data, the upper limit of the last class minus the lower limit of the first class. Uses only the two extreme values. Coefficient of range $=\dfrac{X_{\max}-X_{\min}}{X_{\max}+X_{\min}}$.
• Mean deviation about the mean, $\text{M.D.}(\bar x)=\dfrac1n\sum|x_i-\bar x|$ (ungrouped) or $\dfrac1N\sum f_i|x_i-\bar x|$ (grouped); mean deviation about the median, $\text{M.D.}(M)=\dfrac1n\sum|x_i-M|$, defined analogously. Absolute values are used because signed deviations from the mean always sum to zero.
• Variance, $\sigma^2=\dfrac1n\sum(x_i-\bar x)^2$ (ungrouped) or $\dfrac1N\sum f_i(x_i-\bar x)^2$ (grouped), with the computational shortcut $\sigma^2=\overline{x^2}-(\bar x)^2$ in both cases. Squared (rather than absolute) deviations are used for algebraic convenience and to weight larger deviations more heavily.
• Standard deviation, $\sigma=\sqrt{\sigma^2}$, the positive square root of the variance, restoring the original unit of the data — the single most widely used measure of dispersion.
• The step-deviation method computes the mean and variance of grouped data using small integer step-deviations $u_i=\dfrac{x_i-A}{h}$ from an assumed mean $A$ and class width $h$: $\bar x=A+h\bar u$, and $\sigma_x^2=h^2\sigma_u^2$ — a change of origin leaves variance unchanged, while a change of scale by $h$ scales variance by $h^2$. …