Measures of Dispersion

A measure of dispersion shows the extent to which individual values in a series are scattered or spread out around a measure of central tendency. It is expressed either as an absolute measure (stated in the same units as the original data, such as rupees or marks) or as a relative measure (a unit-free ratio or percentage, usually called a coefficient, used to compare the variability of two series that are in different units or have very different average levels).

1. Range. The simplest measure of dispersion is the difference between the largest ($L$) and smallest ($S$) values in a series:

Its relative counterpart is the Coefficient of Range:

Range is easy to compute but rests on only the two extreme values and ignores everything in between, so a single unusual value at either end distorts it heavily.

2. Quartile Deviation (QD). Also called the semi-interquartile range, QD uses only the middle 50% of the data — between the first quartile $Q_1$ and the third quartile $Q_3$ — dropping the extreme quarter at each end:

3. Mean Deviation (MD). MD improves further by using every value in the series: it is the arithmetic mean of the absolute deviations of each value from a chosen average (usually the mean):

Absolute values are essential here: the signed deviations of any series from its own arithmetic mean always add up to exactly zero, so averaging them without dropping the sign would always give zero and convey nothing about spread.

4. Standard Deviation (SD). Denoted $\sigma$ (sigma), SD is the most widely used and most reliable measure of dispersion — the positive square root of the mean of the squared deviations from the arithmetic mean:

Squaring removes the sign problem (as absolute value does for MD) while keeping the measure mathematically well-behaved for further statistical work — this is exactly why SD, and not MD, underlies almost all advanced statistical analysis, including correlation, taken up next. Its square, $\sigma^2$, is called the variance. Where the actual mean is inconvenient to work with, the equivalent assumed mean (short-cut) method may be used instead, taking deviations $d = X - A$ from any convenient assumed value $A$:

$\sigma = \sqrt{\frac{\Sigma d^2}{N} - \left(\frac{\Sigma d}{N}\right)^2}$ …