Measures of dispersion tell how spread out the data values are around a central value; the main ones are range, quartile deviation, mean deviation, variance, standard deviation and coefficient of variation.
Two data sets can have the same mean but very different spread (e.g. 49, 50, 51 and 0, 50, 100 both have mean 50), so the average alone is not enough.
1. Range = largest − smallest value. Simple but depends only on two extreme values.
2. Quartile deviation (semi-interquartile range) \(= \dfrac{Q_3 - Q_1}{2}\). Not affected by extreme values.
3. Mean deviation \(= \dfrac{\sum |x - \bar x|}{n}\) (can also be taken about the median).
4. Variance and standard deviation
\[\sigma^2 = \frac{\sum (x-\bar x)^2}{n},\qquad \sigma = \sqrt{\sigma^2}\]
(For a sample, divide by \(n-1\) instead of \(n\).) SD is the most widely used measure because it uses every value and is used in further statistics.
5. Coefficient of variation \(= \dfrac{\sigma}{\bar x}\times 100\%\). A relative measure used to compare consistency of data with different units or means; lower CV means more consistent.
Example. Data: 2, 4, 4, 4, 5, 5, 7, 9 (\(n = 8\)).
- Mean \(= 40/8 = 5\); range \(= 9 - 2 = 7\)
- \(\sum (x-\bar x)^2 = 9+1+1+1+0+0+4+16 = 32\)
- \(\sigma^2 = 32/8 = 4\), so \(\sigma = 2\)
- CV \(= 2/5 \times 100 = 40\%\)
Absolute measures (range, QD, MD, SD) are in the units of the data; relative measures (coefficient of range, coefficient of QD, CV) are unit-free.