Measures of Dispersion: Range, Mean Deviation and Standard Deviation

Central tendency describes where the 'centre' of a distribution lies, but says nothing about how spread out — how dispersed — the individual values are around that centre. Two data sets can share an identical mean while one is tightly clustered and the other widely scattered; measures of dispersion capture exactly this difference.

Range is the simplest measure — the difference between the largest and smallest values in a series:

It is easy to compute but uses only two values from the entire data set, ignoring everything in between.

Mean Deviation measures the average absolute distance of every observation from a central value (usually the mean), taking the absolute value of each deviation so that positive and negative deviations do not cancel out:

Standard Deviation ($\sigma$) is the most widely used measure of dispersion. It squares each deviation from the mean (which removes the sign problem and gives larger deviations proportionately more weight), averages these squared deviations, and then takes the square root to bring the measure back to the original unit:

$\sigma = \sqrt{\frac{\sum (X - \bar{X})^2}{n}}$ …