Range in statistics
The range is a measure of variability, calculated by subtracting the smallest value from the largest value in a data set: $$ \omega = x_{max} - x_{min} $$
Here’s a practical example:
Consider the following data set:
$$ X = \{ 4, 6, 3, 8, 2 , 5, 7, 9 \} $$
The maximum value is xmax=9 and the minimum value is xmin=2.
So, the range is ω=7
$$ \omega = x_{max} - x_{min} = 9 - 2 = 7 $$
Key characteristics of the range
The range is one of the simplest measures of variability to compute.
However, it’s an absolute measure and provides limited insight because it only reflects the difference between the highest and lowest values.
It doesn’t take into account the distribution’s structure or other features of the data set.
Example. These three data sets all have a range of 7, yet they are quite different from each other: $$ X = \{ 4, 6, 3, 8, 2 , 5, 7, 9 \} $$ $$ Y = \{ 2, 2, 2, 2, 2 , 9, 2, 2 \} $$ $$ Z = \{ 2, 2, 2, 2, 9 , 9, 9, 9 \} $$
Moreover, the range is highly sensitive to outliers.
For instance, values that are much larger or smaller than the rest of the data can greatly affect the range.
Example. Consider this data set with an outlier (9999): $$ X = \{ 4, 6, 3, 8, 2 , 9999, 7, 9 \} $$ Due to the outlier, the range becomes: $$ \omega = 9999 - 2 = 9997 $$
And so on.
