Range (Statistics)
The range is the difference between the maximum and minimum values in a dataset, measuring overall spread.
Formula
\text{Range} = \text{Maximum value} - \text{Minimum value}
Definition
The range tells you how spread out data is by measuring the distance from the smallest value to the largest value, the simplest measure of variability: $\text{Range} = \max(x_i) - \min(x_i)$. It is easy to compute but is highly sensitive to outliers, since a single extreme value changes it dramatically. Formally, the range $W = X_{(n)} - X_{(1)}$ (the difference between the maximum and minimum order statistics) has a known distribution for common families: for i.i.d. $\text{Uniform}(0,\theta)$ samples, the range is a sufficient statistic for $\theta$ and an MLE, with $E[W] = \frac{n-1}{n+1}\theta$.
Example
Test scores $62$, $74$, $81$, $88$, $95$ have range $= 95 - 62 = 33$, meaning the scores span a $33$-point gap. Temperatures $65$, $68$, $70$, $72$, $74$ have range $9$, but adding one unusually hot day at $105$ changes the range to $105-65 = 40$, nearly quadrupling it from a single outlier. For i.i.d. $\text{Exponential}(\lambda)$ data, the expected range is $E[W] = \frac{1}{\lambda}\sum_{k=1}^{n} \frac{1}{k}$, the $n$th harmonic number divided by $\lambda$, growing like $(\log n)/\lambda$ as $n$ grows.
Key Insight
A small range means values are bunched together, a large range means they are spread out, but the range only looks at the two extremes and ignores everything in between. Because of its sensitivity to outliers, the range is often replaced by the interquartile range (IQR), which measures the spread of the middle $50\%$ of data and ignores extremes. Extreme value theory studies the behavior of the range and related statistics (maxima, minima) as $n$ grows, forming the mathematical foundation of risk management for rare catastrophic events.