Range (Statistics)
Statistics & ProbabilityThe 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.
Example
Test scores: $62$, $74$, $81$, $88$, $95$. Range $= 95 - 62 = 33$. This means the scores span a $33$-point gap.
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.
Definition
The range is the simplest measure of variability, calculated as the difference between the maximum and minimum values: $\text{Range} = \max(x_i) - \min(x_i)$. It is easy to compute but is highly sensitive to outliers, since a single extreme value changes it dramatically.
Example
Temperatures (degrees F): $65$, $68$, $70$, $72$, $74$. Range $= 74 - 65 = 9$. Add one unusually hot day at $105$: Range $= 105 - 65 = 40$. One outlier nearly quadruples the range.
Key Insight
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.
Definition
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
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 $\lambda$. As $n$ grows, the range grows like $(\log n)/\lambda$.
Key Insight
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.