In descriptive statistics, what is the range in math is the total spread of a numerical data set from its minimum value to its maximum value.
Definition of range for a data set
For a set of numbers \(x_1,x_2,\dots,x_n\), the range is defined as the difference between the largest and smallest observations: \[ \text{range}=\max(x_1,\dots,x_n)-\min(x_1,\dots,x_n). \] The range is measured in the same units as the data and summarizes the full width of the observed values.
Core formula: \(\text{range}=\text{maximum}-\text{minimum}\).
Interpretation: A larger range indicates a wider spread of observed values, but it does not describe how values cluster between the endpoints.
Visualization of range and sensitivity to outliers
Example calculation
For the data set \( \{2,4,6,9,11,14\} \), the minimum is \(2\) and the maximum is \(14\), so \[ \text{range}=14-2=12. \]
| Data set | Minimum | Maximum | Range | Comment |
|---|---|---|---|---|
| 2, 4, 6, 9, 11, 14 | 2 | 14 | \(14-2=12\) | Total spread across observed endpoints |
| 2, 4, 6, 9, 11, 14, 30 | 2 | 30 | \(30-2=28\) | Outlier inflates the range |
Statistical interpretation and limitations
The range is a fast measure of dispersion and is easy to compute, but it ignores all interior values between the minimum and maximum. Data sets with very different clustering can share the same range.
Sensitivity to outliers is the main limitation: a single unusually large or small observation can dominate the range and give a misleading impression of typical spread. Measures such as the interquartile range (IQR) and the standard deviation incorporate more of the data and often provide a more stable summary of variability.
Related definitions sometimes confused with “range in math”
In algebra and functions, “range” can mean the set of all possible output values \(y\) of a function \(y=f(x)\). In statistics, “range” means the numerical difference \(\max-\min\) for observed data. Context distinguishes these uses, and descriptive statistics uses the endpoint-difference definition.
Optional scaled version: coefficient of range
A unit-free scaling sometimes used in descriptive summaries is the coefficient of range: \[ \text{coefficient of range}=\frac{\max-\min}{\max+\min}, \] defined when \(\max+\min\ne 0\). This ratio compares spread to overall magnitude, but it retains the same endpoint-only sensitivity as the range itself.