The term Range in mathematics is flexible—it changes its meaning depending on whether you are studying Statistics or Algebra. While both concepts deal with "span" or "coverage," they are calculated and used very differently.
In this article, we will break down both definitions so you can confidently find the range in any math problem.
1. Range in Statistics
In statistics, the range is the simplest measure of variability or "spread." It tells you the distance between the smallest number and the largest number in a dataset.
[Image of range statistics example on number line]How to Calculate:
The formula is straightforward subtraction:
Example:
Consider the test scores of a small class: {75, 90, 60, 85, 100}.
- Identify the Max: 100
- Identify the Min: 60
- Calculate: 100 - 60 = 40
The range is 40. This tells us there is a 40-point gap between the lowest and highest performing students.
Note: The range is very sensitive to outliers. If one student scored a 0, the range would jump to 100, which might distort your view of how the class performed overall.
2. Range of a Function (Algebra)
In algebra and calculus, the range refers to the set of all possible output values (y-values) that a function can produce.
[Image of domain and range mapping diagram]If the Domain is what goes into the function (x-values), the Range is what comes out (y-values).
How to Find the Range:
The easiest way to determine the range is to look at the graph of the function:
- Scan the graph from bottom to top (along the y-axis).
- Where does the graph exist?
- Are there maximum or minimum points?
Examples of Function Ranges:
A. Linear Function: f(x) = 2x + 1
A straight line goes up and down forever. It hits every possible y-value.
Range: All Real Numbers ($-\infty, \infty$).
B. Quadratic Function: f(x) = x²
When you square a number, the result is always positive (or zero). The graph is a parabola that opens upwards from the origin.
Range: $y \ge 0$ (or $[0, \infty)$).
C. Sine Function: f(x) = sin(x)
The sine wave oscillates up and down, but it never goes higher than 1 or lower than -1.
Range: $-1 \le y \le 1$ (or $[-1, 1]$).
Conclusion
Whether you are measuring the spread of data points or identifying the output limits of an equation, Range is fundamentally about boundaries. It answers the question: "How far does this data or function extend?"