By Lisa Maloney, seasoned math educator, Feb 12, 2023 6:08 pm EST
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In mathematics, the term “range” appears in two distinct contexts. In statistics, it refers to the spread between the largest and smallest observations in a dataset. In algebra and calculus, the range of a function denotes the set of all possible output values, also called the codomain, that the function can produce.
When you’re asked to find the range in a statistical context, you simply locate the maximum and minimum values of the data and subtract the latter from the former. The formula is straightforward:
range = maximum – minimum
Include any units (feet, pounds, etc.) that accompany the data.
Suppose a teacher’s notebook lists the following grade percentages for a class: {95, 87, 62, 72, 98, 91, 66, 75}. The highest score is 98 % and the lowest is 62 %. The range is therefore 36 percentage points (98 – 62 = 36).
In the study of functions, you can think of a function as a “math machine.” The domain is the set of inputs, the codomain is the set of all potential outputs, and the actual range is the subset of the codomain that the function actually attains. Each input in the domain corresponds to exactly one output in the range; if an input produced more than one output, the relation would not qualify as a true function.
It is common, however, for distinct inputs to map to the same output. Such many‑to‑one behavior does not violate the definition of a function but simply reflects that the function is not injective.
Consider the function f(x) = x² with the domain restricted to {−3, −2, −1, 1, 2, 3, 4}. Evaluating the function at each domain value yields:
f(−3)=9, f(−2)=4, f(−1)=1, f(1)=1, f(2)=4, f(3)=9, f(4)=16.
Removing duplicates, the range is the set {1, 4, 9, 16}.
Range is a key descriptive statistic that complements measures of central tendency such as the mean and median. While the mean and median indicate where the data cluster, range reveals the overall spread and highlights the influence of outliers. Combined with the standard deviation and interquartile range, it provides a fuller picture of data distribution.
