What Is Range in Math?

Range has two different meanings in mathematics, and which one applies depends entirely on whether you are doing statistics or working with functions. That ambiguity is the single biggest source of confusion around the word, so it is worth settling first.

• In statistics, the range is the largest value minus the smallest. It is one number, and it measures spread.
• For a function, the range is the set of all output values the function can produce. It is a set, not a single number.

Both are standard and both are correct. If a question gives you a list of data, it wants the first. If it gives you f(x) or a graph, it wants the second.

Two steps:

1. Find the largest value and the smallest value.
2. Subtract the smallest from the largest.

For the set 4, 8, 15, 16, 23, 42 the maximum is 42 and the minimum is 4, so the range is 42 − 4 = 38.

You do not need to sort the whole list to do this, though sorting makes the two values easy to spot. Note that the range is reported as a single number, 38, not as “4 to 42” — that pair is the interval the data covers, which is useful but is not what the word range asks for in a statistics question.

The range uses exactly two numbers and ignores everything between them. That makes it quick to compute and easy to break.

Change the 42 in our set to 420 and the range jumps from 38 to 416, an elevenfold increase caused by one value. Every other number in the set is unchanged, and any reasonable description of “how spread out is this data” should barely have moved.

This is why statistics courses introduce the range first and then immediately replace it. The interquartile range uses the middle half of the data and ignores the extremes, and the standard deviation uses every value's distance from the mean. Both are harder to compute and both survive an outlier that destroys the range.

For a function, the range is the set of every output the function actually produces. The set of allowed inputs is called the domain, and the two are always discussed together.

Take f(x) = x² over all real numbers. You can put any real number in, so the domain is all reals. But squaring never produces a negative, so the range is only y ≥ 0. The function has a domain twice the size of its range, in a manner of speaking, and that asymmetry is the point of asking the question.

A second example: f(x) = 2x + 3 has domain all reals and range all reals, because a straight line with a non-zero slope eventually reaches every height. Compare that with f(x) = 1/x, whose range excludes 0, because no input ever makes the output exactly zero.

Domain is what goes in. Range is what comes out. Readers mix them up constantly, and the reliable way to keep them straight is alphabetical: domain before range, input before output, x before y.

The domain calculator works out the allowed inputs for an expression, which is usually the harder half of the pair.

• Answering with an interval. The range of 4, 8, 15, 16, 23, 42 is 38, not “4 to 42”.
• Subtracting in the wrong order. Maximum minus minimum. A range is never negative.
• Using the statistics meaning on a function question, or the reverse. Check whether you were given data or a rule.
• Reporting a range on outlier-heavy data as though it described the typical spread.

• In statistics: range = maximum − minimum, a single number.
• For a function: the set of outputs the rule can produce.
• The statistical range uses only two values, so one outlier can multiply it.
• Domain is input, range is output, in alphabetical order.

The mean, median and mode calculator reports the range alongside the mean, median and mode for any list of numbers.