Averages
How to Find the Range
To find the range of a data set, subtract the smallest value from the largest: range = maximum − minimum. For test scores of 72, 85, 91, 64 and 88, the range is 91 − 64 = 27 points. This guide works through examples with negatives and decimals, explains what the range does and does not tell you, compares it with the interquartile range, and ends with how to find the range of a function.
How to find the range in statistics
The range measures how spread out a set of numbers is. It is the distance from the lowest value to the highest, and it takes two steps to find:
- Find the largest value (the maximum) and the smallest value (the minimum). Sorting the list makes both obvious: they are the last and first numbers.
- Subtract the minimum from the maximum. The result is the range.
The range belongs with the averages rather than replacing them. The mean, median and mode tell you where the data sits; the range tells you how widely it is scattered. Two classes can both average 80% on a test, but a class that scored between 76 and 84 is very different from one that scored between 45 and 100. The mean, median, mode calculator reports the range alongside all three averages, with the working shown.
Worked examples
Example: test scores
Five students scored 72, 85, 91, 64, 88. Find the range.
The scores span 27 points. Pairing that with the mean of 80 gives a much fuller picture than either number alone.
Example: temperatures with negative numbers
The low temperatures for five days were −8, 3, −2, 11, 5 °F. Find the range.
Subtracting a negative adds its size, so the range is 19, not 3. On a thermometer, the distance from −8 up to 0 is 8 degrees and from 0 up to 11 is another 11, for 19 in total. The negative number rules explain why subtracting a negative turns into addition.
Example: decimals
Five rainfall readings, in inches, were 2.4, 3.15, 1.8, 2.95, 3.6. Find the range.
Line up the decimal points when you subtract. Writing 3.6 as 3.60 and 1.8 as 1.80 helps if the values have different numbers of decimal places.
Example: range from a frequency table
A survey records the number of pets per household.
| Pets | Households |
|---|---|
| 0 | 9 |
| 1 | 14 |
| 2 | 6 |
| 3 | 2 |
| 5 | 1 |
Use the values in the first column, never the frequencies. Subtracting 2 from 14 (the household counts) would give 12, which is not a range of anything.
What the range tells you, and what it hides
The range is the simplest measure of spread, and that simplicity is its strength: anyone can compute it in a few seconds and it is in the same units as the data. A range of 27 points or 19 degrees is easy to picture.
Its weakness is that it depends on only two values, the two most extreme ones. Everything in between is ignored, and a single unusual value can change the range enormously.
Example: one outlier
Nine quiz scores: 12, 70, 70, 72, 75, 78, 81, 81, 84. The 12 belongs to a student who missed most of the quiz.
One score multiplies the range by more than five. The other eight students scored within 14 points of each other, but the range alone suggests a class that is all over the place.
So report the range when you want a quick sense of the full span, such as the spread of prices for one product or the high and low of a day’s temperature. Be careful with it whenever the data might contain errors, outliers or one-off events.
Range vs interquartile range
The interquartile range (IQR) fixes the outlier problem by ignoring the top and bottom quarters of the data. It is the range of the middle half.
Example: the quiz scores again
The IQR of 11 describes the typical spread of the class; the range of 72 is mostly describing one student. Quartiles can be calculated in a few slightly different ways, and for this data the common methods agree.
The median calculator finds Q1, Q3 and the IQR along with the range, and the guide to finding the median explains the idea of splitting sorted data into halves and quarters.
| Range | Interquartile range | |
|---|---|---|
| Formula | max − min | Q3 − Q1 |
| Values it depends on | The two extremes | The middle 50% |
| Affected by one outlier? | Heavily | Hardly at all |
| Effort to compute | Very low | Needs sorting and quartiles |
| Best for | Quick span of clean data | Skewed data or data with outliers |
How to find the range of a function
In algebra, “range” means something related but different. The range of a function is the set of all output values it can produce, given the inputs it is allowed to take (its domain). Here you are describing a set of values, not subtracting two numbers.
A reliable approach:
- Find the domain. Which x-values are allowed? Watch for division by zero and square roots of negative numbers.
- Look for a lowest or highest output. Squares, square roots and absolute values are never negative, which often sets a floor.
- Check what outputs can never happen. Try solving
y = f(x)for x. Any y that leaves no valid x is outside the range. - Sketch or test values to confirm the shape.
Example: f(x) = x²
Example: f(x) = −(x − 2)² + 5
The vertex at (2, 5) is the highest point of this downward parabola, so 5 is the maximum output.
Example: f(x) = 1/x
Example: a function on a finite domain
If f(x) = x² and the domain is {−2, −1, 0, 1, 2}, list the outputs: 4, 1, 0, 1, 4. The range is the set {0, 1, 4}. Repeated outputs are listed once.
| Function | Domain | Range |
|---|---|---|
| f(x) = 2x + 3 | all real numbers | all real numbers |
| f(x) = x² | all real numbers | y ≥ 0 |
| f(x) = x² on −1 ≤ x ≤ 3 | −1 ≤ x ≤ 3 | 0 ≤ y ≤ 9 |
| f(x) = √x | x ≥ 0 | y ≥ 0 |
| f(x) = |x| | all real numbers | y ≥ 0 |
| f(x) = 1/x | x ≠ 0 | y ≠ 0 |
| f(x) = −(x − 2)² + 5 | all real numbers | y ≤ 5 |
The restricted-domain row is a common trap. On −1 ≤ x ≤ 3, the lowest output is 0 (at x = 0, inside the interval), not 1 (at the endpoint x = −1), and the highest is 9 (at x = 3). Always check turning points inside the domain as well as the endpoints. The square root calculator and absolute value calculator are handy for testing individual outputs.
Common mistakes when finding the range
- Subtracting the first and last numbers of an unsorted list. In
72, 85, 91, 64, 88the first and last are 72 and 88, but the range uses 91 and 64. Sort first, or scan carefully for the extremes. - Getting the sign wrong with negatives.
11 − (−8)is 19, not 3. A range can never be negative, so a negative answer means the subtraction was done backward. - Writing the range as an interval. In statistics, “the scores ranged from 64 to 91” is a description, but the range itself is the single number 27.
- Using frequencies instead of values in a frequency table.
- Leaving off the units. A range of 19 means 19 °F, or 19 dollars, or 19 points. The units are part of the answer.
- Confusing the range with the average. The range says nothing about where the data is centered; check the guide to averages for that.
- Reading a function’s range off the endpoints only. A curve can dip below or rise above both endpoints, as
x²does on−1 ≤ x ≤ 3.
If you need the most common value as well, the guide to finding the mode covers it, and the average calculator gives the mean, median, mode and range together.
Frequently asked questions
What is the formula for range?
The range equals the maximum minus the minimum: range = max − min. Find the largest and smallest values in the data set and subtract. For the data 4, 9, 15, 2, 11, the range is 15 − 2 = 13. The result is always zero or a positive number.
How do you find the range with negative numbers?
Use the same formula and take care with the signs. Subtracting a negative number is the same as adding its size. For −8, −2, 3, 5, 11 the range is 11 − (−8) = 11 + 8 = 19. If every value is negative, such as −12, −5, −9, the range is −5 − (−12) = 7.
What is the range of 3, 7, 12, 5, 9?
The range is 9. The largest value is 12 and the smallest is 3, so 12 − 3 = 9. The order the numbers are written in does not matter; only the maximum and minimum are used. Sorting first, 3, 5, 7, 9, 12, puts them at the two ends of the list.
Can the range be zero?
Yes. The range is zero when every value in the data set is the same, such as 6, 6, 6, 6, because the maximum and minimum are equal. A range of zero means there is no spread at all. The range can never be negative, because the maximum is never smaller than the minimum.
What is the difference between range and interquartile range?
The range is the spread of all the data, from the minimum to the maximum. The interquartile range is the spread of the middle half only, from Q1 to Q3. Because the IQR ignores the top and bottom quarters, a single outlier barely changes it, while the same outlier can change the range enormously.
How do you find the range of a function?
Work out which output values the function can actually produce. Start from the domain, look for a minimum or maximum output, and check whether any y-value is impossible by solving y = f(x) for x. For f(x) = x² the outputs are never negative and every non-negative value is reached, so the range is y ≥ 0.
Is the range a measure of center or spread?
The range is a measure of spread, also called variability or dispersion. It tells you how far apart the extreme values are, not where the data is centered. Use the mean, median or mode to describe the center, and report the range alongside one of them for a fuller summary.
Calculators for this
Every one of these shows the working, so you can check the method as well as the answer.
More guides
https://fourfunctioncalculator.com/how-to-find-the-range/