How to Find the Range of a Data Set
Largest minus smallest — the one statistic that ignores 98% of a 100-value data set.
Range = maximum − minimum. For 62, 65, 67, 68, 70, 71, 71, 72, 74, 75, 77, 80 the range is 80 − 62 = 18. Keep the unit of the original data. Then check both extremes for typos, because those two numbers are the only ones the answer depends on.
Key takeaways
- The range reads exactly two values however big the data set is. At n = 10 that means it ignores 80% of your data; at n = 100, 98%; at n = 1,000, 99.8%.
- In the worked example below, changing one of twelve values from 80 to 800 multiplied the range by 41 — while the interquartile range stayed at 6.5, completely unmoved.
- The range grows with sample size even when nothing about the data changed. Drawing from the same normal distribution, the average range runs 1.13σ at n = 2, 3.08σ at 10, 5.02σ at 100 and 6.48σ at 1,000.
- That makes ranges from different sample sizes not comparable. A bigger range can mean nothing more than a bigger sample.
- “Range” means two different things. In statistics it is one number; in algebra the range of a function is the set of outputs it can produce. Different question, different tool.
- The range is never negative, and it is zero only when every value is identical.
On this page
How do you find the range?
Find the biggest number, find the smallest number, subtract.
Worked through on that data set — twelve exam scores:
62, 65, 67, 68, 70, 71, 71, 72, 74, 75, 77, 80
- Maximum = 80
- Minimum = 62
- Range = 80 − 62 = 18 marks
Two things about the mechanics. First, the range keeps the unit of the original data — 18 marks here, not 18. Second, you do not have to sort the list; you only have to look at every value once. Sorting helps mostly because it makes the two extremes impossible to miss, and missing one is the only way to get this wrong.
Which “range” are you looking for?
Two different subjects use the word for two unrelated things, and searching for one turns up the other. Check this table before going further.
| Range in statistics | Range of a function | |
|---|---|---|
| What it is | A single number | A set of values |
| Input | A list of data | A function such as f(x) = x2 |
| Answer looks like | 18 | y ≥ 0 |
| How you find it | max − min | Work out every output the function can reach |
| Paired with | Mean, median, IQR | Domain |
| Use | Range Calculator | Domain and Range Calculator |
The quick test: if the question mentions f(x), it is the algebra one. If it hands you a list of numbers, it is this page. The rest of this guide is about the statistics meaning.
What one typo does to the range
Because only two values contribute, the range is completely at the mercy of both. Take the same twelve scores and mistype the last one as 800 instead of 80 — a single stray keystroke.
| Measure | Correct data | With 80 typed as 800 | Change |
|---|---|---|---|
| Range | 18 | 738 | × 41 |
| Standard deviation | 5.10 | 210.72 | × 41 |
| Median | 71 | 71 | unchanged |
| Interquartile range | 6.5 | 6.5 | unchanged |
The range multiplied by 41. So did the standard deviation, which uses every value but squares the distances and so is dominated by the far one. The median and the interquartile range did not move at all, because both describe the middle of the sorted list and neither cares what the outermost value happens to be.
Why the range grows with sample size
This is the property that most often leads people astray, because it makes the range look like it is telling you something when it is not.
Draw values from a fixed normal distribution — the spread of the underlying population never changes — and simply take more of them. Every extra draw is another chance to land further out, so the expected range keeps climbing.
| Sample size n | Average range | Values used | Values ignored |
|---|---|---|---|
| 2 | 1.128σ | 100% | 0% |
| 5 | 2.326σ | 40% | 60% |
| 10 | 3.078σ | 20% | 80% |
| 20 | 3.735σ | 10% | 90% |
| 30 | 4.086σ | 6.7% | 93.3% |
| 50 | 4.498σ | 4% | 96% |
| 100 | 5.015σ | 2% | 98% |
| 250 | 5.638σ | 0.8% | 99.2% |
| 1,000 | 6.483σ | 0.2% | 99.8% |
Read the first two columns together. The population spread σ is identical on every row. Only n changed, and the range nearly sextupled. Meanwhile the third and fourth columns show the cost: by n = 100 the statistic is built from 2% of the data you collected.
The same table read backwards is also why the range remains genuinely useful in one place: quality control. When every sample is the same fixed size — five parts off the line, every hour — the bias is constant, and the range becomes a fast, division-free estimate of σ. That is exactly what a control chart does, dividing the average range by the tabulated constant for that sample size (2.326 for n = 5, matching the second row above).
What to report instead
| Measure | Values it uses | Survives an outlier? | Grows with n? | Use it when |
|---|---|---|---|---|
| Range | 2 | no | yes | The extremes themselves are the point |
| Interquartile range | middle 50% | yes | no | Data is skewed or has outliers |
| Standard deviation | all | no | no | Data is roughly symmetric |
| Mean absolute deviation | all | partly | no | You want an average distance, unsquared |
| Five number summary | 5 | yes | partly | You want the shape, not one number |
For the twelve exam scores, a complete answer reads: median 71, IQR 6.5, range 18 (62 to 80). That is three numbers, and between them they say where the middle sits, how tightly the bulk clusters, and how far the extremes reach. The range alone says only the last of those.
Range of grouped data
When the data has already been sorted into classes you no longer have the actual extremes, so the range becomes an estimate. Subtract the lower boundary of the lowest class from the upper boundary of the highest class.
| Class | Frequency | Boundaries |
|---|---|---|
| 10 – 19 | 4 | 9.5 – 19.5 |
| 20 – 29 | 11 | 19.5 – 29.5 |
| 30 – 39 | 7 | 29.5 – 39.5 |
Using the stated class limits the range is 39 − 10 = 29. Using continuous class boundaries it is 39.5 − 9.5 = 30. Both are accepted; exam boards differ, so follow whichever convention your course uses and say which one you used.
Either way the answer is an overestimate of the true range in the usual case, because the real maximum sits somewhere inside the top class rather than at its edge. Grouping has thrown away exactly the two values the range needs.
When the range is the right answer
Everything above is a warning, so it is worth stating plainly where the range earns its place:
- When the extremes are the subject. Overnight low and daytime high, minimum and maximum blood pressure, the largest drawdown a portfolio suffered. Here you want the endpoints, and the middle is irrelevant.
- Quality control on fixed-size samples. Control charts use the range precisely because it is trivial to compute on the shop floor and the sample size never varies.
- As a first look at new data. A range that is wildly wider than expected is the fastest possible detector of a data-entry error — as the ×41 example above shows, it is the statistic that screams loudest.
- Tiny data sets. At n = 4 or 5 the range uses most of the data anyway, and quartiles are barely meaningful.
What it is not good for is the job it most often gets given: standing alone as the summary of how spread out a data set is.
Terms worth knowing
- Range
- Maximum minus minimum. One number, in the same unit as the data.
- Outlier
- A value far from the rest. Commonly flagged as anything more than 1.5 × IQR beyond the nearest quartile. The range has no defence against one.
- Quartiles
- The three cuts that split sorted data into four equal parts. Q1, the median, and Q3.
- Interquartile range (IQR)
- Q3 − Q1, the width of the middle half. Unmoved by anything outside the quartiles.
- Standard deviation
- The root-mean-square distance from the mean. Uses every value, but squaring gives distant points outsized influence.
- Five number summary
- Minimum, Q1, median, Q3, maximum — the five numbers a box plot draws.
- Population vs sample
- A population is everything you care about; a sample is the part you measured. It changes the standard deviation formula, but not the range, which is the same subtraction either way.
- Grouped data
- Data already sorted into classes with counts. The individual values, including both extremes, are gone.
- Range of a function
- The algebra meaning: the set of all outputs a function produces. Unrelated to anything else in this list.
All 13 statistics tools in this guide
Authoritative sources
- Range (statistics) — Wikipedia — the definition and its dependence on sample size.
- Interquartile range — Wikipedia — the robust alternative used throughout this guide.
- Shewhart X-bar and R charts — NIST/SEMATECH Handbook — the control-chart constants that turn an average range into an estimate of σ.
- Robust statistics — Wikipedia — why some measures survive an outlier and others do not.
- Summarizing quantitative data — Khan Academy — lessons and practice on spread and centre.