Range and interquartile range
Measure total spread and the spread of the middle half of ordered data.
Range and interquartile range is one of 3 spread formulas in the statistics section of this library, and it is used at middle school · high school · ap level.
Why range and interquartile range works
The range is built from the two most extreme values, so it reacts to anything unusual and keeps growing as more data arrive. The interquartile range uses the boundaries of the middle half instead, so shifting or removing the most extreme values leaves it untouched. That resistance is why box plots are drawn from quartiles.
What each symbol means
$Q_1,Q_3$ are the first and third quartiles.
Range and interquartile range: when it holds
Sort the data first; quartile conventions can vary slightly for small samples.
When it stops applying
Quartiles are not defined the same way everywhere, so two tools can disagree on small samples. For the values 1, 2, 3, 4 one common method puts the first quartile at 1.75 while the median-of-the-lower-half method puts it at 1.5, and the interquartile range differs accordingly.
Range and interquartile range: a worked example
For quartiles $Q_1=4,Q_3=11$, $IQR=7$.
The mistake to avoid
What people do: Reporting the range as a pair of endpoints such as 4 to 11.
Why it goes wrong: In statistics the range is a single number measuring width, so those endpoints describe the data but do not answer the question. The range there is 7.
Do this instead: Subtract to get one number, and if you want to report the endpoints as well, call them the minimum and maximum.
Range and interquartile range: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $Q_1,Q_3$ are the first and third quartiles.
- Check the conditions before substituting. Sort the data first; quartile conventions can vary slightly for small samples.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most statistics slips.
Where this formula fits
- Subject
- Statistics formulas — 21 entries in this library
- Topic
- Spread
- Level
- Middle school · High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where range and interquartile range comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Standard Deviation — the lesson behind this formula: measure typical distance from the mean.
- Statistics Calculator — check your substitution and the value it produces.
- Study statistics — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 21 statistics formulas — the full grouped reference, or the complete formula library.
Questions about range and interquartile range
What is the rule that uses 1.5 times the interquartile range?
It sets fences for flagging unusual values. With quartiles of 4 and 11 the width is 7, so the fences fall at -6.5 and 21.5, and anything outside them is marked as a potential outlier.
Does adding one extreme value change the interquartile range much?
Barely, and that is the point. One wild value moves the range immediately but shifts the quartiles by at most one position in the sorted list.
Is the interquartile range the same as the middle 50 percent of the data?
It is the width of that middle half, not the half itself. The values from the first quartile to the third quartile are the middle half; their difference is the number you report.
Which measure of spread pairs with the median?
The interquartile range, because both are based on position rather than on arithmetic with the values. The mean pairs with the standard deviation for the same reason of matching sensitivity.