What this calculator does
Quartile deviation is half the interquartile range, sometimes called the semi-interquartile range. For a data set with Q1 at 12.5 and Q3 at 17.25, the IQR is 4.75 and the quartile deviation 2.375.
Its value is that it depends only on the middle half of the data. Everything below the first quartile and above the third is ignored entirely, so an extreme value can be changed to any number at all without moving the result. That makes it the spread measure of choice for data with suspected outliers or an unknown distribution shape.
The formula
The first and third quartiles are found, their difference gives the interquartile range, and halving it gives the quartile deviation. The coefficient of quartile deviation divides the IQR by the sum of the quartiles instead, producing a relative measure that can be compared across data sets on different scales, in the same way the coefficient of variation does for the standard deviation.
| Term | Meaning |
|---|---|
| Quartile deviation | Half the interquartile range: (Q3 − Q1) / 2. |
| Interquartile range | The spread of the middle 50% of the data. |
| Coefficient of quartile deviation | (Q3 − Q1) / (Q3 + Q1), a relative measure independent of scale. |
| Robust statistic | One that extreme values cannot move, which this is and the standard deviation is not. |
The inputs explained
| Field | What to enter |
|---|---|
| Data (comma or space separated) | Your data, comma or space separated. Only the middle half affects the result. |
When to use it
Describing spread in skewed data
Where a standard deviation would be inflated by a long tail, the quartile deviation describes the bulk of the data.
Summarising with a box plot
A box plot is built from the quartiles, and the quartile deviation is the box half-width.
Comparing spread across scales
The coefficient version is dimensionless, so data sets in different units can be compared directly.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does an outlier affect the quartile deviation?
Including one where a single extreme value is added.
| Data set | Quartile deviation | Coefficient of quartile deviation | Q1, Q3 |
|---|---|---|---|
| 1, 2, 3, 4, 5, 6, 7, 8 | 1.750 | 0.3889 | 2.750, 6.250 |
| 5, 10, 15, 20, 25, 30, 35, 900 | 8.750 | 0.3889 | 13.750, 31.250 |
| 12, 15, 11, 18, 22, 15, 9, 14, 15, 20 | 2.375 | 0.1597 | 12.500, 17.250 |
Questions
What is the difference between quartile deviation and IQR?
The quartile deviation is exactly half the interquartile range. The IQR describes the full width of the middle 50%; the quartile deviation expresses it as a typical distance either side of the centre, which makes it more directly comparable to a standard deviation.
Why is the quartile deviation robust?
Because it depends only on the two quartiles, which are positional. Changing any value above Q3 or below Q1 to any other number on the same side leaves the quartiles unmoved, so extreme values have no effect whatsoever on the result.
What is the coefficient of quartile deviation for?
Comparing spread between data sets measured on different scales. Because it divides the IQR by the sum of the quartiles, it is dimensionless, serving the same purpose for a quartile-based measure that the coefficient of variation serves for the standard deviation.
How do different methods of finding quartiles affect this?
They can change the answer noticeably on small data sets. There are several accepted conventions for interpolating quartiles, and statistical packages do not all agree. On larger data sets the differences become small, but for under twenty points it is worth knowing which convention is in use.
For the interquartile range itself, see the interquartile range calculator. For the cutoffs that define an outlier, see the outlier fences calculator.