Quartile Deviation Calculator

What the quartile deviation calculator does

The quartile deviation is half the distance between the first and third quartiles. It measures how spread out the middle half of your data is, in a single number. This calculator finds it, along with the two quartiles it comes from and your sorted data.

It goes by another name, the semi-interquartile range, and it is a spread measure that shrugs off outliers. Below is how it works and where it fits.

How to use it

1. Enter your numbers in the box, separated by commas, spaces, or line breaks. You need at least four.
2. Press Calculate for the quartile deviation and the quartiles behind it, or Reset to clear it.

How the quartile deviation is worked out

It builds on the quartiles. The calculator sorts your data, finds the first quartile and the third quartile, takes the distance between them, and halves it:

Quartile deviation = (Q3 minus Q1) ÷ 2

The distance between the two quartiles is the interquartile range, the span of the middle half of the data. Halving it gives the quartile deviation, which you can think of as the typical distance from the median out to the edge of that middle half.

Why use the quartile deviation

The quartile deviation earns its place the same way the interquartile range does: by ignoring the extremes. It is built only from the first and third quartiles, so the very smallest and largest values, the ones most likely to be outliers, have no say in it at all. That makes it a steady, robust measure of spread for data that is skewed or has a few stray values, where the standard deviation would be pulled around.

It pairs naturally with the median. Where the standard deviation is the spread that goes with the mean, the quartile deviation is the spread that goes with the median, both of them resistant to outliers. When you report a median, the quartile deviation is often the spread to report beside it.

A worked example

Take the nine numbers 3, 5, 7, 8, 12, 13, 14, 18, 21, already sorted. The first quartile is 6 and the third quartile is 16.

The distance between them is 16 minus 6 = 10, which is the interquartile range. Halving it gives a quartile deviation of 10 ÷ 2 = 5. So the middle half of this data reaches about 5 units either side of the centre.

Entering your data, and the method

You can separate your numbers with commas, spaces, or line breaks, in any mix, and the calculator sorts them for you. It finds the quartiles with the exclusive method, matching our quartile calculator, so the numbers line up between the two tools. For data with an odd number of values, keep in mind that quartile methods differ between calculators, which can nudge the result slightly. The quartile deviation is shown to three decimal places with trailing zeros trimmed.

Questions people ask

What is the quartile deviation?

Half the distance between the first and third quartiles. It measures the spread of the middle half of the data and is also called the semi-interquartile range.

How is it different from the interquartile range?

The interquartile range is the full distance between the quartiles. The quartile deviation is exactly half of that, so it describes the spread from the median out to the edge of the middle half.

Why is it resistant to outliers?

Because it uses only the two quartiles and ignores the smallest and largest values entirely. Extreme values, which pull the standard deviation around, have no effect on it.

When should I use it?

When your data is skewed or has outliers, and especially when you are reporting the median, since the quartile deviation is the matching outlier-resistant measure of spread.

References

A quick note on where the methods here come from. The interquartile range, from which the quartile deviation is taken, is set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering quartiles and spread.

1. NIST/SEMATECH e-Handbook of Statistical Methods (quartiles and the interquartile range). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (measures of the spread of the data). https://openstax.org/details/books/introductory-statistics-2e