Outlier and IQR calculator

The formula

IQR = Q3 − Q1 lower fence = Q1 − 1.5 × IQR upper fence = Q3 + 1.5 × IQR

Q1 is the value a quarter of the way through the sorted data and Q3 three quarters of the way. The distance between them spans the middle 50% of the observations — so the IQR answers "how spread out is the bulk of my data?" while deliberately ignoring the tails.

Why it survives outliers

Take 12, 15, 17, 14, 19, 21, 16, 13, 18 and then replace the 21 with 2100. The standard deviation leaps from about 2.9 to over 690. The IQR does not move at all, because 2100 is still simply "the largest value" and the quartiles are positional.

That robustness is the whole point. It is why box plots are the standard way to compare distributions that may contain anomalies, and why median-and-IQR is the conventional summary for skewed data such as incomes or response times.

Reading the box plot

• The box runs from Q1 to Q3 — the middle half of the data.
• The line inside it is the median. If it sits off-centre, the data is skewed.
• The whiskers reach the furthest values still inside the fences.
• Separate points beyond the whiskers are outliers.

Where 1.5 comes from

John Tukey chose it as a practical compromise rather than deriving it. On perfectly normal data the 1.5 × IQR rule flags roughly 0.7% of observations — rare enough to be worth a look, common enough that the rule is not useless. The 3 × IQR threshold for "extreme" flags about one in half a million.

Because it is a convention rather than a law, expect a few flags on any large clean dataset. A thousand normal observations will produce about seven, none of them errors.

Which outlier test to use

The calculator runs all four and shows each verdict, because they answer slightly different questions:

• Tukey's fences (1.5 × IQR) make no assumption about the shape of the data, so they are the safe default for skewed data such as incomes, reaction times or waiting times.
• The z-score screen flags values more than 3 standard deviations from the mean. It assumes roughly normal data, and it is weak in small samples: an outlier inflates the standard deviation it is measured against, and with 10 or fewer values no point can reach |z| > 3 at all. See the z-score calculator for a single value.
• The modified z-score (Iglewicz and Hoaglin, 1993) replaces the mean with the median and the standard deviation with the median absolute deviation (MAD): M = 0.6745 × (x − median) ÷ MAD, flagging |M| > 3.5. Because neither the median nor the MAD moves much when one value is extreme, an outlier cannot hide by inflating its own yardstick. In the example above, 44 has an ordinary z of about 2.7 — not flagged — but a modified z of 7.42.
• Grubbs' test is a formal significance test (α = 0.05 here) for the single most extreme value, assuming the rest of the data is normal. It is the usual choice in laboratory and quality work where one suspect reading needs a defensible yes or no.

A flag is a reason to look at a value, not to delete it. Check whether it is a recording error or a real observation first: a genuine extreme value removed to tidy the data changes the answer to the question you were asking.

If you already have Q1 and Q3 from a textbook or a software printout, the upper and lower fence calculator gives the fences directly, with any multiplier, without re-entering the data.