IQR Calculator

Why the middle half is the part worth measuring

Spread is easy to measure badly. The two obvious ways both have a specific weakness, and the IQR exists because of them.

The range is the distance between the two most extreme values, which means it is determined entirely by the two least typical numbers you have. A single mistyped figure — a temperature entered in the wrong unit, an extra zero on a price — sets the range by itself, and every other observation becomes irrelevant to it.

The standard deviation uses every value, which sounds strictly better, but it squares each deviation before averaging. Squaring is not neutral: a point ten times further from the mean than another contributes a hundred times as much. Distant points therefore dominate, which is exactly the wrong weighting when the distant points are the suspect ones.

The interquartile range takes the opposite approach and simply refuses to look at the ends. Sort the data, discard the bottom quarter and the top quarter, and measure what is left. Whatever happens outside that middle band — however extreme — cannot change the answer.

You give something up for that. The IQR discards half your data, so on clean, well-behaved measurements it is a less efficient estimate of spread than the standard deviation: it moves around more from sample to sample. The trade is deliberate. You accept a noisier estimate in exchange for one that cannot be captured by a handful of bad values.

Where the 1.5 in the fence rule came from

Values below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR are conventionally flagged. The 1.5 is a choice, not a derivation, and knowing where it came from is the difference between using the rule and obeying it.

John Tukey introduced it in Exploratory Data Analysis in 1977, alongside the box plot itself. The reasoning is calibration against the normal distribution. For normally distributed data the interquartile range is about 1.349 standard deviations, so a fence at 1.5 × IQR beyond each quartile sits roughly 2.7 standard deviations from the centre.

At that distance about 0.7% of normal data falls outside — roughly one observation in 140, split between the two tails. That is the number Tukey was aiming at: rare enough that a flag means something, common enough that the rule finds real problems rather than firing once a decade.

The outer fence at 3 × IQR sits about 4.7 standard deviations out, where normal data essentially never lands. That is why the two are labelled differently: past the inner fence is mild, past the outer fence is extreme, and the second genuinely warrants a different level of suspicion.

None of this makes 1.5 correct in any deeper sense. It is a well-chosen default for roughly-normal data, and it was never claimed to be more.

Robustness, as a number rather than an adjective

“Robust” has a precise definition, and it is worth having because it turns a vague compliment into something you can compare.

The breakdown point of a statistic is the fraction of the data an adversary would have to corrupt before they could push the statistic anywhere they liked. Higher is better, and the ceiling is 50% — past that the corrupted values are the majority and no method can tell which half is real.

The range has a breakdown point of zero. One value out of any number is enough to send it to infinity.

So does the mean, and so does the standard deviation. Both are averages over every observation, so a single value large enough drags either one as far as you want.

The IQR has a breakdown point of 25%. You would have to corrupt a quarter of the observations before Q1 or Q3 moved out from between the values that surround them.

The median reaches 50%, the theoretical maximum, which is why it is the standard example of a resistant statistic.

The slider in the tool above turns this into something you can watch. Multiply one value — one, out of however many you entered — and the range and standard deviation climb without limit while the IQR does not move at all. Not “moves less”: does not move, because the corrupted point is outside the middle half and the IQR never looks there.

What a flagged value has, and has not, been shown to be

This is where the fence rule is most often misused, and the misuse quietly damages real analyses.

A value past the fence has been shown to be far from the middle of this particular dataset. That is the entire claim. It has not been shown to be a mistake, a measurement error, a fraud, or anything else that would justify removing it.

Genuine extremes are common and often the most informative observations you have. The one customer who spends twenty times the median is usually real. So is the one server response that took four seconds, and it is probably the one worth investigating. Deleting them because a fence flagged them removes precisely the signal that mattered.

Worse, the rule flags a fixed share of clean data by construction. Around 0.7% of any normal sample lands outside the fences with nothing wrong at all. In a dataset of ten thousand observations that is about seventy perfectly good values, and removing them makes the sample look tidier while making it less representative.

The defensible sequence is: flag, then look. If a value turns out to be a data-entry error or a unit mix-up, fix or drop it and say so. If it is real, keep it — and if its presence is distorting a mean, that is an argument for reporting the median instead, not for deleting the observation.

IQR against standard deviation: choosing between them

They measure the same idea and are not interchangeable. Which one to report follows from the data, not from preference.

Report the standard deviation when the data is roughly symmetric and clean. It uses every observation, so it is the more efficient estimate, and it is the input almost every further method expects — confidence intervals, t-tests, effect sizes and control limits are all built on it. Pair it with the mean.

Report the IQR when the data is skewed, heavy-tailed, or you do not trust every value. Incomes, house prices, response times, insurance claims and waiting times are all strongly right-skewed, and on those a standard deviation describes the tail more than the typical case. Pair it with the median.

A quick test: compare the mean and the median. If they are close, the distribution is roughly symmetric and the standard deviation is telling you something honest. If the mean sits well above the median, the tail is doing the talking, and the IQR with the median will describe a typical observation far better.

For roughly normal data the two are related by IQR ≈ 1.35 σ, so dividing an IQR by 1.349 gives a standard-deviation-like number that is far harder to corrupt. That quantity has a name — the normalised IQR — and it is one of the standard robust scale estimates.

Reporting both is entirely reasonable, and the gap between them is itself informative: a standard deviation much larger than IQR/1.35 says a few extreme values are carrying it.

Where the fence rule breaks down

Three situations where 1.5 × IQR gives an answer that looks authoritative and is not.

Skewed data. The rule is symmetric, but a right-skewed distribution is not. On income or latency data the upper fence flags a steady stream of ordinary observations simply because the right tail is long, while the lower fence flags nothing at all. The flags are real arithmetic and a poor guide. The standard fix is the adjusted box plot of Hubert and Vandervieren, which shifts the fences using a robust measure of skewness; another is to take logarithms first and apply the ordinary rule to those.

Small samples. With ten or fifteen values the quartiles themselves are unstable, so the fences move noticeably when a single observation is added. Applying an outlier rule at that size mostly measures the noise in your own quartile estimates. Look at the numbers directly instead.

Multiple groups mixed together. If a dataset actually contains two populations — two machines, two sites, two customer segments — the combined IQR describes neither, and the fences will flag members of the smaller group for being ordinary members of the smaller group. Splitting first is the answer, not a wider fence.

One further limit applies to every method here: the fences describe the sample in front of you. A new observation being inside them is not evidence that it is typical of anything wider, and the IQR carries no confidence interval of its own on this page.

Sources and methodology

Where the fence rule and the robustness figures come from.

• Tukey, Exploratory Data Analysis (Addison-Wesley, 1977) — the origin of the box plot and the 1.5 × IQR fence
• Hyndman & Fan (1996) — the quantile definitions the quartiles behind this IQR are computed from
• Hubert & Vandervieren, “An adjusted boxplot for skewed distributions” — the fence correction for skewed data
• Quartile Calculator — the eleven conventions, and why Q1 itself is not a settled number
• How we calculate — methodology

Method. Quartiles use the type 7 rule, which is what Excel, R, NumPy and Google Sheets return, so the IQR here matches the figure those tools give. The 1.349 relationship quoted between the IQR and the standard deviation is the normal-distribution value 2 × Φ⁻¹(0.75), not an approximation chosen for convenience. The contamination panel recomputes all three measures of spread from scratch on the modified dataset rather than adjusting them incrementally, so what it shows is the actual statistic and not a description of one. That engine is verified on every change against 0 assertions, including the quartile values checked against R’s published output. The count and the per-case breakdown are published on the formula verification page.

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