This IQR calculator finds the interquartile range of a data list — Q3 − Q1, the span of the middle 50% — using the Tukey median-of-halves method, and applies the classic 1.5×IQR fences to flag outliers. It also returns the full five-number summary, because quartiles travel in that company.



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The Spread That Outliers Can’t Bully

The interquartile range measures spread the robust way: it ignores the top and bottom quarters entirely and reports how wide the MIDDLE half of the data spreads — Q1 (25th percentile) to Q3 (75th). A single billionaire walking into a bar moves the mean income and the standard deviation violently; the IQR doesn’t even notice him, which is precisely its virtue for skewed real-world data (incomes, house prices, reaction times, hospital stays). The defaults embed the demonstration: nine ordinary values 6–22 plus one 45. Quartiles: median 15, Q1 10, Q3 20, IQR = 10 — and the fences (10 − 15 = −5; 20 + 15 = 35) formally flag the 45 as an outlier, while the IQR itself was computed as if it weren’t there. Pair IQR with the median as the resistant summary duo, the way you’d pair SD with the mean.

Method honesty matters in this corner of statistics: there are at least nine published quartile definitions (Tukey hinges, Moore–McCabe, Excel’s inclusive/exclusive percentiles, numpy’s linear interpolation…), and on small samples they genuinely disagree. This page uses Tukey’s median-of-halves — the boxplot inventor’s own method, and the standard in school statistics — and says so in its output. On large samples all methods converge; on small ones, quote your method beside your quartiles.

Worked Examples (Verified)

  • Defaults: sorted 6…45 → median 15, Q1 10 (median of 6,8,10,12,14), Q3 20 (median of 16,18,20,22,45), IQR 10, fences −5 / 35, outlier: 45.
  • No-outlier set: 3, 5, 7, 9, 11, 13 → median 8, Q1 5, Q3 11, IQR 6, fences −4 / 20 — nothing flagged; the rule stays silent when the data behave.
  • The income sketch: neighborhood incomes (thousands) 28, 32, 35, 38, 41, 44, 47, 52, 58, 240 → Q1 = 35, Q3 = 52, IQR = 17 — seventeen thousand describes the typical spread while the range (212) describes mostly one house. Report median 42.5 with IQR 17 and the picture is honest.

IQR Discipline

  • Sort first — every quartile method assumes ordered data; the textarea sorts for you, but by-hand work dies on unsorted lists.
  • Odd n: Tukey EXCLUDES the median from both halves (our slice logic); Moore–McCabe agrees here, percentile methods differ — method loyalty within one analysis is the rule.
  • Outliers flagged ≠ outliers removed: investigate (typo? different population? genuine extreme?) and report analyses with and without if it changes conclusions.
  • IQR pairs with median; range pairs with nothing (report it, trust it little); SD pairs with mean. Mixing resistant and non-resistant summaries in one sentence confuses readers about which story you’re telling.
  • For boxplots, whiskers conventionally extend to the most extreme NON-outlier points (inside the fences), not to min/max when outliers exist — the five-number display here gives raw min/max; the fence line tells you where whiskers would stop.

Related Tools

Frequently Asked Questions

How do you calculate the IQR?

Sort the data, find the median, then take the medians of the lower and upper halves as Q1 and Q3 (Tukey’s method); IQR = Q3 − Q1. Defaults: halves 6–14 and 16–45 give Q1 = 10 and Q3 = 20, so IQR = 10. This calculator also returns the median, five-number summary and outlier fences in the same pass.

What does the IQR tell you?

The width of the middle 50% of your data — a typical-spread measure immune to extremes. An IQR of 10 on values centered at 15 says half of all observations live inside a 10-wide band (10 to 20) regardless of how wild the tails get. It’s the natural partner of the median and the standard spread for skewed distributions where SD overstates typical variation by chasing the tail.

How do you find outliers with the IQR?

Tukey fences: anything below Q1 − 1.5×IQR or above Q3 + 1.5×IQR is flagged (here: below −5 or above 35, catching the 45). “Extreme” outliers use 3×IQR fences. Flagging is a screening convention, not a verdict — flagged values get investigated (recording errors, mixed populations, genuine rarities) and the decision to keep, correct or exclude them must be reported with the analysis.

Why do different calculators give slightly different quartiles?

Because “quartile” has multiple legitimate definitions for finite samples: Tukey hinges (this page), Excel’s QUARTILE.INC/EXC interpolation, numpy’s linear method and others can differ by a point or two on small datasets. They converge as n grows. Professional practice: state your method. The IQR differences are almost always immaterial to conclusions — but irreproducible numbers erode trust, so the method note travels with the result here.

IQR vs standard deviation — which spread should I use?

Match the pair to the distribution: symmetric, well-behaved data → mean ± SD (more statistical power, feeds parametric tests). Skewed data or data with outliers → median + IQR (resistant: a few wild values can’t inflate them). Many analysts report both. If SD is much larger than IQR/1.35 (their normal-distribution equivalence), skew or outliers are present and the resistant pair deserves the headline.

What is the five-number summary?

Minimum, Q1, median, Q3, maximum — the complete skeleton of a boxplot, printed by this calculator alongside the IQR. Tukey designed the boxplot so these five numbers ARE the graph: box from Q1 to Q3, line at the median, whiskers (conventionally) to the furthest points inside the 1.5×IQR fences, dots beyond. Our Five Number Summary page treats the set on its own.

Can IQR be zero?

Yes — when at least half the data share one value (Q1 = Q3): e.g., survey responses where most answers are “3.” IQR 0 with nonzero range is a signature of heavily piled-up discrete data. Fences then equal that value and EVERYTHING else flags as an outlier — a sign the 1.5× rule, designed for continuous data, is the wrong tool for that dataset.

Is this IQR calculator free?

Yes — free, instant, no sign-up. Paste a list for Q1, median, Q3, IQR, the five-number summary and Tukey outlier flags, method stated.