Load the “two clusters” and “evenly spread” presets in turn. Both have the five-number summary 10 / 20 / 30 / 40 / 50, so they draw the identicalbox — and the strip underneath shows one clumped at three values and the other spread evenly. That is what a five-number summary cannot tell you, and why the points are worth plotting.
What this tool shows
A five-number summary compresses however many observations you have down to five. The tool draws the resulting box plot with every one of your observations plotted underneath it, on the same scale, so you can see exactly what the compression threw away.
Minimum, Q1, median, Q3 and maximum
A box plot drawn from those five numbers
Every observation plotted on the same scale
Where the whiskers actually stop
Two datasets with an identical summary
What the box cannot show you
Box plot drawn live Raw points shown too Correct whisker rule Tukey, cited
Whiskers stop at the last point inside the fence, as Tukey defined them.
Updated 7 September 2026 · Works in any browser, no installation
Minimum, first quartile, median, third quartile, maximum. Five numbers that cut the sorted data into four parts with roughly a quarter of the observations in each. They are enough to draw a box plot, and not enough to know what the data looks like — which is a feature and a limitation at the same time.
At a glance
Formula shown
The five order statistics: min, Q1 (25th percentile), median (50th), Q3 (75th), and max. The box spans Q1 to Q3 with a line at the median; whiskers reach the most extreme observation still within 1.5 × IQR of the box.
Scenario support
Summarising exam marks, salaries or response times in a single line; drawing or reading a box plot; comparing several groups on one axis; giving a distribution's shape without publishing the underlying records.
Educational estimate
Planning support from the values you enter — not professional advice.
Why these five numbers and not some other five
The choice is not arbitrary. Each of the five answers a question the others cannot.
The median says where the centre is in a way the mean cannot when the data is skewed. Half the observations fall on each side of it, whatever the tail is doing.
Q1 and Q3 say how wide the bulk is. The distance between them is the interquartile range, and because it ignores the outer quarters it survives a bad value that would wreck a standard deviation.
The minimum and maximum say how far the data reaches. They are the least stable of the five — each is a single observation, and each will drift outward as you collect more data simply because there are more chances to see something extreme — but the span is often the first thing anyone wants to know.
Together they are five order statistics: values defined purely by position in the sorted list. That is what gives the summary its unusual property of working on any data you can rank, and of being unchanged by any transformation that preserves order. Take logarithms of every value and the five-number summary of the result is the logarithm of the original five, exactly. No summary built on means behaves that way.
Tukey introduced them as a set in 1977 precisely because five numbers is about the most a reader can hold at once, and because those five can be found by hand on a sorted list with nothing but halving.
Two datasets with the same summary and nothing else in common
This is the limitation worth internalising, and the tool above ships a demonstration of it.
Both have the five-number summary 10, 20, 30, 40, 50. Not approximately — exactly. They draw the identical box plot, pixel for pixel: same box, same median line, same whiskers, no outliers in either.
Underneath, they are not remotely the same data. The first is three tight clumps at 20, 40 and the extremes, with a hole through the middle where almost nothing lives. The second walks up the scale in even steps. A box plot cannot tell them apart, because a box plot only ever knew five numbers.
This is the same lesson as Anscombe’s quartet in a different costume: four datasets with the same mean, variance and correlation that look completely different when plotted. Summary statistics are compressions, and every compression has inputs it cannot distinguish. Knowing which ones is most of statistical literacy.
The practical response is not to distrust the summary. It is to plot the points too, which costs nothing, and is why the strip is drawn under the box on this page by default rather than hidden behind a toggle.
Where the whiskers actually stop
This is the commonest box-plot misreading, and it is entirely understandable because software disagrees.
In Tukey’s construction — the one used by R, ggplot2, matplotlib, seaborn and this page — a whisker does not reach the minimum or maximum. It extends to the most extreme observation that is still within 1.5 × IQR of the box. Anything past that is drawn as an individual point.
So the end of a whisker is a real data value, but it is not necessarily the smallest or largest one. If there are no outliers the whisker does land on the extreme, which is why the distinction goes unnoticed until it matters.
Some tools draw a range plot instead, with whiskers running all the way to the minimum and maximum and no outliers marked. Excel’s box-and-whisker chart can be configured either way. Others put the whiskers at fixed percentiles — the 10th and 90th, or the 5th and 95th — which is common in some clinical and finance conventions.
These produce visibly different pictures from identical data. A box plot without a stated whisker convention is genuinely ambiguous, and if you are publishing one, say which you used. This page reports the five-number summary and marks outliers separately, so both readings are available and neither is guessed at.
Reading skew off the box
The box plot is at its best here. Three things carry the signal.
Where the median line sits inside the box. Dead centre means the middle half is symmetric. Pushed towards the left edge means the data bunches at the low end with a longer reach to the right — right-skewed. Pushed right means the opposite.
The relative whisker lengths. A long right whisker with a short left one is a right tail, which is the shape of almost every income, price, latency and waiting-time distribution you will meet.
Which side the outliers are on. Outliers clustered on one side alone is usually skew rather than contamination, and it is a signal to stop using the symmetric fence rule.
A useful cross-check is the gap between the mean and the median. If the mean sits well above the median, the right tail is pulling it, and the median is the more honest single number to quote. Load the right-skewed preset in the tool and both signals appear at once: the median line hugs the left of the box and the right whisker runs long.
What the box will not tell you is how many modes the data has. A bimodal distribution can produce a perfectly ordinary-looking, symmetric box, as the two-cluster preset demonstrates.
Comparing groups, which is what box plots are really for
A single box plot is a mediocre way to look at one distribution — a histogram or a dot plot shows more. Several box plots on a shared axis are excellent, and that is the use the format was designed around.
Because each group collapses to one narrow shape, you can line up eight or twelve of them and read differences in centre, spread and skew at a glance. A histogram cannot be stacked that way without becoming unreadable.
Two cautions apply when you do. Box width means nothing in a standard box plot: a group of nine and a group of nine thousand draw the same width box, and the reader has no way to tell which is which. Label the group sizes, or use a variable-width box plot where the width does encode n.
And overlapping boxes are not a significance test. Two boxes that overlap can still come from populations with genuinely different medians, and two that barely touch can be well within sampling noise at small n. If the question is whether a difference is real, that is a hypothesis test, not a reading of a chart. Notched box plots exist for exactly this: the notch approximates a confidence interval around the median, and non-overlapping notches are a rough indication of a real difference.
When to plot something else
Four situations where a box plot is the wrong chart, with what to reach for instead.
Small samples. With under about ten observations, just plot the points. A box built from five or six values implies a precision in the quartiles that is not there, and the reader can see six numbers directly.
Suspected multiple modes. The box cannot represent them at all. A histogram, a density plot or a violin plot — which is a density curve mirrored around the axis, often with a small box inside it — will show two peaks where a box shows one uninformative rectangle.
Heavily discrete or tied data. Ratings out of five, dice results, Likert scales: quartiles land on the same few values and the box degenerates. A bar chart of counts is both simpler and more informative.
When individual points carry meaning. In small experiments a reader often wants to see each subject. A strip plot or beeswarm, optionally with the box behind it, gives both.
The general principle: a box plot is a comparison tool and a compression. When you have the room to show the data, show the data.
Sources and methodology
Where the construction and the whisker rule come from.
Method. Q1 and Q3 use the type 7 rule, matching Excel, R, NumPy and Sheets. Whiskers are drawn to the most extreme observation still inside the 1.5 × IQR fence rather than to the minimum and maximum, which is Tukey’s definition and whatboxplot.stats returns; values beyond are drawn individually. The box plot is plain SVG with no charting library, so it is present in the page source rather than assembled after load. The claim that the two demonstration presets share an identical five-number summary is not a writing flourish — it is asserted in the verification suite, and it failed there on the first attempt, which is why the presets you see are not the ones first written. That engine is verified on every change against 0 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
QuartileQ1, Q2 and Q3 by all eleven conventions in common use — Excel, R, Tukey hinges and the TI-84 rule — with a warning when they disagree.
IQRInterquartile range, the 1.5 × IQR outlier fences, and exactly which values fall outside — with a slider that corrupts one value so you can watch the range move while the IQR does not.
PercentileThe value at any percentile and the percentile rank of any value, with all nine standard methods — and the round trip that shows the two directions are not inverses.
Standard DeviationSample and population standard deviation, plus variance, mean, median, quartiles, z-scores, outliers, and confidence intervals.
Weighted AverageEach value carries the weight you give it, with every item's share of the total shown as a percentage so you can see what is actually driving the answer.
Geometric MeanThe average for things that compound. Growth of +50% then -50% averages to zero arithmetically and to a real 13.4% loss geometrically, which is what actually happened.
An educational tool for describing a dataset. A five-number summary compresses your data to five values by design; it is a starting point for looking at a distribution, not a substitute for looking at it.
Published a five-number summary calculator that draws the box plot with every one of the reader's observations plotted underneath it on the same scale, so the compression the summary performs is visible on the page rather than described.
Ships two presets with a provably identical five-number summary and visibly different data — the Anscombe argument applied to box plots. The identity is asserted in the verification suite, and the first pair written failed that assertion, which is why the shipped pair is not the one first drafted.
Draws whiskers to the most extreme observation still inside the fence, as Tukey defined them, and says plainly that this differs from the min-to-max whiskers some software draws — the commonest box-plot misreading.
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