Descriptive statistics
Five-Number Summary Calculator
Minimum, first quartile, median, third quartile, maximum — the five numbers a box plot draws.
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How to read this result
Each quarter of the data occupies one of the four gaps. That is what makes the summary informative rather than arbitrary: a wide gap means the values are spread thin there, a narrow one means they are bunched. Comparing the gap from Q1 to the median with the gap from the median to Q3 tells you about skew without computing anything.
The IQR is the spread of the middle half and is the robust counterpart to the standard deviation. Unlike the SD, it does not move when an extreme value gets more extreme.
The fences at Q1 − 1.5×IQR and Q3 + 1.5×IQR are where the box plot stops its whiskers. Values beyond them are drawn separately. That 1.5 is a convention of Tukey's, chosen so that roughly 0.7% of normal data falls outside — not a law of nature.
The five numbers do not show multimodality. Two distinct clusters and one even spread can produce identical summaries. A histogram is the check.
The formula
Tukey's hinges take the median of each half of the sorted data, including the overall median in both halves when n is odd. R's fivenum uses this; spreadsheet QUARTILE functions interpolate instead and can differ slightly.
Worked example
Five numbers from twelve exam scores
Scores: 72, 85, 78, 90, 68, 95, 88, 76, 82, 91, 79, 84.
- Sort them.68 72 76 78 79 82 84 85 88 90 91 95
- The median is the average of the sixth and seventh values.(82 + 84) / 2 = 83
- The lower hinge is the median of the bottom half.68 72 76 78 79 82 → (76 + 78)/2 = 77
- The upper hinge is the median of the top half.84 85 88 90 91 95 → (88 + 90)/2 = 89
- The minimum and maximum complete the summary.68 and 95
- The IQR and fences.IQR = 12; fences at 77 − 18 = 59 and 89 + 18 = 107 — no outliers
The two middle gaps — 77 to 83 and 83 to 89 — are both six points wide, which says the middle of this distribution is symmetric.
Checked against R's fivenum.
The calculator above is loaded with these numbers by the Load the worked example button.
Assumptions, and when to use something else
Only that the values can be sorted. The five-number summary works on skewed data, ordinal scales and data with outliers, which is precisely when the mean and standard deviation stop describing anything useful.
The quartile convention is the one thing to state. Tukey's hinges are the box-plot standard and what this page defaults to; a spreadsheet's QUARTILE function interpolates and can give slightly different values on small samples.
- You want the chartBox plot makerThe five numbers, drawn, with outliers marked and a PNG to download.
- You want other percentiles tooPercentile calculatorAny percentile, under any of the four conventions.
- You want the mean and standard deviation as wellDescriptive statisticsEvery summary statistic at once.
- You want to test the extreme valuesOutlier detectionApplies the IQR fences and two other rules.
Questions people ask
What are the five numbers?
Minimum, first quartile, median, third quartile and maximum. Together they describe the position of every quarter of the data, and they are exactly what a box plot draws.
Why do my quartiles differ from my textbook's?
Because the textbook probably uses Tukey's hinges (the median of each half) while a spreadsheet interpolates. Both are defensible; they differ on small samples. The method selector above lets you match whichever convention you need.
Does the five-number summary include outliers?
The minimum and maximum are the actual extreme values, outliers included. It is the box plot's whiskers that stop at the fences and draw outliers separately — which is why the whisker end and the minimum are not always the same number.