OneTaskBy Tim King · Updated
Five Number Summary Calculator with IQR
Box Plot Values: Quartiles, IQR and Outlier Fences From Your Data
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How the Five Number Summary Calculator works
This five number summary calculator sorts your data and gives the minimum, lower quartile, median, upper quartile and maximum, the five values a box plot is drawn from. It also returns the interquartile range, the outlier fences and any values that fall outside them.
A five number summary describes the shape of a data set without assuming it is bell-shaped. The median marks the centre, the quartiles mark the middle half, and the minimum and maximum mark the ends. Comparing the gaps between them shows at a glance whether the data is skewed.
How the five number summary calculator works
Sort the data and take the median. Q1 is the median of the lower half and Q3 the median of the upper half; with an odd count the middle value is left out of both halves, the Tukey method used in most school courses. The IQR formula is Q3 − Q1, and the fences sit 1.5 × IQR beyond each quartile.
First example. For the default 11 values the summary is 6, 15, 40, 43, 49, with an IQR of 28 and no outliers.
Second example. In 5, 7, 8, 9, 10, 11, 12, 30 the upper fence is 15.25, so 30 is an outlier.
How many values lie between Q1 and Q3? About half of them, which is why the IQR is called the middle 50%.
Five number summary and outliers for small data sets
| Data | Min | Q1 | Median | Q3 | Max | Outliers |
|---|---|---|---|---|---|---|
| 1, 2, 3, 4, 5 | 1 | 1.5 | 3 | 4.5 | 5 | None |
| 1, 2, 3, 4, 5, 6 | 1 | 2 | 3.5 | 5 | 6 | None |
| 2, 4, 6, 8, 10, 12, 14 | 2 | 4 | 8 | 12 | 14 | None |
| 1, 3, 5, 7, 9, 11, 13, 15 | 1 | 4 | 8 | 12 | 15 | None |
| 5, 7, 8, 9, 10, 11, 12, 30 | 5 | 7.5 | 9.5 | 11.5 | 30 | 30 |
| 10, 20, 30, 40, 50 | 10 | 15 | 30 | 45 | 50 | None |
| 3, 3, 4, 5, 6, 6, 7, 8, 9 | 3 | 3.5 | 6 | 7.5 | 9 | None |
| 12, 15, 17, 20, 22, 25, 28 | 12 | 15 | 20 | 25 | 28 | None |
| 1, 1, 1, 1, 2 | 1 | 1 | 1 | 1.5 | 2 | None |
| 0, 5, 10, 15, 20, 100 | 0 | 5 | 12.5 | 20 | 100 | 100 |
| 55, 60, 62, 65, 70, 72, 75, 80, 95 | 55 | 61 | 70 | 77.5 | 95 | None |
| 2, 2, 3, 3, 4, 4, 5, 5 | 2 | 2.5 | 3.5 | 4.5 | 5 | None |
| 1, 10, 100, 1000 | 1 | 5.5 | 55 | 550 | 1,000 | None |
| 7, 8, 9 | 7 | 7 | 8 | 9 | 9 | None |
| 18, 21, 22, 24, 25, 27, 29, 31, 50 | 18 | 21.5 | 25 | 30 | 50 | 50 |
| 4, 8, 15, 16, 23, 42 | 4 | 8 | 15.5 | 23 | 42 | None |
| 100, 102, 104, 106, 300 | 100 | 101 | 104 | 203 | 300 | None |
| -5, -3, 0, 2, 4, 6 | -5 | -3 | 1 | 4 | 6 | None |
| 1.5, 2.5, 2.5, 3, 4.5, 6 | 1.5 | 2.5 | 2.75 | 4.5 | 6 | None |
| 30, 31, 32, 33, 34, 35, 36, 37, 38, 39 | 30 | 32 | 34.5 | 37 | 39 | None |
| 6, 47, 49, 15, 42, 41, 7, 39, 43, 40, 36 | 6 | 15 | 40 | 43 | 49 | None |
| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | 1 | 3.5 | 6.5 | 9.5 | 12 | None |
Tips
- Quartile methods vary. Calculators and spreadsheets can differ slightly on Q1 and Q3; this one uses medians of halves.
- Outliers are flags, not errors. Check them, but do not delete them without a reason.
- IQR resists outliers. Unlike the range, one extreme value barely moves it.
FAQ
How do you find Q1 and Q3?
Sort the numbers and split them at the median. Q1 is the median of the lower half and Q3 is the median of the upper half. If there is an odd number of values, leave the median itself out of both halves. The calculator shows which values went into each half in its working.
What counts as an outlier?
The usual rule flags any value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. These are the fences drawn as the ends of the whiskers on a box plot. Values beyond 3 × IQR are sometimes called extreme outliers. The rule is a convention for spotting values worth checking, not a law.
Why does Excel give different quartiles?
Excel's QUARTILE.INC interpolates between values, using the same method as PERCENTILE.INC, rather than taking medians of halves. On small data sets the results can differ by a fraction. Both are valid; school and exam questions usually expect the median-of-halves method used here.
See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.
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