Five Number Summary Calculator
This five number summary calculator turns a list of numbers into its minimum, Q1, median, Q3 and maximum. Example: 7 9 12 15 18 21 24 31 80 gives 7 · 10.5 · 18 · 27.5 · 80, and 80 is an outlier.
Source: Wikipedia – Five-number summary. Updated: .
How it is calculated
The five number summary describes a data set with five values: the minimum, the lower quartile Q1, the median, the upper quartile Q3 and the maximum. It is the basis of the box plot.
How the five numbers arise
The values are sorted. The minimum and maximum are the smallest and largest value, and the median splits the data into two halves. Q1 is the median of the lower half, Q3 the median of the upper half. With an odd count the calculator’s default method leaves the middle value out of both halves; Tukey keeps it in both, and Excel interpolates. The interquartile range IQR = Q3 − Q1 measures the spread of the middle half.
Worked example
The list 7, 9, 12, 15, 18, 21, 24, 31, 80 has n = 9 values. The median is 18. The lower half 7, 9, 12, 15 has median 10.5, the upper half 21, 24, 31, 80 has median 27.5. The five number summary is 7 · 10.5 · 18 · 27.5 · 80, the range is 73 and the IQR 17. The fences are 10.5 − 25.5 = −15 and 27.5 + 25.5 = 53; 80 lies above and is an outlier. With Tukey or Excel (QUARTILE.INC), Q1 = 12 and Q3 = 24.
- The median and quartiles resist outliers; the mean does not.
- If the median sits close to Q1 or Q3, the distribution is skewed.
- For cut-offs at every tenth of the data use the decile calculator.
Frequently asked questions
What is in the five number summary?
The minimum, first quartile, median, third quartile and maximum. For 7 9 12 15 18 21 24 31 80 these are 7, 10.5, 18, 27.5 and 80.
Why does Q1 depend on the method?
There is no single definition of quartiles. The textbook method, Tukey and the Excel functions can differ for short lists; here Excel QUARTILE.INC gives Q1 = 12 instead of 10.5.
How do you spot outliers?
By the 1.5 × IQR rule: values below Q1 − 1.5 · IQR or above Q3 + 1.5 · IQR. In the example the limit is 53, so 80 is an outlier.
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