How to use the five number summary calculator
- Paste or type your numbers, separated by commas, spaces or new lines. At least two are needed.
- Pick a quartile method if your course or software uses a particular one. The default is inclusive, the same as Excel QUARTILE.INC.
- Read the five numbers in order: minimum, first quartile, median, third quartile, maximum.
- Check the table of methods to see how much Q1 and Q3 change if you switch. The minimum, median and maximum never change.
Formula
Five numbers: minimum, Q1, median, Q3, maximum
Range = maximum - minimum
IQR = Q3 - Q1
Inclusive quartile at p: position (n - 1) x p, with linear interpolation
The median is the middle value of the sorted list, or the average of the two middle values when n is even.
Ten delivery times in minutes
The sorted times are 4, 7, 9, 10, 12, 15, 18, 21, 25 and 40. The minimum is 4 and the maximum is 40. The median is the average of 12 and 15, which is 13.5. With the inclusive method Q1 is at position 9 x 0.25 = 2.25, which is 9 + 0.25 x (10 - 9) = 9.25, and Q3 is at position 6.75, which is 18 + 0.75 x (21 - 18) = 20.25.
The five number summary is 4, 9.25, 13.5, 20.25, 40. The range is 36 and the IQR is 11. With the median-of-halves method Q1 would be 9 and Q3 would be 21. For the numbers 1 to 9 the inclusive summary is 1, 3, 5, 7, 9.
Reading the five numbers
Between the minimum and Q1 lies the lowest quarter of the data. Between Q1 and the median lies the next quarter, and so on. Half of the values sit between Q1 and Q3. If the median is closer to Q1 than to Q3, the upper half of the data is more stretched out than the lower half.
These five numbers are exactly what a box plot draws: a box from Q1 to Q3, a line at the median and whiskers toward the extremes.
Why quartiles differ between tools
Software and textbooks define the quartiles in different ways, so Q1 and Q3 can vary slightly for the same data. The table in the results shows all four methods side by side for your list. For large data sets the methods agree closely, and for small ones the difference can be noticeable.
Pair this summary with the outlier calculator to flag unusual values, or with the frequency distribution calculator to see the shape of the data.
Assumptions and limits
- The quartile method is stated with the result. The default is inclusive.
- The data are numbers. Entries that are not numbers are ignored and listed in a warning.
- The exclusive method cannot place Q1 and Q3 for very small lists, and the tool says so instead of guessing.
- Up to 5,000 numbers are accepted.
Frequently asked questions
What is the five number summary?
It is the minimum, the first quartile, the median, the third quartile and the maximum of a data set. Together they describe the center and spread without assuming any particular shape.
How do I find Q1 and Q3 by hand?
Sort the data and find the median. Q1 is the median of the lower half and Q3 is the median of the upper half, with a rule for whether the middle value is included. Spreadsheets instead interpolate, which can give slightly different values.
Is the interquartile range part of the five number summary?
It is calculated from it. The IQR is Q3 minus Q1 and measures the spread of the middle half of the data. It is shown beside the five numbers.
Can I use the five number summary for a box plot?
Yes. The box runs from Q1 to Q3 with a line at the median, and the whiskers reach toward the minimum and maximum. Some box plots stop the whiskers at the fences and plot outliers as points.