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Five number summary calculator

Paste a list of numbers. The minimum, first quartile, median, third quartile and maximum appear instantly, with the working shown and a box plot drawn from them.

Min
12
Q1
18
Median
22
Q3
27
Max
48
1020304050Data — min 12, Q1 18, median 22, Q3 27, max 48Outlier 48Data

Separate with commas, spaces or new lines, or paste a spreadsheet column. 14 numbers read.

Quartile method

Textbooks and TI calculators take medians of the lower and upper halves. Excel interpolates. Results can differ on small data sets.

Full results
Count (n)14
Minimum12
First quartile Q118
Median22
Third quartile Q327
Maximum48
Range36
Interquartile range (IQR)9
Lower fence Q1 − 1.5 × IQR4.5
Upper fence Q3 + 1.5 × IQR40.5
Outliers48
Mean23.5714
Standard deviation (sample)8.8118

Step by step

  1. Sort the 14 values: 12, 15, 17, 18, 19, 21, 22, 22, 24, 25, 27, 29, 31, 48.
  2. The median is the middle pair (values 7 and 8), averaged: 22.
  3. Q1 is the median of the lower half (12, 15, 17, 18, 19, 21, 22): 18. Q3 is the median of the upper half (22, 24, 25, 27, 29, 31, 48): 27.
  4. IQR = Q3 − Q1 = 27 − 18 = 9.
  5. Fences: 18 − 1.5 × 9 = 4.5 and 27 + 1.5 × 9 = 40.5. Values outside them are outliers: 48.

What the five-number summary is

The five-number summary describes a data set with five values: the minimum, the first quartile (Q1), the median, the third quartile (Q3) and the maximum. Together they show the centre, the spread of the middle half, and the full range. John Tukey popularised it in Exploratory Data Analysis (1977) as the basis of the box plot.

How to find it by hand

  1. Sort the numbers from smallest to largest.
  2. The first and last values are the minimum and maximum.
  3. The middle value is the median (average the two middle values if the count is even).
  4. Q1 is the median of the lower half and Q3 the median of the upper half. With an odd count, leave the overall median out of both halves.

Example: for 3, 5, 7, 8, 12, 13, 14, 18, 21 the summary is min 3, Q1 6, median 12, Q3 16, max 21.

Turn it into a chart

A box plot is the five-number summary drawn to scale. Use “Style & download box plot” above, or paste several groups into the box plot maker to compare them side by side.

When does a value become an outlier?

Ten delivery times in days. Drag the slowest one further out. It turns into a separate dot once it passes Q3 + 1.5 × IQR.

fence 15051015202530

Q1 5 · median 7 · Q3 9 · IQR 4

Upper fence = Q3 + 1.5 × IQR = 9 + 1.5 × 4 = 15. 12 is inside, so the whisker reaches it.

Questions

What are the five numbers in a five-number summary?

Minimum, first quartile (Q1), median, third quartile (Q3) and maximum.

How do you find Q1 and Q3?

Sort the data and split it at the median. Q1 is the median of the lower half and Q3 the median of the upper half. If there is an odd number of values, the middle value belongs to neither half (the method used on TI-83/84 calculators).

Why does Excel give a different Q1?

Excel’s QUARTILE.INC interpolates between values at position 1 + (n − 1) × p. Switch the quartile method above to “Excel QUARTILE.INC” to match it. Both methods are valid; they converge on large data sets.

Does the five-number summary include the mean?

No. It uses only order-based values, which makes it resistant to outliers. The mean and standard deviation are listed underneath for convenience.