Five number summary
Also known as: 5 number summary, five-number summary
A five number summary describes a data set using five values: the minimum, first quartile, median, third quartile, and maximum. Together they show the center, spread, and shape of the data, and they define the box plot.
The five number summary is a compact description of a data set built from five order statistics: the minimum, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum. Reported in that order, they divide the sorted data into four groups of roughly equal size, so each quartile boundary marks off about 25% of the observations.
To find one, sort the data, take the smallest and largest values as the minimum and maximum, and find the median. Q1 is the median of the lower half and Q3 is the median of the upper half. For the data set 4, 8, 11, 15, 18, 22, 27, the median is 15, the lower half is 4, 8, 11 with a median of 8, and the upper half is 18, 22, 27 with a median of 22. The five number summary is therefore 4, 8, 15, 22, 27.
The summary drives two common tools. The interquartile range (IQR = Q3 − Q1) measures the spread of the middle 50% of the data — 22 − 8 = 14 in the example above — and is resistant to extreme values in a way the range and standard deviation are not. A box plot draws a box from Q1 to Q3 with a line at the median and whiskers extending toward the minimum and maximum, making it easy to compare distributions and spot skew. Values more than 1.5 × IQR beyond Q1 or Q3 are commonly flagged as outliers.
Five number summaries and box plots are standard content in the data analysis, statistics, and probability section of the Praxis Core Math exam. Expect to compute quartiles from a small data set, read values off a box plot, or compare two distributions using their summaries.
Key takeaways
- The five values are minimum, Q1, median, Q3, and maximum, reported in that order.
- Q1 is the median of the lower half of the data and Q3 is the median of the upper half.
- The interquartile range, Q3 − Q1, measures the spread of the middle 50% of the data.
- A box plot is the graphical display of a five number summary.
- Values beyond 1.5 × IQR from Q1 or Q3 are typically treated as outliers.