Seven-number summary

In descriptive statistics, the seven-number summary is a collection of seven summary statistics, and is a modification or extension of the five-number summary. There are two common forms.

(Parametric) Seven-number summary

The following numbers are parametric statistics for a normally distributed model:
# the 2^nd percentile
# the 9^th percentile
# the 25^th percentile or lower quartile or "first quartile"
# the 50^th percentile or median (middle value, or "second quartile")
# the 75^th percentile or upper quartile or "third quartile"
# the 91^st percentile
# the 98^th percentile

The unusual percentiles are used because the locations of all seven values will be equally spaced if the data is normally distributed. Some statistical tests require normally distributed data, so the plotted values provide a convenient visual check for validity of later tests, simply by scanning to see if the locations of those seven percentiles appear to be equally spaced.

Notice that whereas the five-number summary makes no assumptions about the distribution of the data, the seven-number summary is based on the normal distribution, and is not especially appropriate when normal data is not expected.

The values can be represented using a modified box plot. The 2^nd and 98^th percentiles are represented by the ends of the whiskers, and hatch-marks across the whiskers mark the 9^th and 91^st percentiles.

Bowley’s seven-figure summary

Arthur Bowley used a set of non-parametric statistics, called a "seven-figure summary", including the extremes, deciles and quartiles, along with the median - see his "Elementary Manual of Statistics" (3rd edn., 1920), p.62 - he defines "the maximum and minimum, median, quartiles and two deciles" as the "seven positions").

Thus the numbers are:
# the minimum
# the 10^th percentile (first decile)
# the 25^th percentile or lower quartile or "first quartile"
# the 50^th percentile or median (middle value, or "second quartile")
# the 75^th percentile or upper quartile or "third quartile"
# the 90^th percentile (last decile)
# the maximum

ee also

*Stanine