The five-number summary is a descriptive statistic

Descriptive statistics

Descriptive statistics quantitatively describe the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics , in that descriptive statistics aim to summarize a data set, rather than use the data to learn about the population that the data are...

 that provides information about a set of observations. It consists of the five most important sample percentiles

Percentile

In statistics, a percentile is the value of a variable below which a certain percent of observations fall. For example, the 20th percentile is the value below which 20 percent of the observations may be found...

:

1. the sample minimum (smallest observation)
2. the lower quartile
   Quartile
   In descriptive statistics, the quartiles of a set of values are the three points that divide the data set into four equal groups, each representing a fourth of the population being sampled...
   
    or first quartile
3. the median
   Median
   In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
   
    (middle value)
4. the upper quartile
   Quartile
   In descriptive statistics, the quartiles of a set of values are the three points that divide the data set into four equal groups, each representing a fourth of the population being sampled...
   
    or third quartile
5. the sample maximum (largest observation)

In order for these statistics to exist the observations must be from a univariate

Univariate

In mathematics, univariate refers to an expression, equation, function or polynomial of only one variable. Objects of any of these types but involving more than one variable may be called multivariate...

 variable that can be measured on an ordinal, interval or ratio scale

Level of measurement

The "levels of measurement", or scales of measure are expressions that typically refer to the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 Science article titled "On the theory of scales of measurement"...

.

Use and representation

The five-number summary provides a concise summary of the distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 of the observations. Reporting five numbers avoids the need to decide on the most appropriate summary statistic. The five-number summary gives information about the location (from the median), spread (from the quartiles) and range (from the sample minimum and maximum) of the observations. Since it reports order statistic

Order statistic

In statistics, the kth order statistic of a statistical sample is equal to its kth-smallest value. Together with rank statistics, order statistics are among the most fundamental tools in non-parametric statistics and inference....

s (rather than, say, the mean) the five-number summary is appropriate for ordinal measurements, as well as interval and ratio measurements.

It is possible to quickly compare several sets of observations by comparing their five-number summaries, which can be represented graphically using a boxplot.

The five-number summary is sometimes represented as in the following table:

median
1st quartile	3rd quartile
Minimum	Maximum

Example

This example calculates the five-number summary for the following set of observations: 0, 0, 1, 2, 63, 61, 27, 13.
These are the number of moons of each planet in the Solar System

Solar System

The Solar System consists of the Sun and the astronomical objects gravitationally bound in orbit around it, all of which formed from the collapse of a giant molecular cloud approximately 4.6 billion years ago. The vast majority of the system's mass is in the Sun...

.

It helps to put the observations in ascending order: 0, 0, 1, 2, 13, 27, 61, 63. There are eight observations, so the median is the mean of the two middle numbers, (2 + 13)/2 = 7.5. Splitting the observations either side of the median gives two groups of four observations. The median of the first group is the lower or first quartile, and is equal to (0 + 1)/2 = 0.5. The median of the second group is the upper or third quartile, and is equal to (27 + 61)/2 = 44.
The smallest and largest observations are 0 and 63.

So the five-number summary would be 0, 0.5, 7.5, 44, 63.

Example in R

It is possible to calculate the five-number summary in the R programming language using the fivenum function. The summary function, when applied to a vector, displays the five-number summary together with the mean (which is not itself a part of the summary).

> moons <- c(0, 0, 1, 2, 63, 61, 27, 13)
> fivenum(moons)
[1] 0.0 0.5 7.5 44.0 63.0
> summary(moons)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.0 0.5 7.5 20.88 44.0 63.0

See also

• Sample maximum and minimum
  Sample maximum and minimum
  In statistics, the maximum and sample minimum, also called the largest observation, and smallest observation, are the values of the greatest and least elements of a sample....
  
  
• Quartile
  Quartile
  In descriptive statistics, the quartiles of a set of values are the three points that divide the data set into four equal groups, each representing a fourth of the population being sampled...
  
  
• Median
  Median
  In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
  
  
• Seven-number summary
  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...
  
  
• Three-point estimation
  Three-point estimation
  The three-point estimation technique is used in management and information systems applications for the construction of an approximate probability distribution representing the outcome of future events, based on very limited information...