Five-number summary

<h2 id="use-and-representation">Use and representation</h2>
<p>The five-number summary provides a concise summary of the <a href="/facts/Probability_distribution/EpsKKVRu">distribution</a> 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 <a href="/facts/Order_statistic/QS3RnGcC">order statistics</a> (rather than, say, the mean) the five-number summary is appropriate for <a href="/facts/Level_of_measurement/8GAxPxFv">ordinal measurements</a>, as well as interval and ratio measurements.
</p><p>It is possible to quickly compare several sets of observations by comparing their five-number summaries, which can be represented graphically using a <a href="/facts/Boxplot/9jIxvjym">boxplot</a>.
</p><p>In addition to the points themselves, many <a href="/facts/L-estimator/JCk6uEru">L-estimators</a> can be computed from the five-number summary, including <a href="/facts/Interquartile_range/tRk6u8w2">interquartile range</a>, <a href="/facts/Midhinge/zm06bL9o">midhinge</a>, <a href="/facts/Range_(statistics)/Yu0HgeRj">range</a>, <a href="/facts/Mid-range/B4XTaulj">mid-range</a>, and <a href="/facts/Trimean/0uPDsLYF">trimean</a>.
</p><p>The five-number summary is sometimes represented as in the following table: 
</p>
<table><tbody><tr><td colspan="2">median</td></tr><tr><td>1st quartile</td><td>3rd quartile</td></tr><tr><td>Minimum</td><td>Maximum</td></tr></tbody></table>
<h2 id="example">Example</h2>
<p>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 <a href="/facts/Solar_System/QIcLSazQ">Solar System</a>.
</p><p>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.
</p><p>So the five-number summary would be 0, 0.5, 7.5, 44, 63.
</p>
<h3>Example in R</h3>
<p>It is possible to calculate the five-number summary in the <a href="/facts/R_programming_language/LSrkr8K8">R programming language</a> 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 five-number summary). The fivenum uses a different method to calculate percentiles than the summary function.
</p>
> 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. 
   0.00    0.75    7.50   20.88   35.50   63.00

<h3>Example in Python</h3>
<p>This python example uses the percentile function from the numerical library numpy and works in Python 2 and 3. 
</p>
import numpy as np

def fivenum(data):
    """Five-number summary."""
    return np.percentile(data, [0, 25, 50, 75, 100], method="midpoint")

>>> moons = [0, 0, 1, 2, 63, 61, 27, 13]
>>> print(fivenum(moons))
[  0.    0.5   7.5  44.   63. ]

<h3>Example in SAS</h3>
<p>You can use PROC UNIVARIATE in <a href="/facts/SAS_(software)/7y2ulIMX">SAS</a> to get the five number summary:
</p>
data fivenum;
input x @@;
datalines;
1 2 3 4 20 202 392 4 38 20
;
run;

ods select Quantiles;
proc univariate data = fivenum;
 output out = fivenums min = min Q1 = Q1 Q2 = median Q3 = Q3 max = max;
run;

proc print data = fivenums;
run;

<h3>Example in Stata</h3>

input byte y
0 
0 
1 
2 
63 
61 
27 
13
end 
list

tabstat y, statistics (min q max)

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Seven-number_summary/c8BP0qNu">Seven-number summary</a></li>
<li><a href="/facts/Three-point_estimation/4dfKwhTN">Three-point estimation</a></li>
<li><a href="/facts/Box_plot/9jIxvjym">Box plot</a></li></ul>

<ul><li>Hoaglin, David C.; <a href="/facts/Frederick_Mosteller/LdkVt2Tj">Mosteller, Frederick</a>; <a href="/facts/John_Tukey/2a5FbPWb">Tukey, John W.</a>, eds. (21 December 1982). <a href="https://archive.org/details/understandingrob0000unse"><i>Understanding Robust and Exploratory Data Analysis</i></a>. Wiley Series in Probability and Statistics (1st ed.). <a href="/facts/Wiley_(publisher)/53g3LqJc">Wiley</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0471097778. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/82008528">82008528</a>. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/473252998">473252998</a>. <a href="/facts/OL_(identifier)/A13NLNNh">OL</a> <a href="https://openlibrary.org/books/OL3488838M">3488838M</a> – via <a href="/facts/Internet_Archive/PHStU8Lo">Internet Archive</a>.</li>
<li>Greenwood, David; Woolley, Sara; Goodman, Jenny; Vaughan, Jennifer; Palmer, Stuart (8 November 2019). "Chapter 9: Statistics". <i>Essential Mathematics for the Australian Curriculum Year 10</i> (3rd ed.). <a href="/facts/Cambridge_University_Press_%2526_Assessment/TPeYA1rK">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1108773461. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/1231440374">1231440374</a>. <a href="/facts/OL_(identifier)/A13NLNNh">OL</a> <a href="https://openlibrary.org/books/OL33037157M">33037157M</a>.</li></ul>

Five-number summary open-in-new

Five-number summary