Box plot: Difference between revisions

Revision as of 14:59, 21 February 2025

Revision as of 20:45, 15 April 2025

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[[File:Michelsonmorley-boxplot.svg|thumb|upright=1.5|Figure 1. Box plot of data from the [[Michelson–Morley experiment#Michelson experiment (1881)|Michelson experiment]]]]

In [[descriptive statistics]], a '''box plot''' or '''boxplot''' is a method for demonstrating graphically the locality, spread and skewness groups of numerical data through their [[quartile]]s.<ref>{{Cite book|last=C.|first=Dutoit, S. H.|url=http://worldcat.org/oclc/1019645745|title=Graphical exploratory data analysis.|date=2012|publisher=Springer|isbn=978-1-4612-9371-2|oclc=1019645745}}</ref>

In addition to the box on a box plot, there can be lines (which are called ''whiskers'') extending from the box indicating variability outside the upper and lower quartiles, thus, the plot is also called the '''box-and-whisker plot''' and the '''box-and-whisker diagram'''. [[Outlier]]s that differ significantly from the rest of the dataset<ref>{{Cite journal|last=Grubbs|first=Frank E.|date=February 1969|title=Procedures for Detecting Outlying Observations in Samples|url=http://dx.doi.org/10.1080/00401706.1969.10490657|journal=Technometrics|volume=11|issue=1|pages=1–21|doi=10.1080/00401706.1969.10490657|issn=0040-1706}}</ref> may be plotted as individual points beyond the whiskers on the box-plot. Box plots are [[non-parametric]]: they display variation in samples of a [[statistical population]] without making any assumptions of the underlying [[probability distribution|statistical distribution]]<ref>{{Cite book|last=Richard.|first=Boddy|url=http://worldcat.org/oclc/940679163|title=Statistical Methods in Practice : for Scientists and Technologists.|date=2009|publisher=John Wiley & Sons|isbn=978-0-470-74664-6|oclc=940679163}}</ref> (though Tukey's boxplot assumes symmetry for the whiskers and normality for their length).

Box plots are [[non-parametric]]: they display variation in samples of a [[statistical population]] without making any assumptions of the underlying [[probability distribution|statistical distribution]]<ref>{{Cite book|last=Richard.|first=Boddy|url=http://worldcat.org/oclc/940679163|title=Statistical Methods in Practice : for Scientists and Technologists.|date=2009|publisher=John Wiley & Sons|isbn=978-0-470-74664-6|oclc=940679163}}</ref> (though Tukey's boxplot assumes symmetry for the whiskers and normality for their length). The spacings in each subsection of the box-plot indicate the degree of [[statistical dispersion|dispersion]] (spread) and [[skewness]] of the data, which are usually described using the [[five-number summary]]. In addition, the box-plot allows one to visually estimate various [[L-estimator]]s, notably the [[interquartile range]], [[midhinge]], [[range (statistics)|range]], [[mid-range]], and [[trimean]]. Box plots can be drawn either horizontally or vertically.

== History ==

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Another popular choice for the boundaries of the whiskers is based on the 1.5 IQR value. From above the upper quartile ('''''Q''₃'''), a distance of 1.5 times the IQR is measured out and a whisker is drawn ''up to'' the largest observed data point from the dataset that falls within this distance. Similarly, a distance of 1.5 times the IQR is measured out below the lower quartile ('''''Q''₁''') and a whisker is drawn ''down to'' the lowest observed data point from the dataset that falls within this distance. Because the whiskers must end at an observed data point, the whisker lengths can look unequal, even though 1.5 IQR is the same for both sides. All other observed data points outside the boundary of the whiskers are plotted as '''outliers'''.<ref>{{Cite book |title=A Modern Introduction to Probability and Statistics |url=https://archive.org/details/modernintroducti00dekk_722 |url-access=limited |last=Dekking |first=F.M. |publisher=Springer |year=2005 |isbn=1-85233-896-2 |pages=[https://archive.org/details/modernintroducti00dekk_722/page/n240 234]–238 }}</ref> The outliers can be plotted on the box-plot as a dot, a small circle, a star, ''etc.'' (see example below).

[[File:Box Plot Picture.png|thumb|377x377px|This is a picture of a box plot representing data.]]

There are other representations in which the whiskers can stand for several other things, such as:

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=== Example with outliers ===

[[File:Boxplot with outlier.png|thumb|Figure 6. The generated boxplot of the example on the left with outliers]]

Above is an example without outliers. Here is a followupfollow-up example for generating box-plot with outliers:

The ordered set for the recorded temperatures is (°F): 52, 57, 57, 58, 63, 66, 66, 67, 67, 68, 69, 70, 70, 70, 70, 72, 73, 75, 75, 76, 76, 78, 79, 89.