Editing Box plot (section)

==Variations==
[[File:Fourboxplots.svg|thumb|300px|Figure 4. Four box plots, with and without notches and variable width]]

Since the mathematician [[John W. Tukey]] first popularized this type of visual data display in 1969, several variations on the classical box plot have been developed, and the two most commonly found variations are the variable-width box plots and the notched box plots shown in Figure 4.

'''Variable-width box''' plots illustrate the size of each group whose data is being plotted by making the width of the box proportional to the size of the group.  A popular convention is to make the box width proportional to the square root of the size of the group.<ref name="mcgill tukey larsen">{{Cite journal|last1=McGill|first1=Robert|last2=Tukey|first2=John W.|author2-link=John W. Tukey|last3=Larsen|first3=Wayne A.|date=February 1978|title=Variations of Box Plots|journal=[[The American Statistician]]|volume=32|issue=1|pages=12–16|doi=10.2307/2683468|jstor=2683468}}</ref>

'''Notched box''' plots apply a "notch" or narrowing of the box around the median. Notches are useful in offering a rough guide of the significance of the difference of medians; if the notches of two boxes do not overlap, this will provide evidence of a statistically significant difference between the medians.<ref name="mcgill tukey larsen" /> The height of the notches is proportional to the interquartile range (IQR) of the sample and is inversely proportional to the square root of the size of the sample. However, there is an uncertainty about the most appropriate multiplier (as this may vary depending on the similarity of the variances of the samples).<ref name="mcgill tukey larsen" /> The width of the notch is arbitrarily chosen to be visually pleasing, and should be consistent amongst all box plots being displayed on the same page.

One convention for obtaining the boundaries of these notches is to use a distance of <math alt="±1.58×IQR/sqrt(n)">\pm \frac{1.58 \text{ IQR}}{\sqrt n}</math> around the median.<ref name="Rboxplotstats">{{Cite web | title = R: Box Plot Statistics | work = R manual | url = http://stat.ethz.ch/R-manual/R-devel/library/grDevices/html/boxplot.stats.html | access-date = 26 June 2011}}</ref>

'''Adjusted box''' plots are intended to describe [[skewness|skew distributions]], and they rely on the [[medcouple]] statistic of skewness.<ref name="Hubert2008">{{cite journal
|first1=M. |last1=Hubert | author1-link = Mia Hubert
|first2=E. |last2=Vandervieren
|date=2008
|title=An adjusted boxplot for skewed distribution
|journal=Computational Statistics and Data Analysis
|volume=52
|issue=12
|pages=5186–5201
|doi=10.1016/j.csda.2007.11.008|citeseerx=10.1.1.90.9812
}}</ref> For a medcouple value of MC, the lengths of the upper and lower whiskers on the box-plot are respectively defined to be:
:<math>\begin{matrix}
1.5 \text{IQR} \cdot e^{3 \text{MC}}, &  1.5 \text{ IQR} \cdot e^{-4 \text{MC}} \text{ if } \text{MC} \geq 0, \\
1.5 \text{IQR} \cdot e^{4 \text{MC}}, & 1.5 \text{ IQR} \cdot e^{-3\text{MC}} \text{ if } \text{MC} \leq 0.
\end{matrix}
</math>
For a symmetrical data distribution, the medcouple will be zero, and this reduces the adjusted box-plot to the Tukey's box-plot with equal whisker lengths of <math>1.5 \text{ IQR}</math> for both whiskers.

'''Other kinds of box plots''', such as the [[violin plot]]s and the bean plots can show the difference between single-modal and [[Multimodal distribution|multimodal]] distributions, which cannot be observed from the original classical box-plot.<ref name=":0" />