Variations Box plot

figure 4. 4 box plots, , without notches , variable width

since mathematician john w. tukey introduced type of visual data display in 1969, several variations on traditional box plot have been described. 2 of common variable width box plots , notched box plots (see figure 4).

variable width box plots illustrate size of each group data being plotted making width of box proportional size of group. popular convention make box width proportional square root of size of group.

notched box plots apply notch or narrowing of box around median. notches useful in offering rough guide significance of difference of medians; if notches of 2 boxes not overlap, offers evidence of statistically significant difference between medians. width of notches proportional interquartile range (iqr) of sample , inversely proportional square root of size of sample. however, there uncertainty appropriate multiplier (as may vary depending on similarity of variances of samples). 1 convention use



±



1.58
i
q
r


n





{\displaystyle \pm {\frac {1.58iqr}{\sqrt {n}}}}

.

adjusted box plots intended skew distributions. rely on medcouple statistic of skewness. medcouple value of mc, lengths of upper , lower whiskers respectively defined be

1.5
i
q
r
⋅

e

3
m
c


,


1.5
i
q
r
⋅

e

−
4
m
c



 if 

m
c
≥
0




1.5
i
q
r
⋅

e

4
m
c


,


1.5
i
q
r
⋅

e

−
3
m
c



 if 

m
c
≤
0






{\displaystyle {\begin{matrix}1.5iqr\cdot e^{3mc},&1.5iqr\cdot e^{-4mc}{\text{ if }}mc\geq 0\\1.5iqr\cdot e^{4mc},&1.5iqr\cdot e^{-3mc}{\text{ if }}mc\leq 0\end{matrix}}}

.

observe symmetrical distributions, medcouple zero, , reduces tukey s boxplot equal whisker lengths of



1.5
i
q
r


{\displaystyle 1.5iqr}

both whiskers.