# Box plot

Figure 1. Box plot of data from the Michelson–Morley experiment

In descriptive statistics, a box plot or boxplot is a method for graphically depicting groups of numerical data through their quartiles. Box plots may also have lines extending vertically from the boxes (whiskers) indicating variability outside the upper and lower quartiles, hence the terms box-and-whisker plot and box-and-whisker diagram. Outliers may be plotted as individual points. Box plots are non-parametric: they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution (though Tukey's boxplot assumes symmetry for the whiskers and normality for their length). The spacings between the different parts of the box indicate the degree of dispersion (spread) and skewness in the data, and show outliers. In addition to the points themselves, they allow one to visually estimate various L-estimators, notably the interquartile range, midhinge, range, mid-range, and trimean. Box plots can be drawn either horizontally or vertically. Box plots received their name from the box in the middle.

## Types

Figure 2. Boxplot with whiskers from minimum to maximum
Figure 3. Same Boxplot with whiskers with maximum 1.5 IQR

Box and whisker plots quartiles, and the band inside the box is always the second quartile (the median). But the ends of the whiskers can represent several possible alternative values, among them:

• the minimum and maximum of all of the data[1] (as in figure 2)
• the lowest datum still within 1.5 IQR of the lower quartile, and the highest datum still within 1.5 IQR of the upper quartile (often called the Tukey boxplot)[2][3][4] (as in figure 3)
• one standard deviation above and below the mean of the data
• the 9th percentile and the 91st percentile
• the 2nd percentile and the 98th percentile.

Any data not included between the whiskers should be plotted as an outlier with a dot, small circle, or star, but occasionally this is not done.

Some box plots include an additional character to represent the mean of the data.[2][5]

On some box plots a crosshatch is placed on each whisker, before the end of the whisker.

Rarely, box plots can be presented with no whiskers at all.

Because of this variability, it is appropriate to describe the convention being used for the whiskers and outliers in the caption for the plot.

The unusual percentiles 2%, 9%, 91%, 98% are sometimes used for whisker cross-hatches and whisker ends to show the seven-number summary. If the data are normally distributed, the locations of the seven marks on the box plot will be equally spaced.

## Variations

Figure 4. Four box plots, with and without notches and variable width

Prior to the introduction of the box plot, the range bar chart was used to convey the interquartile range via a box by Mary Eleanor Spear in 1952[6] and again in 1969[7].

Since the mathematician John W. Tukey introduced this type of visual data display in 1969, several variations on the traditional box plot have been described. Two of the most common are variable width box plots and notched box plots (see 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.[1]

Notched box plots apply a "notch" or narrowing of the box around the median. Notches are useful in offering a rough guide to significance of difference of medians; if the notches of two boxes do not overlap, this offers evidence of a statistically significant difference between the medians.[1] The width of the notches is proportional to the interquartile range (IQR) of the sample and inversely proportional to the square root of the size of the sample. However, there is uncertainty about the most appropriate multiplier (as this may vary depending on the similarity of the variances of the samples).[1] One convention is to use ${\displaystyle \pm {\frac {1.58IQR}{\sqrt {n}}}}$.[3]

Adjusted box plots are intended for skew distributions. They rely on the medcouple statistic of skewness.[8] For a medcouple value of MC, the lengths of the upper and lower whiskers are respectively defined to be

${\displaystyle {\begin{matrix}1.5IQR\cdot e^{3MC},&1.5IQR\cdot e^{-4MC}{\text{ if }}MC\geq 0\\1.5IQR\cdot e^{4MC},&1.5IQR\cdot e^{-3MC}{\text{ if }}MC\leq 0\end{matrix}}}$.

For symmetrical distributions, the medcouple will be zero, and this reduces to Tukey's boxplot with equal whisker lengths of ${\displaystyle 1.5IQR}$ for both whiskers.

## Visualization

Figure 5. Boxplot and a probability density function (pdf) of a Normal N(0,1σ2) Population

The box plot allows quick graphical examination of one or more data sets. Box plots may seem more primitive than a histogram or kernel density estimate but they do have some advantages. They take up less space and are therefore particularly useful for comparing distributions between several groups or sets of data (see Figure 1 for an example). Choice of number and width of bins techniques can heavily influence the appearance of a histogram, and choice of bandwidth can heavily influence the appearance of a kernel density estimate.

As looking at a statistical distribution is more commonplace than looking at a box plot, comparing the box plot against the probability density function (theoretical histogram) for a normal N(0,1σ2) distribution may be a useful tool for understanding the box plot (Figure 5).