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Figure 1. Box plot of data from the Michelson-Morley Experiment

In descriptive statistics, a box plot or boxplot (also known as a box-and-whisker diagram or plot) is a convenient way of graphically depicting groups of numerical data through their five-number summaries: the smallest observation (sample minimum), lower quartile (Q1), median (Q2), upper quartile (Q3), and largest observation (sample maximum). A boxplot may also indicate which observations, if any, might be considered outliers.

Boxplots display differences between populations without making any assumptions of the underlying statistical distribution: they are non-parametric. The spacings between the different parts of the box help indicate the degree of dispersion (spread) and skewness in the data, and identify outliers. Boxplots can be drawn either horizontally or vertically.

Contents

Alternative forms

Boxplot with whiskers from minimum to maximum
Same Boxplot with whiskers with maximum 1.9 IQR

Box and whisker plots are uniform in their use of the box: the bottom and top of the box are always the 25th and 75th percentile (the lower and upper quartiles, respectively), and the band near the middle of the box is always the 50th percentile (the median). But the ends of the whiskers can represent several possible alternative values, among them:

  • the minimum and maximum of all the data[1]
  • 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
  • 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 dot or a cross is plotted inside of the box, to represent the mean of the data in addition to the median.

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

Fairly 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.

Visualization

Figure 2. Boxplot and a probability density function (pdf) of a Normal N(0,1σ2) Population

The boxplot is a quick way of examining one or more sets of data graphically. Boxplots 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 intuitive than looking at a boxplot, comparing the boxplot against the probability density function (theoretical histogram) for a normal N(0,1σ2) distribution may be a useful tool for understanding the boxplot (Figure 2).

See also

Notes

References

External links

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