boxplot
Visualize summary statistics with box plot
Description
boxplot( creates a box plot of the data in
x)x. If x is a vector,
boxplot plots one box. If x is a
matrix, boxplot plots one box for each column of
x.
On each box, the central mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. The whiskers extend to the most extreme data points not considered outliers, and the outliers are plotted individually using the '+' symbol.
boxplot( creates
a box plot using the axes specified by the axes graphic object ax,___)ax,
using any of the previous syntaxes.
boxplot(___, creates
a box plot with additional options specified by one or more Name,Value)Name,Value pair
arguments. For example, you can specify the box style or order.
Examples
Create a Box Plot
Load the sample data.
load carsmallCreate a box plot of the miles per gallon (MPG) measurements. Add a title and label the axes.
boxplot(MPG) xlabel('All Vehicles') ylabel('Miles per Gallon (MPG)') title('Miles per Gallon for All Vehicles')
The boxplot shows that the median miles per gallon for all vehicles in the sample data is approximately 24. The minimum value is about 9, and the maximum value is about 44.
Create Box Plots for Grouped Data
Load the sample data.
load carsmallCreate a box plot of the miles per gallon (MPG) measurements from the sample data, grouped by the vehicles' country of origin (Origin). Add a title and label the axes.
boxplot(MPG,Origin) title('Miles per Gallon by Vehicle Origin') xlabel('Country of Origin') ylabel('Miles per Gallon (MPG)')
Each box visually represents the MPG data for cars from the specified country. Italy's "box" appears as a single line because the sample data contains only one observation for this group.
Create Notched Box Plots
Generate two sets of sample data. The first sample, x1, contains random numbers generated from a normal distribution with mu = 5 and sigma = 1. The second sample, x2, contains random numbers generated from a normal distribution with mu = 6 and sigma = 1.
rng default % For reproducibility x1 = normrnd(5,1,100,1); x2 = normrnd(6,1,100,1);
Create notched box plots of x1 and x2. Label each box with its corresponding mu value.
figure boxplot([x1,x2],'Notch','on','Labels',{'mu = 5','mu = 6'}) title('Compare Random Data from Different Distributions')
The boxplot shows that the difference between the medians of the two groups is approximately 1. Since the notches in the box plot do not overlap, you can conclude, with 95% confidence, that the true medians do differ.
The following figure shows the box plot for the same data with the maximum whisker length specified as 1.0 times the interquartile range. Data points beyond the whiskers are displayed using +.
figure boxplot([x1,x2],'Notch','on','Labels',{'mu = 5','mu = 6'},'Whisker',1) title('Compare Random Data from Different Distributions')
With the smaller whiskers, boxplot displays more data points as outliers.
Create Compact Box Plots
Create a 100-by-25 matrix of random numbers generated from a standard normal distribution to use as sample data.
rng default % For reproducibility x = randn(100,25);
Create two box plots for the data in x on the same figure. Use the default formatting for the top plot, and compact formatting for the bottom plot.
figure subplot(2,1,1) boxplot(x) subplot(2,1,2) boxplot(x,'PlotStyle','compact')
Each plot presents the same data, but the compact formatting may improve readability for plots with many boxes.
Box Plots for Vectors of Varying Length
Create box plots for data vectors of varying length by using a grouping variable.
Randomly generate three column vectors of varying length: one of length 5, one of length 10, and one of length 15. Combine the data into a single column vector of length 30.
rng('default') % For reproducibility x1 = rand(5,1); x2 = rand(10,1); x3 = rand(15,1); x = [x1; x2; x3];
Create a grouping variable that assigns the same value to rows that correspond to the same vector in x. For example, the first five rows of g have the same value, First, because the first five rows of x all come from the same vector, x1.
g1 = repmat({'First'},5,1);
g2 = repmat({'Second'},10,1);
g3 = repmat({'Third'},15,1);
g = [g1; g2; g3];Create the box plots.
boxplot(x,g)
Input Arguments
More About
Box Plot
A box plot provides a visualization of summary statistics for sample data and contains the following features:
The bottom and top of each box are the 25th and 75th percentiles of the sample, respectively. The distance between the bottom and top of each box is the interquartile range.
The red line in the middle of each box is the sample median. If the median is not centered in the box, the plot shows sample skewness.
The whiskers are lines extending above and below each box. Whiskers go from the end of the interquartile range to the furthest observation within the whisker length (the adjacent value).
Observations beyond the whisker length are marked as outliers. By default, an outlier is a value that is more than 1.5 times the interquartile range away from the bottom or top of the box. However, you can adjust this value by using additional input arguments. An outlier appears as a red + sign.
Notches display the variability of the median between samples. The width of a notch is computed so that boxes whose notches do not overlap have different medians at the 5% significance level. The significance level is based on a normal distribution assumption, but comparisons of medians are reasonably robust for other distributions. Comparing box plot medians is like a visual hypothesis test, analogous to the t test used for means.
Tips
boxplotcreates a visual representation of the data, but does not return numeric values. To calculate the relevant summary statistics for the sample data, use the following functions:min— Find the minimum value in the sample data.max— Find the maximum value in the sample data.median— Find the median value in the sample data.quantile— Find the quantile values in the sample data.iqr— Find the interquartile range in the sample data.grpstats— Calculate summary statistics for the sample data, organized by group.
You can see data values and group names using the data cursor in the figure window. The cursor shows the original values of any points affected by the
datalimparameter. You can label the group to which an outlier belongs using thegnamefunction.To modify graphics properties of a box plot component, use
findobjwith theTagproperty to find the component's handle.Tagvalues for box plot components depend on parameter settings, and are listed in the following table.Parameter Settings Tag Values All settings 'Box''Outliers'
When 'PlotStyle'is'traditional''Median''Upper Whisker''Lower Whisker''Upper Adjacent Value''Lower Adjacent Value'
When 'PlotStyle'is'compact''Whisker''MedianOuter''MedianInner'
When 'Notch'is'marker''NotchLo''NotchHi'
Alternative Functionality
You can also create a BoxChart object by using the boxchart
function. Although boxchart does not include all the functionality
of boxplot, it has some advantages. Unlike
boxplot, the boxchart function:
Allows for categorical rulers along the group axis
Provides the option of a legend
Works well with the
hold oncommandHas an improved visual design that helps you see notches more easily
To control the appearance and behavior of the object, change the BoxChart Properties.
References
[1] McGill, R., J. W. Tukey, and W. A. Larsen. “Variations of Boxplots.” The American Statistician. Vol. 32, No. 1, 1978, pp. 12–16.
[2] Velleman, P.F., and D.C. Hoaglin. Applications, Basics, and Computing of Exploratory Data Analysis. Pacific Grove, CA: Duxbury Press, 1981.
[3] Nelson, L. S. “Evaluating Overlapping Confidence Intervals.” Journal of Quality Technology. Vol. 21, 1989, pp. 140–141.
[4] Langford, E. “Quartiles in Elementary Statistics”, Journal of Statistics Education. Vol. 14, No. 3, 2006.
See Also
anova1 | grpstats | kruskalwallis | max | median | min | multcompare | quantile