LESSON 12: Error bars

FOCUS QUESTION: How can I depict uncertainty and variability in data?

This lesson discusses various ways of putting error bars on graphs.

In this lesson you will: Understand visual methods for conveying confidence in measurements. Display error bars on different types of charts.

Contents

DATA FOR THIS LESSON

File

Description

fisheriris

This data set contains the famous Fisher iris data set. The data set consists of measurements of 150 flower samples from each of three species of flowers: Iris setosa, Iris virginica, and Iris versicolor. The measurements are in mm. Four features were measured for each sample: The length of the flower sepal The width of the flower sepal The length of the flower petal The width of the flower petal All 150 samples from the Fisher iris data are stored in a single table called meas: The four columns correspond to the four types of measurements: sepal length, sepal width, petal length and petal width, respectively. The first 50 rows contain data for Iris setosa The second 50 rows contain data for Iris virginica The third 50 rows contain data for Iris versicolor. The species information is kept in a separate vector called species. The data is sometimes referred to as Anderson's Iris data in honor of Edgar Anderson, the biologist who collected the data. See http://en.wikipedia.org/wiki/Iris_flower_data_set for additional information. Note: This dataset comes with the MATLAB distribution so you don't have to download it separately.

SETUP FOR LESSON 12

EXAMPLE 1: Load the Fisher iris data (comes with MATLAB)

Create a new cell in which you type and execute:

   load fisheriris;

You should see the following 2 variables in your Workspace Browser:

EXAMPLE 2: Compute the mean and standard deviation of the sepal lengths for 3 species

Create a new cell in which you type and execute:

   sepalLens = reshape(meas(:, 1), 50, 3);  % Make column 1 into a 50 x 3 array
   sLenMeans = mean(sepalLens);  % Calculate the means for the 3 species
   sLenSDs = std(sepalLens);     % Calculate the standard deviations for the 3 species

You should see the following varibles in your Workspace Browser:

Note: std(sepalLens) is population estimate of standard deviation, not the sample standard deviation.

EXAMPLE 3: Plot mean sepal length using standard deviation (SD) error bars

Create a new cell in which you type and execute:

   fisherTitle = 'Comparison of three species in the Fisher Iris data';
   irisSpecies = {'Setosa', 'Virginica', 'Versicolor'}; % Use for legend
   figure                                      % Label the top
   errorbar(sLenMeans, sLenSDs, 'ks');         % Error bars use black squares
   set(gca, 'XTick', 1:3, 'XTickLabel', irisSpecies) % Set ticks and tick labels
   xlabel('Species of Iris')
   ylabel('Sepal length in mm')
   title(fisherTitle)
   legend('Mean (SD error bars)', 'Location', 'Southeast') % Put in lower right

You should see the following 2 variables in your Workspace Browser:

You should see a Figure Window with a labeled error bar plot:

Create a new cell right here (beginning of a cell starts with %%). Write MATLAB code to display the mean petal widths for the three species, showing standard deviation errorbars.

EXAMPLE 4: Plot the SD error bars on a bar chart

Create a new cell in which you type and execute:

   figure
   hold on
   bar(sLenMeans, 'FaceColor', [0.5, 0.5, 1])  % Lighter so error bars show up
   errorbar(sLenMeans, sLenSDs, 'ks');            % Error bars use black squares
   set(gca, 'XTick', 1:3, 'XTickLabel', irisSpecies) % Set ticks and tick labels
   xlabel('Species of Iris');
   ylabel('Sepal length in mm')
   title(fisherTitle)
   legend('Mean (SD error bars)', 'Location', 'Northwest') % Put in lower right
   box on                                         % Force box around axes
   hold off

You should see a Figure Window with a labeled error bar plot:

EXAMPLE 5: Compute the standard error of the mean (SEM) for sepal lengths

Create a new cell in which you type and execute:

   numSamples = length(sepalLens);  % Length along the longest dimension
   sLenSEMs = sLenSDs./sqrt(numSamples); % Compute the standard error of the mean (SEM)

You should see the following 2 variables in your Workspace Browser:

The SEM estimates the standard deviation of sample means from the true population mean.

EXAMPLE 6: Plot mean sepal length using (SEM) error bars

Create a new cell in which you type and execute:

   figure
   errorbar(sLenMeans, sLenSEMs, 'rd') % Red diamonds
   set(gca, 'XTick', 1:3, 'XTickLabel', irisSpecies) % Set ticks and tick labels
   xlabel('Species of Iris');
   ylabel('Sepal length in mm')
   title(fisherTitle)
   legend('Mean (SEM error bars)', 'Location', 'Southeast')

You should see a Figure Window with a labeled error bar plot:

EXAMPLE 7: Plot SD and SEM error bars on the same graph

Create a new cell in which you type and execute:

   xPositions = [1, 2, 3];  % Use these as base x-axis error bar positions
   figure
   hold on
   errorbar(xPositions-0.1, sLenMeans, sLenSDs, 'g^')  % Green up triangles
   errorbar(xPositions+0.1, sLenMeans, sLenSEMs, 'bv') % Blue down triangles
   set(gca, 'XTick', 1:3, 'XTickLabel', irisSpecies) % Set ticks and tick labels
   xlabel('Species of Iris');
   ylabel('Sepal length in mm')
   title(fisherTitle)
   legend({'Mean (SD error bars)', 'Mean (SEM error bars)'}, 'Location', 'Southeast')
   hold off

You should see a Figure Window with two sets of error bars:

EXAMPLE 8: Compute the median and inter quartile range (IQR) for sepal lengths

Create a new cell in which you type and execute:

   sLenMedians = median(sepalLens);        % Species median sepal lengths
   sLenIQR = prctile(sepalLens, [25, 75]); % 25th and 75th percentiles

You should see the following 2 variables in your Workspace Browser:

The rows of sLenIQR correspond to the percentiles, and the columns correspond to the species.

EXAMPLE 9: Plot median sepal length using the inter quartile range (IQR) for error bars

Create a new cell in which you type and execute:

   lowerDist = sLenMedians - sLenIQR(1, :);      % Size of bottom bar
   upperDist = sLenIQR(2, :) - sLenMedians;      % Size of top bar
   figure
   errorbar(xPositions, sLenMedians, lowerDist, upperDist, 'm*') % Magenta asterisks
   set(gca, 'XTick', 1:3, 'XTickLabel', irisSpecies)
   xlabel('Species of Iris');
   ylabel('Sepal length in mm')
   title(fisherTitle)
   legend('Median (IQR error bars)', 'Location', 'Northwest') % Upper left

You should see the following 2 variables in your Workspace Browser:

You should see a Figure Window with median/IQR error bars:

EXAMPLE 10: Calculate the means and standard deviations of all characteristics

Create a new cell in which you type and execute:

    setosa = meas(1:50, :);               % First 50 rows are setosa
    virginica = meas(51:100, :);          % Second 50 rows are viginica
    versicolor = meas(101:150, :);        % Third 50 rows are versicolor
    irisMeans = [mean(setosa); mean(virginica); mean(versicolor)];
    irisSTDs =  [std(setosa); std(virginica); std(versicolor)];

*You should see the following 2 variables in your Workspace Browser:

EXAMPLE 11: Draw a grouped bar chart of mean iris characteristics

Create a new cell in which you type and execute:

   irisMeas = {'Sepal length', 'Sepal width', 'Petal length', ...
                               'Petal width'}; % Characteristics for legend
   figure
   bar(irisMeans, 'grouped')  % Rows are groups columns are group members
   set(gca, 'XTickLabel', irisSpecies);
   legend(irisMeas, 'Location', 'Northwest')
   xlabel('Species of Iris');
   ylabel('Mean size in mm')
   title(fisherTitle)

*You should see the following variable in your Workspace Browser:

You should see a Figure Window with a labeled group bar chart (groups correspond to species):

EXAMPLE 12: Plot the means of all characteristics using SD error bars

Create a new cell in which you type and execute:

   figure
   errorbar(irisMeans, irisSTDs, 's') % Use default colors and square markers
   set(gca, 'XTick', 1:3, 'XTickLabel', irisSpecies);
   legend(irisMeas, 'Location', 'Northwest')
   xlabel('Species of Iris');
   ylabel('Mean size in mm')
   title(fisherTitle)

You should see a Figure Window with multiple labeled error bar plots:

EXAMPLE 13: Plot the means of all characteristics using connected SD error bars

Create a new cell in which you type and execute:

   figure
   hold on
   errorbar(irisMeans(:, 1), irisSTDs(:, 1), ':sk')
   errorbar(irisMeans(:, 2), irisSTDs(:, 2), ':og')
   errorbar(irisMeans(:, 3), irisSTDs(:, 3), ':vb')
   errorbar(irisMeans(:, 4), irisSTDs(:, 4), ':^r')
   set(gca, 'XTick', 1:3, 'XTickLabel', irisSpecies);
   legend(irisMeas, 'Location', 'Northwest')
   xlabel('Species of Iris');
   ylabel('Mean size in mm')
   title(fisherTitle)
   hold off

You should see a Figure Window with multiple labeled error bar plots (with bars representing the same characteristic being connected):

SUMMARY OF SYNTAX

MATLAB syntax	Description
errorbar(Y, E)	creates a plot of the values of Y similar to plot(Y). The corresponding values in E show error bars at +/- that amount above and below the corresponding values in Y.
errorbar(X, Y, E)	creates a plot similar to errorbar(Y, E) except that this function uses the values of X for the x positions rather than using the integers 1, 2, ... .
errorbar(X, Y, L, U)	creates a plot similar to errorbar(X, Y, E) except that this function uses the values of L and U to determine the span of the error bars. The L array gives the distances below the corresponding values in Y, and the U array gives the distances above the corresponding values of Y.
Y = prctile(X, p)	returns a vector of the percentiles of the vector X. The vector p specifies the percentiles. When X is a 2D array, the i-th row of Y contains the percentiles p(i).
set(gca, PropertyName, PropertyValue)	sets a property of the current axis. The PropertyName argument is a string representing a property. The PropertyValue argument gives the value of the property.

_This lesson was written by Kay A. Robbins of the University of Texas at San Antonio and last modified on 31-Dec-2010. Please contact krobbins@cs.utsa.edu with comments or suggestions. The image is a reproduction of a drawing by James Sowerby that appeared in The Botanical Magazine vol. 1. no. 1 (1792). The drawing is available on Wikipedia at http://en.wikipedia.org/wiki/File:Iris_persica_%28Sowerby%29.jpg._

Published with MATLAB® 7.14