10.5: Plots with Error Bars

Experimental data often contains uncertainty. For example, if you measure light intensity at different distances, repeated measurements may not give exactly the same value every time. A plotted point may represent an average measurement, but the actual measured values may have varied above and below that average.

Error bars help show this variation. An error bar is usually drawn as a short vertical line passing through a data point. The length of the bar shows the amount of uncertainty, scatter, or possible measurement error associated with that point.

Note

Interpretation tip: Shorter error bars usually suggest less variation in the data. Longer error bars suggest more uncertainty or more spread in the measurements.

For symmetric error bars, where the error extends the same distance above and below each point, use:

 

errorbar(x, y, e)

 

Here x contains the x-values, y contains the plotted y-values, and e contains the error amount for each point. The error bar at each point extends from y(i) - e(i) to y(i) + e(i).

For asymmetric error bars, where the lower and upper errors are different, use:

 

errorbar(x, y, d, u)

 

In this form, d contains the lower error values and u contains the upper error values. The error bar at point i extends from y(i) - d(i) to y(i) + u(i).

Note

Important: The x vector, y vector, and error vector must have the same length. MATLAB needs one error value for each data point.

 

Example \(\PageIndex{1}\)

Light Intensity Data

Suppose we measure the intensity of light at different distances from a light source. The values below represent average intensity measurements, and ydErr stores the uncertainty associated with each measurement.

clear; clc; clf;

xd = 10:2:22;
yd = [950 640 460 340 250 180 140];
ydErr = [30 20 18 35 20 30 10];


errorbar(xd, yd, ydErr, 'o-')
xlabel('Distance (cm)')
ylabel('Intensity (lux)')
title('Light Intensity with Error Bars')
grid on

 

 

Solution

 

This plot shows not only the average intensity at each distance, but also the uncertainty in each measurement. This is much more informative than plotting the average values alone.

 

 

What Should the Error Values Represent?

Error bars are commonly used in laboratory reports, experimental engineering, biology, chemistry, physics, and any situation where measurements have uncertainty. They help the reader judge how reliable the trend appears to be.

The error vector can represent different things depending on the experiment. It might represent measurement uncertainty, standard deviation, standard error, tolerance, or a confidence interval. MATLAB does not decide what the error means; it only draws the bars using the values you provide.

Note

Reporting habit: When using error bars in a report, explain what the bars represent. For example, say whether the bars show standard deviation, standard error, or instrument uncertainty.