Error Analysis in Experimental Physical Science

Graphical Analysis

So far we have been using arithmetic and statistical procedures for most of our analysis of errors. In this section we discuss a way to use graphs for this analysis.

First, we introduce the use of "error bars" for the graphical display of a data point including its errors. We illustrate for a datapoint where (x, y) = (0.6 ± 0.1, 0.5 ± 0.2). The value of the datapoint, (0.6, 0.5), is shown by the dot, and the lines show the values of the errors. The lines are called error bars.

 

To the right we show data used in the analysis of a Boyle's Law experiment in the introductory Physics laboratory at the University of Toronto. Note the error bars on the graph. Instead of using a computer to fit the data, we may simply take a straight edge and a sharp pencil and simply draw the line that best goes through data points, as shown. Note we have used a red pencil. Recall that the slope is defined as the change in the dependent variable, pV in this case, divided by the change in the independent variable, 1/V in this case. The intercept is defined as the value of the dependent variable when the independent variable is equal to zero. In the graph to the right, the point where the independent variable is equal to zero is not shown. From the drawn line we can calculate that the intercept is 334 and the slope is -49.

The errors in the determination of the intercept and slope can be found by seeing how much we can "wiggle" the straight edge and still go through most of the error bars. To the right we draw those lines with a blue pencil. The intercepts and slopes of the blue lines, then, allows us to estimate the error in the intercept and slope of the red best match to the data. For this example, this procedure gives an estimate of the error in the intercept equal to ± 4 and the error in the slope equal to ± 10. So finally: .

intercept = 334 ± 4
slope = -49 ± 10

Question 14.2. In the first graph above we can not read the intercept directly off the graph because of the scale we have chosen. In the example to the right we can: it is just the point where the line intercepts the pV axis. Why is this graph not as good as the first one?

GRAPHS:

title graph, title each axis, use whole page, use straight edge, scale must be appropriate and consistent

To make a graph there are several rules that you need to follow.

A graph should be well organized and easy to read.

Use a descriptive title for every graph.

The most difficult decision in plotting a graph is the selection of the scale. The scale of the graph should be chosen such that the graph itself (data points that you are plotting) will fill most of the page. By increasing the size of the graph it will be much easier to see relationships and trends between the different points.

The scales on the different axis do not need to start at zero and do not need to be the same increments. The scale on an axis must be consistent. (each square represent the same increment)  The key is the data points should fill the graphing space efficiently. Do not have your axis fill the entire page and then have your points grouped together in only a small portion. Be sure your points FILL the page.

Both the horizontal axis and the vertical axis should be clearly labeled to explain what is being plotted and should include the necessary units, NEVER forget the units on either axis!

The data points themselves should be clearly marked, never just a "pin point" on the page. Be sure that each graph point is large enough and dark enough to see easily.

If the data points that have been plotted appear to fall along what seem to be a straight line we say that there is a possible linear relationship between these different data points. You can then use a ruler to draw a single straight line that DOESN'T go from point to point, but is a best AVERAGE fit of all the data points on the graph.

If instead the data points do not represent a linear relationship but seem to follow a curve, then you should try to draw the best smooth curve through the points on your graph. Once again the line that you draw will most likely not go through all the points (in some cases it might not go through any points). The line or the curve is an average value representing the data that has been plotted.

Relationship Between Volume and Temperature of a Gas at Constant Pressure