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Basic Skills and Principles: Statistics and Graphing

Presenting Data
Choosing the Values to Graph

Students can also have difficulty with graphing because they are unsure exactly what data to graph. Let's examine an example to show you what I mean.

Let's assume that a researcher is attempting to determine the effects of pesticide pollution on the hatching success of fish eggs. He has noticed that fish eggs in streams near agricultural fields fare poorly, while those is undisturbed areas hatch successfully. The researcher sets up a lab experiment in which ten groups of fish eggs are allowed to develop in unmanipulated stream water, and ten groups of eggs develop in stream water with the addition of pesticide. The proportion of eggs hatching in each group is tallied (Table 3), and descriptive statistics for the two groups compared.

 Statistic No Pesticide Pesticide Mean proportion eggs hatching 0.87 0.59 Standard deviation 0.07 0.09

 Group Proportion eggs hatching No Pesticide Pesticide 1 0.80 0.50 2 0.76 0.45 3 0.81 0.68 4 0.90 0.77 5 0.95 0.64 6 0.84 0.60 7 0.88 0.54 8 0.99 0.57 9 0.86 0.62 10 0.93 0.57

Table 3. Effects of pesticide pollution on
hatching success of fish eggs.

Now that we have the data, how do we graph it? First, we determine that a bar graph is most appropriate, as the independent variable is the two groups, and this is categorical, not continuous. Second, we must determine what to graph. If we graph all of the values, we get a graph like the one below. Figure 12. Effects of pesticide pollution on
hatching success of fish eggs.

But what does this graph tell you? There are 20 bars, 2 categories, and lots of different group numbers of mix things up. If we take a step back and look at the study, what we're really interested in doing is comparing the hatching success in groups exposed to pesticide and the groups not exposed to pesticide. So what we really want to do is to graph the mean of the two groups and compare them to one another. If we do that, we get a graph that looks like: Figure 13. Effects of pesticide pollution on
hatching success of fish eggs.

Now while this graph is useful in that it readily allows us to compare the means of the two groups, it doesn't provide us with any indication of the spread of the data in each group. Knowing the spread of the data can be important in making conclusions, so you need to provide the reader with this information. This is done by adding "error bars" to data points or bars on a graph that indicate the standard deviation of the data. In this example, we would add the standard deviation error bar to the bar for each group. In bar and line charts, error bars are drawn both above and below the mean to a distance equal to their value. Figure 14. Effects of pesticide pollution on
hatching success of fish eggs.

Realize that error bars are only appropriate when graphing a group's mean. If you are graphing individual data points, you cannot add error bars as there is only one data point per point on the graph (hence, no mean or standard deviation). Error bars are useful in that they allow you to see the overlap in the data points from two groups. Remembering that all of the data points in a distribution are within two standard deviations of the mean, you can visually estimate the data's distribution by doubling the height of the error bar (because it's only drawn to one standard deviation) for each group and seeing the degree to which their distributions overlap. As shown in the examples below, including error bars can make a big difference in your interpretation of a graph. Although the means of the two groups are identical in each of the graphs, the two groups in the first figure appear to be more different from one another than the two in the second figure due to the smaller error bars and lesser degree of overlap.  Figure 15. The role of error bars in estimating differences in treatment groups

Creating Graphs
You may choose to make your graphs by hand or with the assistance of a computer software application. In either case, you need to make sure your choice of graphs, the data to graph, and the formatting are all appropriate. There are several software applications that enable you to create graphs (e.g., Microsoft Excel) that are available in the campus computer labs. There are also some web sites that allow you to create graphs online, and they are listed below. Providing directions for graph preparation in each would be prohibitively long, so you will need to become familiar with their operation on your own if you opt to use them.

StatCrunch: http://www.statcrunch.com/

National Center for Education Statistics: http://www.nces.ed.gov/nceskids/graphing/