Reporting Statistical Results in Your Paper [Table of Contents] [ PDF Version ] | Overview | Descriptive Statistics | Inferential Statistics | Graphical Summary Overview
The results of your statistical analyses help you to understand the outcome of your study, e.g., whether or not some variable has an effect, whether variables are related, whether differences among groups of observations are the same or different, etc. Statistics are tools of science, not an end unto themselves. Statistics should be used to substantiate your findings and help you to say objectively when you have significant results. Therefore, when reporting the statistical outcomes relevant to your study, subordinate them to the actual biological results.
Reporting Descriptive (Summary) Statistics
Means: Always report the mean (average value) along with a measure of variablility (standard deviation(s) or standard error of the mean ). Two common ways to express the mean and variability are shown below:
"Total length of brown trout (n=128) averaged 34.4 cm (s = 12.4 cm) in May, 1994, samples from Sebago Lake."
s = standard deviation (this format is preferred by Huth and others (1994)
"Total length of brown trout (n=128) averaged 34.4 ± 12.4 cm in May, 1994, samples from Sebago Lake."
This style necessitates specifically saying in the Methods what measure of variability is reported with the mean.
If the summary statistics are presented in graphical form (a Figure), you can simply report the result in the text without verbalizing the summary values:
"Mean total length of brown trout in Sebago Lake increased by 3.8 cm between May and September, 1994 (Fig. 5)."
Frequencies: Frequency data should be summarized in the text with appropriate measures such as percents, proportions, or ratios.
"During the fall turnover period, an estimated 47% of brown trout and 24% of brook trout were concentrated in the deepest parts of the lake (Table 3)."
Reporting Results of Inferential (Hypothesis) Tests
In this example, the key result is shown in blue and the statistical result, which substantiates the finding, is in red.
"Mean total length of brown trout in Sebago Lake increased significantly (3.8 cm) between May (34.4 ± 12.4 cm, n=128) and September (38.2 ± 11.7 cm, n = 114) 1994 (twosample t-test, p < 0.001)."
NOTE: AVOID writing whole sentences which simply say what test you used to analyze a result followed by another giving the result. This wastes precious words (economy!!) and unnecessarily increases your paper's length.
Summarizing Statistical Test Outcomes in Figures
If the results shown in a figure have been tested with an inferential test, it is appropriate to summarize the outcome of the test in the graph so that your reader can quickly grasp the significance of the findings. It is imperative that you include information in your Materials and Methods, or in the figure legend, to explain how to interpret whatever system of coding you use.
Several common methods for summarizing statistical outcomes are shown below.
Examples: Comparing groups (t-tests, ANOVA, etc)
Comparison of the means of 2 or more groups is usually depicted in a bar graph of the means and associated error bars.
For two groups, the larger mean may have 1-4 asterisks centered over the error bar to indicate the relative level of the p-value. In general, "*" means p< 0.05, "**" means p< 0.01, "***" means p< 0.001, and "****" means p<0.0001. In all cases, the p-value should be reported as well in the figure legend
The asterisk may also be used with tabular results as shown below. Note how the author has used a footnote to define the p-values that correspond to the number of asterisks. (Courtesy of Shelley Ball)
For three or more groups there are two systems typically used: lines or letters. The system you use depends on how complicated it is to summarize the result. The first example below shows a comparison of three means. The line spanning two adjacent bars indicates that they are not significantly different (based on a multiple comparisons test), and because the line does not include the pH 2 mean, it indicates that the pH 2 mean is significantly different from both the pH 5.3 (control) and the pH 3.5 group means. Note that information about how to interpret the coding system (line or letters) is included in the figure legend.
When lines cannot easily be drawn to summarize the result, the most common alternative is to use capital letters placed over the error bars. Letters shared in common between or among the groups would indicate no significant difference.
Example: Summarizing Correlation and Regression Analyses
For relationship data (X,Y plots) on which a correlation or regression analysis has been performed, it is customary to report the salient test statistics (e.g., r, r-square) and a p-value in the body of the graph in relatively small font so as to be unobtrusive. If a regression is done, the best-fit line should be plotted and the equation of the line also provided in the body of the graph.
Modified 1-11-2012
Department of Biology, Bates College, Lewiston, ME 04240
