## 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.*

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**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)."

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Page

**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