##
Analysis of Variance
from Summary Data

(updated April 17 -- handles up to 10 groups)

This web page performs a one-way ANOVA from summary data -- that is, from the
counts, means, standard deviations (or standard errors) for each group.
This can be useful if you don't have the individual numbers for the members of
each group, but only the summarized data. This might happen, for example,
if you're analyzing data that has been summarized in a book or published
article.

A one-way ANOVA can be thought of as an extension of the
unpaired Student
t-test to more than two groups. Or, you can think of the Student t-test
as a special case of the ANOVA for only two groups (or "levels" in ANOVA
terminology). A two-level ANOVA is algebraically equivalent to a t-test,
and produces exactly the same p values. This page can handle up to 10 groups.

To use this page you must know **how many observations**
are in each group,
and you must know the **average** (arithmetic mean) and either the
**standard deviation (SD)** or the **standard error of the mean (SEM)**
for the observations in each group. Enter them in the table below,
replacing the sample data that's in the table. Leave all unused
groups' cells completely
blank). You can also put short meaningful group names
into the first column, replacing the generic "Group 1", "Group 2", etc.
Indicate, by the pop-up menu above the last column,
whether you've provided SD's or SEM's. Then click the "Compute" button.
The results will appear in the conventional "ANOVA Table" below. A p-value
less than 0.05 indicates that there is a significant difference somewhere
among the various groups; that is, they do not appear to have all come from
the same population.

The program also performs the Tukey HSD
("Honestly Significant Difference") post-hoc
test, to indicate which groups were significantly different from which
others. And it provides 95% confidence intervals around
the differences between the groups. If you want some other confidence
level (like 90% or 97.5% confidence intervals), you can type the
desired confidence level (without the % sign) in the box above the
Compute button.

Technical Note:
Performing the Tukey HSD test requires evaluating the Studentized Range
Distribution. This page uses a JavaScript routine from David Lane's Hyperstat web page,
modified to provide more accurate results for very large samples.
(Many thanks, David!)