## 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. If you need to evaluate more than 10 groups go here.

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!)

An updated version of this page, allowing more than 10 groups, graphic data visualization and sortable Tukey HSD output available here

 Group Name N (count) Mean Std. Dev. Std. Error

Desired confidence level for post-hoc confidence intervals:

 Source of Variation Sum of Squares d.f. Variance F p Between Groups: Within Groups: Total:

Post-hoc tests...

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