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# Tukey’s Method for Means Help

Tukey’s Method is used to compare means from multiple processes. The method compares all pairs of means. It controls the family error rate (α). Both the family error and the individual error rates are given in the output. The output from this method also includes the ANOVA table, the table of mean comparisons, and a graph containing all pairs of means.

There are two options for entering the data: stacked or unstacked. Stacked data have each treatment in a single column. Unstacked data have the treatment labels in one column and the results in the adjacent column. The example below uses unstacked data.

Five different treatments (A to E) were used to control the weight of a coating in grams. There were four samples for each treatment. We want to use Tukey’s method to determine if there are any significant differences in the treatments.

#### Tukey’s Method for Means Output

There are two new worksheets added for to your workbook for this test. One worksheet contains the numerical results. The other worksheet contains the confidence interval plots for each pair of treatments. The output is described below starting with the numerical results.

Numerical Results

The data are summarized as stacked data along with some statistics for each treatment level. The statistics include the number of results (count), the average, the standard deviation, and the variance for each treatment level. The value of alpha is also given.

The ANOVA table is given next. This table provides the sum of squares of the treatments, error and total as well as the degrees of freedom. The mean square results are given along with the F value. The key number is the p value. If this is less than or equal to alpha, it is turned red. This means that there is significant difference in the treatment means.

The table of means is then given. The family error rate and the individual rate are given. These are expressed as confidence limits.

The columns in the table are:

where qα(a,f) = the studentized range statistic, f = degrees of freedom associated with MSE, a = number of treatment levels, MSE = the mean square error, and n = the sample size for the individual treatment levels.

Confidence Interval Chart

This chart plots each pair of treatment levels. Those that do not include 0 are significantly different.

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