SixSigma First - Article |Analysis Of Means

The Analysis of Variance (ANOVA) is a good tool to determine if there is a difference between several sample means, but it does not determine where the difference comes from (if there is any difference), It does not show what samples are so disparate that the null hypothesis has to be rejected. To know where the difference originates from, it is necessary to conduct further analyses after rejecting the null hypothesis. Tukey’s, Fisher’s and Dunnett’s are examples of comparisons that can help situate the sources of variations between means.
A simpler way to determine if the sample means are equal and at the same time visually determine where the difference is coming from (if there is any) would be the Analysis Of Means (ANOM).
The Analysis Of Means is a lot simpler and easier to conduct than ANOVA and it provides an easy to interpret visual presentation of the results.
When conducting ANOM, what we want to achieve is to determine the Upper and Lower Decision Limits. If all the sample means fall within those boundaries we can say with confidence that there is no ground to reject the null hypothesis, i.e. there no significant difference between the samples’ means. If at least one mean falls outside the limits, we reject the null hypothesis.

The Upper and Lower Decision Limits depend on several factors:

· The samples’ means

· The mean of all the observed data (The mean of the samples’ means)

· The standard deviation

· The Alpha level

· The number of samples

· The sample sizes (to determine the degree of freedom)

ANOM compares the natural variability of every sample mean with the mean of all the sample means.

If we have j samples and n treatment levels then the sample means are given as:

And the mean of all the sample means is:

Let’s call N the number of all observed data and s is the standard deviation.

Then the variance for the treatments would be

the overall Standard deviation

and the Upper and Lower Decision Limits would be:

Where represents the significance level and is found on the ANOM table at the intersection between the degree of freedom and the number of samples.

In our ANOVA example, we wanted to know if there was a difference between the productivity of the three shifts; after conducting the test, we concluded that there was not a significant difference between them. Let’s take the same example and this time use the Analysis Of Means.

Unfortunately, Excel does not have the capabilities to conduct ANOM, so we will use Minitab.

Note that we did not select Column C1. Click OK and a new worksheet appears with two stacked columns.

Now that we have the stacked data, we can conduct the ANOM. We click on Stat, we select ANOVA and then “Analysis of means”

For “Response” we select C2 because that’s where the data we are looking for resides and we double-click on “Subscripts” for Factor 1 if we have selected “Normal”. Remember that “Subscript is the default column title for the treatments titles.

The default for the Alpha level is 0.05. We can change it, but for the sake of this example, we leave it as it is
When we click on OK, the following graph pops up.

Since all the dots are within the decision boundaries, we conclude that there is not enough evidence to reject the null hypothesis. The difference between the three means is insignificant at Alpha level of 0.05.
This is the same conclusion we reached when we conducted and Analysis Of Variance with the sane data.