*The Business Professor*, updated February 24, 2019, last accessed August 9, 2020, https://thebusinessprofessor.com/lesson/analysis-of-means-definition/.

### Analysis of means (ANOM) Definition

Analysis of means is a systematic statistical procedure used in depicting significant differences among groups of information in a visual form. It is active mostly in quality control. ANOM methodology compares the average of each group to the mean of the overall process to discover statistical differences of significance.

### A Little More on What is Analysis of Means

It allows data to be graphically visualized. The technique was developed in 1967 by R. Ellis Ott, who observed statisticians were not challenged with comprehension of the analysis of variance. Edward G. Schilling extended the concept further in 1973 by allowing ANOM to be used with a large number of statistical tests based on the assumption of normality and connected data in a way that the mean and variances of the number of successes in repeated trials of a binomial experiment when only two outcomes are possible does not apply.

ANOM is a graphical likeness of analysis of variance which is a statistical procedure for deciding the amount of similarity or difference between two groups of data used to evaluate the balance of population averages. The graph shows the decision limits, total mean, and mean for each determinant. If a point in the chart falls outside of the decision limit for any given factor, it shows there is an important difference between the factor’s mean value and the population mean, which is the average of all elements that meet the criteria of selection for a group.

Mean analysis is the same as the analysis of variance but can be used for both normal distribution, which is a bell-shaped symmetrical curve representing the number of times a given sum of objects or events occur in a data set and binomial distribution, which is the number of times one of two possible outcomes occurs in a set of data.

ANOM is equal to the null hypothesis of the analysis of variances (ANOVA) which states all factors are the same in averages. When it comes to the alternative hypothesis of analysis of means, it indicates that the mean value of an element is not the same as the population mean. However, the alternative hypothesis of ANOVA states not all factor means are equal. Because of this difference, the ANOM and ANOVA can end up with different conclusions.

For example, if the averages of one factor group is higher than the population mean, and the other group is lower, then the F test which is a statistical test that determines if two populations with normal distribution have identical variances or standard deviation, for ANOVA will show there are differences, but the F test for ANOM will show none. In another example, if a one-factor level has a mean different than the other means, the analysis of variances F test may not show a difference, but for analysis of means, it could explain a variation of the group from the population mean.

### References for Analysis of Means

- http://www.businessdictionary.com/definition/autoregressive-moving-average-ARMA-model.html
- http://support.sas.com/documentation/cdl/en/qcug/63964/HTML/default/viewer.htm#qcug_anom_a0000000463.htm
- http://www.businessdictionary.com/definition/analysis-of-means-ANOM.html
- https://support.minitab.com/en-us/minitab/18/help-and-how-to/modeling-statistics/anova/supporting-topics/basics/understanding-analysis-of-means/

### Academic Research for Analysis of Means

- Exact critical values for use with the analysis of means, Nelson, L. S. (1983). Journal of Quality Technology, 15(1), 40-44. A simulation of analysis of means for samples of different samples is discussed and serve as an example. Estimates of critical values are presented, and the root is addressed.
- Applications of the analysis of means, Ramig, P. F. (1983). Journal of Quality Technology, 15(1), 19-25. When factors are at fixed levels, the analysis of means serves as an alternative to the analysis of variance with experimental designs. For simple interpretation and to have a graph of results, analysis of means is highly recommended. This paper is an introductory guide to how the analysis of means is used. It is tested on one-way and two-way factor designs..
- Additional uses for the analysis of means and extended tables of critical values, Nelson, P. R. (1993). Technometrics, 35(1), 61-71. The purpose of this article is to demonstrate how the critical values of analysis of means are appropriate for axial mixture designs, balanced, complete designs, Youden squares, Latin squares, balanced incomplete block designs, and Graeco-Latin squares. Although there are different designs, there is a situation when particular parameter combinations of the levels of freedom for error are lower than the number of levels for possibly all factors. Therefore, this article is also meant to provide critical values for this specific situation.
- Factors for the Analysis of Means, Nelson, L. S. (1974). Journal of Quality Technology, 6(4), 175-181. When examining fixed outcomes, ANOM or analysis of means is an alternative to ANOVA, the analysis of variance. This paper makes a comparison of the two methods. The analysis of means for samples unequal in size is used as an example for discussion.
- A comparison of sample sizes for the analysis of means and the analysis of variance, Nelson, P. R. (1983). Journal of Quality Technology, 15(1), 33-39. The sample sizes for the analysis of means compared to the sample sizes for the analysis of variance doesn’t show any remarkable differences.
- Power curves for the analysis of means, Nelson, P. R. (1985). Technometrics, 27(1), 65-73. This paper provides a short definition of analysis of means and then shows how the method was used to calculate the power curves for discovering the differences among a few different treatments.
- Analysis of means—a review, Rao, C. V. (2005). Analysis of means—a review. Journal of Quality Technology, 37(4), 308-315. Originally, ANOM or analysis of means was created to test the identicalness of different population means. Then the use of analysis of means was expanded to include tests for the equality of different correlation coefficients, treatment effects, variances, interaction effects, counts, proportions, and linear contrasts. ANOM shows not only statistical importance but also practical importance of the compared samples, which makes an analysis of means techniques a vital help for the search-and-identification process. ANOM is also helpful in the decision-making process.
- Modern experimental design, Ryan, T. P., & Morgan, J. P. (2007). Journal of Statistical Theory and Practice, 1(3-4), 501-506. This article serves as a guiding model of statistical concepts in the construction of experimental design. It acts as an applied introduction and short review of the elemental kinds of experimental designs and their uses. The paper covers designs having at minimum, one blocking factor, designs with multiple factors, split-unit, and Plackett-Burman designs.
- An analysis-of-means-type test for variances from normal populations, Wludyka, P. S., & Nelson, P. R. (1997). Technometrics, 39(3), 274-285. This paper presents a test for homogeneity of variances or the assumption that differences among all populations are the same. The test can be shown as a graph for the deviations of analysis of means. With the chart, it will be easier for practitioners to analyze statistical importance and significance of practicality. The critical values are shown for balanced designs.
- Exact analysis of means with unequal variances, Nelson, P. R., & Dudewicz, E. J. (2002). Technometrics, 44(2), 152-160. The analysis of means is a method for giving a comparison of treatment means to the overall mean to see if there is any significant difference present among them. It can be looked at as an alternate option for assessing fixed primary outcomes in a designed experiment than using the analysis of variance. In this paper looks at a population that has the same standard deviation or variances.
- Testing for interactions using the analysis of means, Nelson, P. R. (1988). Technometrics, 30(1), 53-61. Nelson suggests using analysis of means or ANOM technique to test the importance of two-factor interplay in balanced designs with fixed effects. When at least one of the two factors is only at two levels, the test uses identical critical points. When the two factors are at levels fewer than six but more than two, new critical points are given. Assessing the practical significance for quantities of interactions with higher order is of little use.
- A systematic approach to the analysis of means: part I. Analysis of treatment effects, Schilling, E. G. (1973). Journal of Quality Technology, 5(3), 93-108. This paper has a series of three parts which as a whole is an extension of the analysis of treatment technique. Part I provides a formula for make applying the analysis of means straightforward and considers the part of random factors. Part II extends the analysis of means to linear comparisons. Part III applies the method to non-normal distributions and explains the data for analysis of attributes.
- The analysis of means for balanced experimental designs, Nelson, P. R. (1983). Journal of Quality Technology, 15(1), 45-54. The plots of fixed effects for either mixed or fixed models of analysis of means is produced by a computer program. The main results use exact critical points, and the interaction effects have improved decision lines.