# A Discussion of R-Square and S

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*Article Revised: *March 27, 2019

## Question from a Black Belt student

When selecting a model two of the criteria say that the Standard Error be small and R Square be large.

Since R Square is a proportion, one might think that the usual 95% would be the threshold — correct?

## My Response

It may be surprising, but most of the cutoff values used in statistics are quite arbitrary. This includes the p-value cutoff of 5% for hypothesis tests and confidence intervals. This discussion of p-values is important because in the discussions of R^{2} and *S* below I assume that the p-value for the model is “statistically significant.” In other words, I discuss the cutoffs for R^{2} and *S* values based on the assumption that both are from statistical models that meet an arbitrary cutoff for the model’s p-value!

## R^{2}

With R^{2} no arbitrary cutoff has ever become the accepted norm. Thus the advice that R^{2} should be “large.” In this case, *large* is in the eye of the experimenter. Indeed, what is considered large varies a great deal according to the type of experiment being conducted. In social science experiments researchers are delighted with statistically significant R^{2} values as low as 0.2 or even lower. In hard science and engineering experiments R^{2} values greater than 0.9 is often expected. As a general rule, the more the researcher knows about the science, the better controlled the experiment can be and the expectation for R^{2} increases. Obviously, humans behavior is poorly understood, to the point where some question the usage of the term “Social Science.” So R^{2} values that physicists and engineers would dismiss out-of-hand are acceptable in that field.

In Lean Six Sigma we usually find ourselves somewhere in the middle of these two extremes. If our projects involve customer responses, then statistically significant R^{2} values around 0.5 might be enough to give us the direction we need for improvement. But if we are improving, say, cycle time through a a process, then our threshold would be higher, perhaps 0.7. Still, as you can tell, these are arbitrary. The point is that we want our data to point us in the right direction for making improvements. What R^{2} value will do this for us varies on a case-by-case and project-by-project basis.

*S*

The proper value for the standard error, *S*, is also subject-matter and experiment or project specific. In fact, *S* and R^{2} are just two different ways of describing the same thing: how well the statistical model fits the data. R^{2} is a proportion or percentage, while *S* is in the units of the response variable. S is the standard deviation of the residuals (model errors.) Since residuals from good models are normally distributed, the *S* value can be used to model the distribution of modeling errors in the same units as the response variable. This often makes it easier for subject-matter experts to tell you, the Black Belt, what an acceptable value of *S* should be.

One last thing, there are published papers that treat various statistical cutoff values, such as p-values, much more rigorously. For example, p-value cutoffs based on economic or risk considerations. If you have a deeper interest in the subject these papers are worth looking up. Look in journals such as *The Journal of Quality Technology* or *Quality Engineering*. Expect to see a bit of math in these papers.

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