How can r-squared be very high in all groups except one where it is very low?

I have run a regressions relating metal revenue to a metal index in each of 25 different factories. In 24 of the factorias $R^2$ is greater than .75. In one factory the $R^2$ is .061.

Why is that $R^2$ is so low in one factory?

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What do you mean by "high regression"? –  Macro Jun 15 '12 at 17:17
The R-squared is like 99% –  AgainstASicilian Jun 15 '12 at 17:18
I'm guessing you have a small sample size. Is that right? –  Macro Jun 15 '12 at 17:19
Also, if $R^2 = .99$, you have very high correlation, since $R^2$ is the square of the correlation between the observed and predicted values. –  Macro Jun 15 '12 at 17:20
Making scatterplots of each of your 25 sets of $(x,y)$ data would likely show you instantly what's going on, Against. The discrepancy you note in the comments could have a simple explanation, such as one mistyped $y$ value in the dataset, or it could reveal some fundamentally different behavior for that factory. You will learn much more from the graphics than from the regressions. –  whuber Jun 15 '12 at 21:51

You can think of the R-squared as a property of the factory. You should have a conceptual understanding of why there should be a relationship between your two variables in each factory and what leads to stronger or weaker relationships.

You have encountered an outlier observation (i.e., the factory with low $R^2$).

It's up to you to analyse the data and think about what might have caused the outlier.

As @whuber notes a useful strategy is to plot the scatterplots of your two variables in each organisation, and particularly in the outlier organisation. There may be a mistyped datapoint in the outlier. Alternatively, something unique may have been occurring in the factory to prevent the relationship from holding (e.g., the factory broke down or their was industrial action, etc.).

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The $R^2$ is small which just means that the regression is not a very good fit. You can have a poorly fitting regression and still have the regression coefficient statistically significantly different from 0. This could happen if you had a lot of data so that the variance of the regression coefficient is small and hence the coefficient itself can be statistically significantly different from 0 even though it is small. Nevertheless you could still have large residuals meaning that the regression model only explains a small percentage of the total variance, 6.1% in your case.

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