# Regression to the mean puzzle

In the "Regression to the Mean" chapter of "Thinking, Fast and Slow" by Daniel Kahneman, an example is given and the reader is asked to forecast the sales of individual stores given the overall sales forecast and the sales numbers from the previous year. For instance(the book's example has 4 stores, I use 2 here for simplicity):

Store    2011    2012
1        100      ?
2        500      ?
Total    600     660


The naive forecast would be 110 and 550 for stores 1 and 2, 10% increase for each. However, the author claims this naive approach is wrong. It is more likely for the poorer-performing store to increase more than 10%, and the better-performing store to increase(or even decrease) by less than 10%. So perhaps a forecast of 115(15% increase) and 535(7% increase) would be "more correct" than the naive forecast.

What I don't understand is how we can conclude that sales of 100 of store 1 is necessarily the poorer-performing store? Perhaps, due to location differences, the true time-series means of stores 1 and 2 are 10 and 550, and store 1 had a super year in 2011, and store 2 had a disastrous year in 2011. Then wouldn't it make sense to forecast a decrease for store 1 and increase for store 2?

I know that time series information was not given in the original example, but I am under the impression that "regression to the mean" refers to the cross-sectional mean and therefore time-series information does not matter. What am I misunderstanding?

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I happen to be reading that book. You have not adequately transcribed the key information. It says that "all stores are similar in size and merchandise selection, but their sales differ because of location, competition and random factors." That is key, especially that last bit. Random factors are necessary for regression to the mean to occur (if sales grew by a fixed amount, then the 10% gain equally dispersed across stores would be right).

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Are you saying that the "all stores are similar" assumption implies that their time series means are the same? Otherwise, two identical stores can still have very different means due to location. –  ezbentley Feb 21 '13 at 3:11
I admit it's not the greatest wording of a problem, but it's a lot clearer than what you had in your original question. –  Peter Flom Feb 21 '13 at 10:49

With so few data points, the answer will be almost entirely dictated by the prior (or implied equivalent). If the author has seen a lot this kind of data before, they may well have good reason to think their answer is more likely to be correct, given their past observations. I think it's a stretch to suggest this is an example of regression to the mean though, at least not without specifying some more information. For instance, are the stores in comparable locations or not? If they are and there are no other obvious differences between the stores then we may feel justified in thinking they are part of a comparable population and we can think about regression to the mean. If there are obvious differences between the stores that could explain a systematic difference in sales, then it becomes less sensible to do so.

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I think a better (hypothetical) illustration might be something like this:

Store    2011    2012
1        100      ?
2        180      ?
3        190      ?
4        210      ?
5        235      ?
6        300      ?


Barring systematic reasons we'd expect the worst performer (from random causes) to not be so again. And so also for the best performer.

Hence with 10% average growth I'd expect #1 to do better than 110 and #6 to do worse than 330.

I feel the iffy part is the assumptions. It is very rare IMHO that the laggard of the pack is truly just a random fluke and not some underlying heterogeneity.

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