# Applications of "Regression Towards the Mean" in Real Life

I was reading about "Regression Towards the Mean": (https://en.wikipedia.org/wiki/Regression_toward_the_mean). Over here, an explanation of this concept is provided:

"Consider a simple example: a class of students takes a 100-item true/false test on a subject. Suppose that all students choose randomly on all questions. Then, each student's score would be a realization of one of a set of independent and identically distributed random variables, with an expected mean of 50. Naturally, some students will score substantially above 50 and some substantially below 50 just by chance. If one selects only the top scoring 10% of the students and gives them a second test on which they again choose randomly on all items, the mean score would again be expected to be close to 50. Thus the mean of these students would "regress" all the way back to the mean of all students who took the original test. No matter what a student scores on the original test, the best prediction of their score on the second test is 50. If choosing answers to the test questions was not random – i.e. if there were no luck (good or bad) or random guessing involved in the answers supplied by the students – then all students would be expected to score the same on the second test as they scored on the original test, and there would be no regression toward the mean."

This is an interesting example - if I were to summarize this (assuming I understand this correctly), it seems that the regression towards the mean indicates that "luck runs out". Students who were lucky (i.e. guessed the correct answers to questions on the exam) are likely to perform poorly in future exams if they keep guessing - and students were unlucky on the exam (i.e. guessed the incorrect answers to questions on the exam) are likely to perform better on future exams.

My Question: What are some examples of the "Regression Towards the Mean" in real life modelling situations? In supervised learning models (e.g. regression, classification) applied on data from the healthcare and finance industries, what are some examples of "Regression Towards the Mean" that can appear in these problems?

The only (very general) example I can think of is "outliers influencing and skewing the results of a model". Does anyone know of any other real world examples where "Regression Towards the Mean" can appear and complicate statistical modelling?

• At stats.stackexchange.com/a/24649/919 I describe a famous example where a leading statistician failed to recognize hundreds of cases of regression to the mean and based an entire (but erroneous) theory on this omission.
– whuber
Commented Oct 15, 2021 at 15:34
• Stock returns are an almost perfect example of regression towards the mean. So are most games where luck plays a significant role Commented Oct 15, 2021 at 22:26

## 1 Answer

What are some examples of the "Regression Towards the Mean" in real life modelling situations?

In many (most?) cases where random effects coefficients are preferred over fixed-effects coefficients (BLUP versus BLUE): The former are shrunken towards the mean, compared to the latter, and tend to provide lower generalization error. Observations with higher values on the response variable are more likely to have positive error. Observations with lower values on the response variable are more likely to have negative error. Error being the observation's deviation from the population's conditional mean. One interpretation of that could be 'luck' (in case of positive error; and 'bad luck' in case of negative error; of course also depending on whether having more of the quantity is favorable). To reduce the influence of these errors on the estimated coefficients, shrinking the coefficients of units with more extreme values towards the mean yields lower generalization error.