## What is Regression to the Mean?

Regression to the mean is the statistical tendency for an extreme sample or observed value to be followed by a more average one. It is also known as reverting to the mean, highlighting the propensity for a later observation to move closer to the mean after an extreme value. The concept applies only to random variation in a process or system and does not pertain to interventions or events that affect the outcome.

In short, outliers and flukes are likely to be followed by more typical events.

For example, suppose classes of students take a standardized test, and the average score for a class is 75%. If the students in one class average 90%, the next class will likely be lower and closer to the mean. Conversely, if a class averages 60%, the next class will likely be higher.

In statistics, “regression to the mean” is an important concept that explains why extreme results are often followed by more average ones in subsequent measurements. It’s a natural occurrence requiring no intervention. If you observe something unusually high or low, chances are it will be closer to the average the next time you measure it.

Regression to the mean is not just a theoretical concept, but a practical one with significant implications. Whether it’s a surprising performance in sports, a sudden market shift in stocks, or a change in standardized test scores, the concept of regression to the mean is at play, bringing results closer to the average.

Let’s learn more about this intriguing statistical concept and why it occurs, explore its implications, and why it matters in research.

## Regression to the Mean Examples

The following examples illustrate the statistical concept of regression to the mean in various contexts. In all these cases, someone might search for an underlying cause that produced the change, but it could just be a natural byproduct of random variation.

### Sports Performance

Sports can be an excellent source for regression to mean with all the statistics they record!

A basketball player scores unusually high in one game but returns to their average scoring in the subsequent match.

The idea can also apply to entire seasons because you can count a season as a sample of games. A rookie can have an unusually good first season and then experience the “sophomore slump,” where performance declines toward the average. Conversely, baseball players with subpar batting averages one season tend to improve toward the mean the next season.

### Investment in High-Performing Stocks

An investor selects a range of stocks for their portfolio because they performed unusually well in one quarter. However, in the following quarter, the performance of these stocks regresses to more average levels, reflecting the typical market behavior. Regression to the mean in action!

### Hospital Emergency Room Visits

Over a week, a hospital’s emergency room sees significantly fewer patients than average, seemingly without any specific cause. The following week, the number of patients returns to the usual average, aligning with the typical random variations in ER visits.

### Standardized Test Score Policy

Massachusetts’s 1999 effort to improve standardized test scores is an example of regression to the mean. That year, schools were given goals to improve their average test scores. Many of the lowest-performing schools achieved their targets. The policy looked successful at first. However, many top schools failed to meet their goals. This situation appears to be regression to the mean, where extreme scores naturally move closer to the average.

### Praise vs. Criticism

From fighter pilots to educational settings, trainers often criticize poor performers and praise high performers. Additionally, observers have noticed that the worst performers do better subsequently while the top ones do worse. This pattern leads to the mistaken conclusion that criticism boosts performance more than praise.

By now, you can see that this pattern was likely due to regression to the mean!

## Regression to the Mean Fallacy in Research

The regression to the mean fallacy arises in research when analysts mistakenly attribute changes in the outcome to manipulations of an experimental factor rather than observations reverting to the mean. This fallacy overlooks how unusual outcomes tend to shift towards the mean due to random chance, producing an apparent improvement.

To illustrate, consider a study in which researchers measure blood pressure and select individuals with unusually high readings. The initial higher-than-average readings could be due in part to random chance making the underlying condition appear worse. Consequently, when they retest these individuals later, their blood pressure readings are likely to shift towards more typical levels, creating the appearance of an improvement.

If researchers hastily conclude that this change is entirely due to their intervention, they fall prey to the regression to the mean fallacy. The subsequent reduction could be partially due to natural, random fluctuations in blood pressure. That random chance is separate from any intentional changes in the study’s variables.

In experiments targeting subjects with extreme characteristics, including a control group with similar extreme traits is crucial to account for regression to the mean. This approach helps differentiate between effects due to the intervention and those resulting from regression to the mean. Additionally, repeated measurements over time can help determine whether the initial extreme values were random fluctuations or part of a consistent trend. That’s better than relying on a single extreme measurement!

Understanding and accounting for the regression to the mean fallacy is a crucial aspect of research. It’s about avoiding the trap of drawing incorrect conclusions from data that is simply regressing to its mean.

## Why Does Regression to the Mean Occur?

You’ve seen the regression to the mean examples and know it occurs due to random variation, but how exactly? Let’s take a look!

First and foremost, this propensity is based on probabilities relating to sampling distributions. It does not reflect a memory or intentional adjustment in the system. People can get confused because it seems to counter the Gambler’s Fallacy by invoking a balancing mechanism that remembers past events. But it doesn’t!

Regression to the mean is the product of a consistent distribution of values where extreme values are less likely to occur than more central values. Consequently, probability distributions can model this phenomenon. We’ll do that in the next section!

## Regression to the Mean Worked Example

Let’s see how regression to the mean occurs with a worked example. For this example, I’ll use the distribution of IQ scores, which follow a normal distribution with a mean of 100 and a standard deviation of 15.

Imagine that researchers draw 5 participants randomly from this distribution and calculate a mean IQ of 90. We can use a sampling distribution of the means to calculate the probability that the next sample of 5 will have a mean closer to the population mean of 100.

### Results

Below is the sampling distribution for IQ scores for a sample size of five. Because it is a sampling distribution, each point on the curve relates to sample means, not individual values. I’m performing a two-tailed analysis because a sample mean of 110 is equally extreme as a sample mean of 90, just in the other direction from the mean.

The shaded region in the graph represents less extreme sample means than our sample mean of 90. Visually, you can see how the bulk of sample means will be less extreme because the shaded area is so much larger than the two extreme regions. That, in a nutshell, is how regression to the mean works.

But I also love how these probability distributions can quantify it! This graph shows that the probability of the second sample being less extreme is 0.8639.

Comparing the probability of the second sample being less extreme (0.8639) to it being at least as extreme (1 – 0.8639 = 0.1361) reveals that it is 6.3 times more likely to be less extreme, as shown below.

A sample mean of 90 might not seem that extreme compared to a population mean of 100, but you can already see regression to the mean in action.

Now, suppose the first sample mean is 85, a bit more extreme. Using the same calculation process, the following sample mean is a whopping 38.4 times more likely to be less extreme!

Regression to the mean is real even though it relies on random chance. It’s the natural outcome for distributions where extreme values are less likely than central values. The worked example shows the tendency can be substantial. Regressing to the mean is powerful enough that you must account for it when designing experiments and using data with extreme values!

## Reference

Mitchell, R. J. (2021). Regression to the Mean. In *Evidence-Based Policing: An Introduction (pp. 135-138)*. Routledge.

Adrian says

I view this ‘concept’ as an observation … many examples obviously can support it … but it’s a stretch to consider it as predictive. I’d say when the outlier presents itself, we view that event as worthy of observation … will the ‘regression-to-the-mean’ apply here? A ‘new mean’ could be in the making.

Jim Frost says

Hi Adrian,

That’s a valid point. You do need to be aware if a process or phenomenon is changing. I do thoroughly agree that studying outliers is crucial. Is there an assignable cause for why it occurred or was it part of the natural variation? As with any statistical concept, you need to know when to apply it.

However, for stable situations that you can model with a single probability distribution where extreme values occur less frequently than middle values, it’s actually extremely predictive as I show in the worked example.

Indeed, in some studies, it can be a combination of regression to the mean and a true effect. For example, in the blood pressure study in the post, the results for the treatment group could be a combined treatment effect plus regression to the mean where ROM exaggerates the true effect. So, you need the control group with the same extreme properties to account for the regression to the mean portion so you have a better estimate of the treatment effect.