Regression to the Mean Calculator
What is regression to the mean, and how do you calculate it?
An extreme reading is expected to be less extreme next time, because part of what made it extreme does not repeat. The prediction is mean + r × (observed − mean), where r is how much one period tells you about the next. On the twelve readings here, a record of 84 against an average of 52.6 is expected to be followed by about 61 — a fall of 23 with nothing whatsoever having changed.
Reward your best month and the next one disappoints. Punish your worst and the next one improves. Both happen whether or not you do anything at all.
That is regression to the mean, and it is the single most common reason people believe something worked.
Built by Muhammad Shahzeb Anwar, MSc Industrial and Organizational Psychology, who also writes the paid version of this tool. This is the arithmetic behind most claims that an intervention worked.
Where the next reading is expected to land
Paste your readings, then the one you want to explain. Twelve monthly figures is a reasonable minimum.
What regression to the mean actually is
Any measurement is partly the thing you are measuring and partly noise: luck, weather, who was on shift, which customers happened to call. When a reading comes out unusually high, it is usually because both parts happened to be favourable at once.
The skill repeats. The luck does not. So the next reading is expected to sit closer to the average, and nothing has to cause that - it is what the arithmetic does on its own.
The prediction is mean + r × (observed − mean). The r is the lag-one correlation: how much one period tells you about the next. At r = 1 nothing pulls back. At r = 0 the whole distance closes and the next reading is simply the average.
Why this makes interventions look like they worked
Almost every intervention is aimed at an extreme. You retrain the worst-performing branch. You investigate the month with the most complaints. You have a word with the salesperson who had a dreadful quarter.
All of those are, by construction, measurements at the far end of the distribution - and the next reading was going to move toward the average whatever you did. The improvement arrives on schedule and the intervention takes the credit.
It runs the other way too, and more cruelly. Reward the record month and the next one disappoints, which teaches managers that praise makes people complacent and criticism works. Kahneman described exactly this happening with flight instructors, who had concluded from the pattern that praise was harmful.
The only way to know whether something worked is to compare against what would have happened anyway. That needs a control group, and this tool cannot supply one.
What the correlation is telling you
A high r means the process has memory: a good month really does predict another good month, and an extreme reading is less likely to be luck.
A low r means each period is close to independent, and almost the whole distance from average is expected to close by itself. That is the case where interventions look most impressive and are least likely to have done anything.
Below about twenty readings, treat r as a rough indication. The tool computes it from three, and says so, but a correlation from a handful of points is not a stable number.
What this will not tell you
This predicts where a reading is expected to land. It cannot tell you whether anything you did worked, because that needs a comparison with what would have happened otherwise. It is not a forecast either: it is the expected value, and the actual next reading will have its own noise on top.
Questions people actually ask
What is regression to the mean in simple terms?
An unusually high or low measurement tends to be followed by one closer to average, because part of what made it unusual was chance and chance does not repeat. Nothing causes the move back; it is what the arithmetic does on its own.
How do you calculate regression to the mean?
Predicted next value = mean + r x (observed - mean), where r is the lag-one correlation of the series: how much one period tells you about the next. At r = 0 the whole distance to the average is expected to close; at r = 1 none of it is.
Is regression to the mean the same as linear regression?
No, and searching for one returns the other. Linear regression fits a line through points to describe a relationship. Regression to the mean is the tendency of extreme measurements to be followed by less extreme ones.
Does this prove my training or intervention did not work?
No. It shows how much improvement was expected without any intervention at all, which is the figure any claim has to beat. Proving something worked needs a comparison group, and no calculator can supply that.
Cite this tool
Free to use and free to quote. If a figure here is useful in something you are writing, a link back is all that is asked.
Muhammad Shahzeb Anwar, MSc Industrial and Organizational Psychology. Regression to the Mean Calculator. Shahzeb Speaks. https://www.shahzebspeaks.com/regression-to-the-mean-calculator/
The other free tools
Each one answers a single question and refuses the ones it cannot. All free, none asks for an email address.
Want the control limits as well?
The paid Regression to the Mean Calculator is part of a set that also asks whether a reading is a signal at all, using individuals control limits at two and three sigma - so you can tell an extreme reading from a changed process before deciding anything about it.
See the full tool