Note: there is something special about these datasets: x and y measure the same thing (number of strokes needed for a round). In such a case it is often a good idea to draw the scatterplot with the same x and y scale.
Consider the dataset golf scores. In the next graph we have the four fitted line plots for the first and second round of the 2006 AT&T Peeble Beach National Pro-AM, 2006 Honda Classic, 2006 Byron Nelson Classic and the 2007 Sony Open:
Note: there is something special about these datasets: x and y measure the same thing (number of strokes needed for a round). In such a case it is often a good idea to draw the scatterplot with the same x and y scale.
In all four cases (and these were randomly chosen from all the tournaments) the slope of the regression line is positive but less than 1. Is this surprising?
• That the slope is positive is explained by the fact that golfers (even pros) have long term ups and downs: for weeks or months they play well, and then they play not so well. So a golfer who plays bad in the first round also likely plays bad in the second, and vice versa.
• That the slope is less than one is a general phenomenan often observed in real life called regression to the mean. Here it is this: a golfer at any one moment in time has a given scoring ability. It will change over time but not from one day to the next. So if somebody shot a high score one day they are likely not playing well overall but they also likely had an exceptionally bad day as well. The next day their overall ability is still the same but they likely won't have a bad day on top of that, and their score should go down. If somebody shot a very low score one day they are not likely to be able to do it again the next day.
This is one of the most missunderstood principles of statistics. Say a player has done very well in the first round of a major tournament, but then he plays not so well in the second. Almost always commentators will say the he "felt the pressure" and so played worse. In reality it is likely just regression to the mean: the first day he played well above his natural ability, and the second day he came back to it.
Example let's illustrate this effect with a simulation. Say a group of 1000 people takes a true-false test with 25 questions, and assume that none of them has any idea what the correct answers are, so they just guess. We can simulate this as follows:
Calc > Random Data > Binomial, generate 1000 rows of data, store in c1, Number of trials: 25, probability of success=0.5
How did they do?
Stat > Basic Statistics > Display Descriptive Statistics, Variable= c1 Shows they got about 12 or 13 questions correct on average, just what one would expect (μ=np=25·0.5=12.5!)
Now they all take a second similar but different test:
Calc > Random Data > Binomial, generate 1000 rows of data, store in c2, Number of trials: 25, probability of success=0.5
Here is a question: those who did well on the first test, how did they do on the second?
Data > Subset Worksheet > rows that match c1>=17
Stat > Basic Statistics > Display Descriptive Statistics, Variable= c2 shows they got about 12 or 13 questions correct on average, same as the overall average
Although they did well on the first test, they have now regressed to the mean.
Here is a general description of when and how regression to the mean happens:
a) there is an "experiment" that results in a "score"
b) the scores are due in part to "skill" and in part to luck
c) the "high scorers" are "selected" and repeat the experiment
d) they still have the "skill" but will they again have the luck? certainly not all of them
Example Golf
a) "experiment"=play a round of golf
b) "skill"=being a good player and in form, luck=get some lucky bounces
c) let's concentrate on those who did well in the first round, repeat the experiment = play another round
d) they are still playing good but some will not be as lucky as in the first round
Example Doctors visit
a) "experiment"=a person is ill, score is how long it takes to get healthy
b) "skill"=if person goes to a doctor hopefully he knows what he is doing, luck=lot's of other things effect how long an illness takes, eventually everybody is gets better
c) some people go to a doctor, repeat the experiment = are they still sick after some time, say a week?
d) everybody is eventually cured, but those who went to the doctor will think he did it!
This by the way is also one of the explanations for the famous placebo effect.
Example Students take tests
a) "experiment"=take a test
b) "skill"=student know material (or not)
c) repeat the experiment = take another test
d) the good students will do good again, but some will not be lucky again
Example Traffic lights
a) "experiment"=how many accidents are there in some intersections?
b) "skill"=some intersections are more dangerous than others, luck=the number of accidents on any one intersection will fluctuate.
c) repeat the experiment = see how many accidents happen now
d) the lights will help prevent accidents, but likely there would not have been as many anyway.
Note that this does not say that there is no effect here, just that it may not be as great as it appears
Example Golf: there really are player that are better than others (Tiger Woods!)
Example Doctors: yes, they really can help
Example Students: there are students who really know the material
Example traffic lights: they do prevent accidents (mostly)
One feature of regression to the mean is that it works both ways: change "high" for "low" above and everything is still true.