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Month: April 2021

Simpson’s paradox and real wages

Simpson’s paradox and real wages

I have good news and bad news. First the good news: after a decade of stagnation, real wages have been rising since 2010. The following figure shows weekly wages for full-time employees (source), which I adjusted for inflation and indexed so the series starts at 100.

Real wages in 2019 Q3 were about 5% higher than in 2010.

Now here’s the bad news: at every level of education, real wages are lower now than in 2000, or practically the same. The following figure shows real weekly wages grouped by educational attainment:

For people with some college or an associate degree, real wages have fallen by about 5% since 2000 Q1. People with a high school diploma or a bachelor’s degree are making less money, too. People with advanced degrees are making about the same, and high school dropouts are doing slightly better.

But the net change for every group is substantially less than the 5% increase we see if we put the groups together. How is that even possible?

The answer is Simpson’s paradox, which is when a trend appears in every subgroup, but “disappears or reverses when these groups are combined”. In this case, real wages are declining or stagnant in every subgroup, but when we put the groups together, wages are increasing.

In general, Simpson’s paradox can happen when there is a confounding variable that interacts with the variables you are looking at. In this example, the variables we’re looking at are real wages, education level, and time. So here’s my question: what is the confounding variable that explains these seemingly impossible results?

Before you read the next section, give yourself time to think about it.

Credit: I got this example from a 2013 article by Floyd Norris, who was the chief financial correspondent of The New York Times at the time. He responded very helpfully to my request for help replicating his analysis.

The answer

The key (as Norris explained) is that the fraction of people in each educational level has changed. I don’t have the number from the BLS, but we can approximate them with data from the General Social Survey (GSS). It’s not exactly the same because:

  1. The GSS represents the adult residents of the U.S.; the BLS sample includes only people employed full time.
  2. The GSS data includes number of years of school, so I used that to approximate the educational levels in the BLS dataset. For example, I assume that someone with 12 years of school has a high school diploma, someone with 16 years of school has a bachelor’s degree, etc.

With those caveats, the following figure shows the fraction of GSS respondents in each educational level, from 1973 to 2018:

During the relevant period (from 2000 to 2018), the fraction of people with bachelor’s and advanced degrees increased substantially, and the fraction of high school dropouts declined.

These changes are the primary reason for the increase in median real wages when we put all educational levels together. Here’s one way to think about it:

  1. If you compare two people with the same educational level, one in 2000 and one in 2018, the one in 2018 is probably making less money, in real terms.
  2. But if you compare two people, chosen at random, one in 2000 and one in 2018, the one in 2018 is probably making more money, because the one in 2018 probably has more education.

These changes in educational attainment might explain the paradox, but the explanation raises another question: The same changes were happening between 2000 and 2010, so why were real wages flat during that interval?

I’m not sure I know the answer, but it looks like wages at each level were falling more steeply between 2000 and 2010; after that, some of them started to recover. So maybe the decreases within educational levels were canceled out by the shifts between levels, with a net change close to zero.

And there’s one more question that nags me: Why are real wages increasing for people with less than a high school diploma? With all the news stories about automation and the gig economy, I expected people in this group to see decreasing wages.

The resolution of this puzzle might be yet another statistical pitfall: survivorship bias. The BLS dataset reports median wages for people who are employed full-time. So if people in the bottom half of the wage distribution lose their jobs, or shift to part-time work, the median of the survivors goes up.

And that raises one final question: Are real wages going up or not?

Berkson Goes to College

Berkson Goes to College

Suppose one day you visit Representative College, where the student body is a representative sample of the college population. You meet a randomly chosen student and you learn (because it comes up in conversation) that they got a 600 on the SAT Verbal test, which is about one standard deviation above the mean. What do you think they got on the SAT Math test?

  • A: 600 or more
  • B: Between 500 and 600 (above the mean)
  • C: Between 400 and 500 (below the mean)
  • D: 400 or less

If you chose B, you are right! Scores on the SAT Math and Verbal tests are correlated, so if someone is above average on one, they are probably above average on the other. The correlation coefficient is about 0.7, so people who get 600 on the verbal test get about 570 on the math test, on average.

Now suppose you visit Elite University, where the average score on both tests is 700. You meet a randomly chosen student and you learn (because they bring it up) that they got a 750 on the verbal test, which is about one standard deviation above the mean at E.U. What do you think they got on the math test?

  • A: 750 or more
  • B: Between 700 and 750 (above the mean)
  • C: Between 650 and 700 (below the mean)
  • D: 650 or less

If you chose B again, you are wrong! Among students at E.U., the correlation between test scores is negative. If someone is above average on one, they are probably below average on the other.

This is an example of Berkson’s paradox, which is a form of selection bias. In this case, the selection is the college admission process, which is partly based on exam scores. And the effect, at elite colleges and universities, is a negative correlation between test scores, even though the correlation in the general population is positive.


To see how it works in this example, let’s look at some numbers. I got data from the National Longitudinal Survey of Youth 1997 (NLSY97), which “follows the lives of a sample of [8,984] American youth born between 1980-84”. The public data set includes the participants’ scores on several standardized tests, including the SAT and ACT.

About 1400 respondents took the SAT. Their average and standard deviation are close to the national average (500) and standard deviation (100). And the correlation is about 0.73. To get a sense of how strong that is, here’s what the scatter plot looks like.

Since the correlation is about 0.7, someone who is one standard deviation above the mean on the verbal test is about 0.7 standard deviations above the mean on the math test, on average. So at Representative College, if we select people with verbal scores near 600, their average math score is about 570.

Elite University

Now let’s see what happens when we select students for Elite University. Suppose that in order to get into E.U., your total SAT score has to be 1320 or higher. If we select students who meet or exceed that threshold, their average on both tests is about 700, and the standard deviation is about 50.

Among these students, the correlation between test scores is about -0.33, which means that if you are one standard deviation above the E.U. mean on one test, you are about 0.33 standard deviations below the E.U. mean on the other, on average.

The following figure shows why this happens:

The students who meet the admission requirements at Elite University form a triangle in the upper right, with a moderate negative correlation between test scores.

Specialized University

Of course, most admissions decisions are based on more than the sum of two SAT scores. But we get the same effect even if the details of the admission criteria are different. For example, suppose another school, Specialized University, admits students if either test score is 720 or better, regardless of the other score.

With this threshold, the mean for both tests is close to 700, the same as Elite University, and the standard deviations are a little higher. But again, the correlation is negative, and a little stronger than at E.U., about -0.38, compared to -0.33.

The following figure shows the distribution of scores for admitted students.

There are three kinds of students Specialized University: good at math, good at language, and good at both. But the first two groups are bigger than the third, so the overall correlation is negative.

Sweep the Threshold

Now let’s see what happens as we vary the admissions requirements. I’ll go back to the previous version, where admission depends on the total of the two tests, and vary the threshold.

As we increase the threshold, the average total score increases and the correlation decreases. The following figure shows the results.

At Representative College, where the average total SAT is near 1000, test scores are strongly correlated. At Elite University, where the average is over 1400, the correlation is moderately negative.

Secondtier College

But at a college that is selective but not elite, the effect might be even stronger than that. Suppose at Secondtier College (it’s pronounced “seh con’ tee ay'”), a student with a total score of 1220 or more is admitted, but a student with 1320 or more is likely to go somewhere else.

In that case, the average total score would be about 1260. So, based on the parameter sweep in the previous section, we would expect a weak positive correlation, but the correlation is actually strongly negative, about -0.8! The following picture shows why.

At Secondtier, if you meet a student who got a 690 on the math test, about one standard deviation above the mean, you should expect them to get a 580 on the verbal test, on average. That’s a remarkable effect.


Among the students at a given college or university, verbal skills and math skills might be strongly correlated, anti-correlated, or uncorrelated, depending on how the students are selected. This is an example of Berkson’s paradox.

If you enjoy this kind of veridical paradox, you might like my previous article “The Inspection Paradox Is Everywhere“. And if you like thinking about probability, you might like the second edition of Think Bayes (affiliate link), which will be published by O’Reilly Media later this month.

If you want to see the details of my analysis and run the code, click here to run the notebook on Colab.

Finally, if you have access to standardized test scores at a college or university, and you are willing to compute a few statistics, I would love to compare my results with some real-world data. For students who enrolled, I would need

  • Mean and standard deviation for each section of the SAT or ACT.
  • Correlations between the sections.

The results, if you share them, would appear as a dot on a graph, either labeled or unlabeled at your discretion.