Via the Economist journalist Daniel Knowles  comes a good example of why its important to look below the surface of statistics. American economist Steve Landsburg addresses a commonly heard refrain  – that the wage of the median worker has barely risen in the past thirty years – and shows that all is not as it seems.
Landsburg cites a book by economist Edward Conard  (first chapter, containing what we're talking about, here ), which itself cites the Census Bureau. I confess that, without a more specific citation, I can't find the exact data Conard uses, but have found similar enough data  (pdf, table A-5) to confirm the overall thrust of the argument.
Conard shows that from 1980 to 2005, median income in the US rose just 3 per cent once inflation is taken into account, from $25,000 to $25,700. 2005 is pre-crash, as well, so this isn't a tale of the recession.
But when you break the data down by race and gender, a very different story appears:
|1980 Median||2005 Median||Increase|
For every single demographic group, there was a much bigger increase in the median wage than we see when the groups are combined. The reason for this is obvious when it's pointed out: demographic change in the US means that there are far more (low-salaried) women and people of colour working now than there were in 2005, which pushes the overall average down.
Imagine a farmer with a few 100-pound goats and a bunch of 1000-pound cows. His median animal weighs 1000 pounds. A few years later, he’s acquired a whole lot more goats, all of which have grown to 200 pounds, while his cows have all grown to 2000. Now his median animal weighs 200 pounds.
A very silly person could point out to this farmer that his median animal seems to be a lot scrawnier these days. The farmer might well reply that both his goats and his cows seem to be doing just fine, at least relative to where they were.
This is almost an example of Simpson's Paradox, a well-known (to stats nerds) effect where the direction of a correlation disappears when that correlation is disaggregated. I was taught it with an example involving racial discrepancies in application of the death penalty:
Sixty per cent of white murderers are executed for their crimes, and fifty per cent of black murderers. Are black people discriminated against in the application of the death penalty?
Now suppose that we break down the murder victims by race as well. We find the common pattern that people tend to attack victims of their own race:
Number of murders where death penalty is applied
White Murderer Black Murderer White Victim 50/70=71% 25/30=83% Black Victim 10/30=33% 25/70=36%
What about now? Does it begin to look like black people are discriminated against? In this example, black people are more likely to be executed for the murder of black or white victims; but because the murder of black victims isn't taken as seriously by the courts, the fact that murderers predominantly attack people of their race makes it look like black people are less likely to be executed than white people.
The median income example isn't quite a case of Simpson's Paradox, because there is still a positive increase in wage whether or not the statistics are disaggregated. But it's still an example of a time when it is best to dig beneath the surface.
But there is more to be said on the story of wage stagnation. Because a second claim normally accompanies the belief that US wages have stagnated, and that is that there has been a "decoupling" of wages . Due to rising inequality, the median household wage hasn't risen as fast as the mean wage:
If we've seen that the median wage grows faster when disaggregated, then the solid red line is likely to take a steeper ascent. But what happens to the dashed red line when disaggregated?
Sadly, I cannot find the data required to answer the question. If anyone knows where to look, tell me, and maybe we can put the issue to rest.