# Families and percentiles

As I have written here before, looking at slices of the population over time is a very misleading indicator of what happens to particular families over time, particularly when family composition is changing. Arnold Kling makes the same point and does it superbly:

In his new book Unequal Democracy, Larry Bartels writes (p.7),

families at the 20th percentile experienced declining real incomes in 20 of the 58 years…by comparison, families at the 95th percentile have experienced only one decline of 3% or more in their real incomes since 1951.

I have a nit to pick, which is that Census department percentiles are not families.

Suppose that we start out with 20 families, and the 4th-lowest family (the 20th percentile) has an income of \$10,000, while the 3rd family has an income of \$9500. Next year, suppose that everyone’s family income rises by 2 percent, but we add a new family at the bottom of the income distribution, with an income of \$6000. As a result, the new 20th percentile is now somewhere between the income of the original 3rd family (now the 4th family out of 21) and the original 4th family (now the 5th family). The income of the 20th percentile goes down, even though the income of every family has gone up.

Next, consider what happens when you have millions of families, and you add lots of new families each year. Because new families (immigrants and young families) tend to join the income escalator at the bottom, it should be no surprise that the bottom percentile shows declines more frequently than the top percentile.

I do not want to succumb to disconfirmation bias, which is the tendency to find one thing wrong with something you disagree with and then dismiss the whole idea. But I have a hard time buying into stories about income inequality that look at the behavior of census percentiles over time. At the very least, the author ought to be clear that movements in census percentiles are not the same as movements in families. Bartels is the opposite of clear on that point.

Another issue that people raise with Census data is that the basic unit is the household. If a household breaks into two households, due to divorce, average household income plunges by 50 percent, even though nobody’s income has changed. Trends in household income tend to look worse than trends in income per person.

Arnold has it exactly right. To get an idea of the magnitudes, here are some numbers:

Here’s what has happened to the number of households in the US:

2000   105 million
1990   93 million
1980   81 million
1970   63 million
1960   53 million

So between 1960 and 2000, the number of households has doubled. What happened to population over that same period?  Again from the Census:

2000   282 million
1990   250 million
1980   228 million
1970   205 million
1960   181 million

The average American household has gotten a lot smaller:

2000   2.7
1990   2.7
1980   2.8
1970   3.2
1960   3.4

Why did this happen?  The obvious answer is that people are having fewer children.  That would lower average household size.  But that is not much of the story.  The real story is a change in the composition of families due to an explosion in divorce in the 60s and 70s.  Here’s a breakdown of the proportion of total families headed by women:

Year     Single Mothers   Single Woman w/o kids      Total

2000      12.1%                     17.2%                            29.3%
1990      11.7%                     16.8%                            28.5%
1980      10.8%                     15.4%                            26.2%
1970        8.7%                     12.4%                            21.1%
1960        8.4%                       9.8%                            18.2%

So over the last half-century, the number of households has increased at a much faster rate than the number of people, mainly because of divorce. That totally contaminates the comparison of percentiles over time and makes it appear that people are falling behind or standing still when in, fact, particular families are seeing their standard of living rise. Arnold calls a nitpick. I call it a massive structural flaw.