Half of Marriages, and Other Arithmetic

Half of marriages end in divorce. Everybody knows it. It is in newspaper articles, wedding speeches and sermons, and it is produced by a mistake you can see in one line of arithmetic.

Here is the mistake. Take the number of divorces granted in a year. Take the number of marriages registered in the same year. Divide the first by the second. In some years in the United States that ratio has come out near a half.

But the people divorcing in 2024 are not the people who married in 2024. They married in 1998, 2007, 1984. The two numbers describe different populations who happen to have appeared in the same annual statistics, and dividing one by the other does not produce a probability of anything.

It gets worse, in an instructive way. The ratio moves when the denominator moves. If fewer people get married, the ratio rises even if existing marriages are becoming more durable. That is roughly what has happened in several rich countries: marriage rates fell, the crude ratio looked alarming, and marriages were in fact lasting longer than they had a generation earlier.

The measure demographers use is a cohort measure. Follow people who married and ask how many of them later divorced. On that basis, lifetime divorce risk for first marriages rose through the 1960s and 1970s, peaked somewhere near forty per cent, levelled off in the 1980s and has been drifting down since. Projections of current patterns put it below the famous half.

Two further details stop this becoming a different kind of over-claim.

That estimate is for first marriages. Second and third marriages dissolve more often, so a figure for first marriages quoted as a figure for all marriages understates the overall rate.

And a projection assumes current conditions hold. It is a conditional statement about people who are not finished being married yet, not a measurement of a completed fact.

Now the other famous number in this area, which has the same shape of problem.

A well-known research programme reported identifying couples who would later divorce with accuracy in the eighties and nineties of per cent. The figure travelled widely. The difficulty is that in the headline studies the model was built on the same couples it was then scored against, which makes the figure a measure of fit rather than of prediction. The research group's own later paper described one such analysis as admittedly speculative and post hoc, and noted that they had not known in advance who would divorce.

When specific mechanisms from that work were tested on a separate sample of couples, several of them did not predict relationship outcomes.

And then there is the base rate, which is the mathematics lesson from another book in this app arriving here. Suppose a test really is eighty per cent accurate and sixteen per cent of couples in the population divorce. Work through the arithmetic and a prediction of divorce turns out to be wrong more often than it is right, because there are so many more couples who will not divorce for the test to be wrong about.

So the pattern repeats. A statistic with an arithmetic error, a prediction that was a fit, and in both cases a number that sounded authoritative enough that nobody checked the construction.