Age Patterns in Divorce
MARK K. ALTSHULER
Matrimonial lawyers observe many divorce cases in a career and get a feel for certain patterns in divorce, such as the sense that most divorces occur in middle age. Younger couples go through their honeymoon period and then raise children and older couples have a very low rate of divorce. The peak of divorce would then occur at mid life, after the children are raised and one or both of the parties may be going through mid life crisis. If the couple gets through middle age and does not divorce, it seems unlikely they will later on. It is hard to imagine 75 year olds getting divorced (although it does happen). This study quantifies those intuitions on a statistical basis, analyzing age at divorce among general population males, general population females, and police officers from 1983 to 2004.
Starting with a sample of 15,938 records, the applicable data was divided into the three categories. In the male sample there are 12,097 records with divorce ages ranging from 23 to 86. In the female sample there are 2,759 records with divorce ages ranging from 21 to 80. Finally, in the police sample there are 861 records with divorce ages ranging from 25 to 66. The data was analyzed with basic statistical methods, including graphing the raw data and smooth curve fitting. The analysis is on a basic level, along the lines of the percentile analysis of the SATs.
Since for every divorced man there is a divorced woman, the question may arise as to why the sample for divorced women is smaller. The reason is because this sample is based on people with pensions that are equitably distributed in divorce cases, and the number of women with pensions is smaller than the number of men.
BACKGROUND OF ANALYSIS
The first step was developing the frequency distribution for each class. This is the percent of divorces as a function of age, where the age is given as a whole number (integer), using rounding. For example, if someone is age 45.6 at time of divorce, the actuarial age is 46.
We took the total number of divorces at a given age and divided them by the number of total records in a given category, in order the get the percentage. For example, at age 45 the number of divorces for males is 661, and the total number of records is 12,097 in this class. The percentage is then 661/12,097 x100, or 5.46 percent.
PERCENTAGE AND PROBABILITY
The concept of probability is intuitively similar to a percentage. If the percent of males getting divorced at age 45 is 5.46 percent and the sample size is large enough (in this case 12,097), we can think of the percentage as a probability. For example, consider the opposite process.
We know that for a perfect coin, the probability of getting heads is 50 percent (.5) and the probability of getting tails is 50 percent. Therefore, if a perfect coin is flipped 1,000 times, the number of heads should be very close to 500 and the number of tails will be close to 500. Similarly, if we have a large enough sample, the percentage of divorces at a given age will be very close to the probability. Thus, the frequency distributions for each class are shown as a probabilities, rather than percentages. At age 45, the probability of divorce is 661/12,097, or .0546.
If the probabilities on each graph are summed, over all the ages, the total is one.
This is because the probabilities are actually percentages and percentages always total to 100 percent. For example, if there were 100 divorce cases and 4 ages, with 25 divorces at each age, there would be 25 percent chance of divorce at each age and the sum would be 100 percent.
Similarly, if we flip a coin, we know with 100 percent probability that the outcome will be either heads or tails.
The plot of the raw (empirical) data of probabilities as a function of age is called a histogram. This is a plot with the vertical columns at each age. We then created smooth curve fits that were the best fit to the empirical probability of our data. With sufficient data, the smooth curve becomes the theoretical probability, as opposed to the empirical (measured) probability that is based on the given data set.
In this study, the largest dataset is for males, and the empirical histogram for males is closest to the smooth curve fit. There is less data for women, and still less for police officers. In each case, as the size of the sample declines, the histogram is increasingly different than the smooth curve.
Again, consider the case of the' coin flip. Experimental results are shown in Exhibit 1. For just 2 flips, it is not surprising that the result is 2 heads. For ten flips, the result was 60 percent heads. For 100 flips, the result was 51 percent heads. As the number of flips increases, the result will get closer and closer to the theoretical result of 50:50. Now turning to the divorce exhibits, note that the theoretical smooth curve in the divorce study is similar to the classic bell shaped curve, except that there is a "tail" at the high end. The classic bell shape curve is symmetric.
An understanding of percentiles will help to clarify exactly what a tail in a probability distribution is. Going back to the SATs, results were given in percentiles.
Being in the 98th percentile means scoring in the top 2 percent, which would be the top 2 out of 100, for example. Similarly, being in the 2nd percentile means 98 percent has higher scores.
The graphs show the 98th and 2nd percentile, along with mean, median, and mode.
Mean, median and mode are all measures of central tendency. The median is merely the 50th percentile. Half of the divorces are above the median age and half below. Arithmetic mean, more commonly known as the average, is the sum of results from the set of data divided by the size of the set. For example, in adding up all the ages for divorced men, the total is 553,628, dividing this by the size of the set, 12,097, gives the rounded average age of 46 for divorced men. The mode is the result that appears most frequently, which in the case of divorced men is age 45.
In the classic symmetric bell shaped curve, the median and the mean (average) are the same. Most probability distributions in nature follow the bell curve. However, social and economic distributions often are skewed (also known as having tails). The classic example is the distribution of wealth. Suppose half the people in a sample have income less than $50,000 per year and half have more. The median income is then $50,000. However, if just a few of those above $50,000 have incomes over a million dollars annually, the average (mean) will be greater than $50,000 per year. Similarly, the divorce distributions are skewed.
Analysis of Data
The tail for divorced men can be quantified as follows. The 2nd percentile is age 31 and the median is age 45. Thus, the distance from the 2nd percentile to the median (50th percentile) is 14 years. If the curve were sym':' metric, the 98th percentile would be at age 59 (14 years from the median), but is actually at age 63. Thus, there is a "tail," to the curve, which is a range of ages well over 59 where men are still getting divorced. There are two reasons for this. First, there is the obvious honeymoon period early after marriage where people ignore faults in the spouse. This honeymoon period does not occur at the other end of marriage.
Second, going 25 years from the median on the high end is age 70. There are significant numbers of men living past age 70 and married, so that they can possibly get divorced. At the low end, 25 years from the median is age 20. Very few people get married before age 20, so few can get divorced. Thus, the data is inherently asymmetric, due to the human lifespan. Of the few who do get married, few are likely to get divorced so soon after marriage. These are the two reasons for the skew: the honeymoon period and the inherent asymmetry due to lifespan.
The mean (average) is higher than the median, due to the skewness of the distribution. In an extreme case such as one millionaire in a roomful of people, the average would be much higher than the median (50th percentile).
The distribution for the women is also skewed. If it were symmetric, the 98th percentile would be at age 58, not age 62, since the 2nd percentile is age 30, making both 14 years from the median at age 44. Since the 98th percentile is actually at age 62, the distribution has a tail and is skewed. However, the median and mean (average) are equal in this case, which implies symmetry, although the distribution is skewed. This is due to the fact that the empirical data is choppier, due to the smaller size of the data set. The spike at age 42 ensures that the median and average are the same for the class of divorced women. With more data in this set, it is likely the median and average would not be equal and the average (mean) would be higher, due to the skew. Note that the median and mode are lower for women than for men. This is probably due to women marrying men who are older.
The police distribution is also skewed. If symmetric, the 98th percentile would be at age 57, 14 years from the median, since the 2nd percentile is at age 29, 14 years from the median at age 43. However, the 98th percentile is age 60 and the distribution is skewed. Again, the median and average (mean) are the same due to the choppiness of the data. However, in the case of police, the smooth curve is close to symmetric. This suggests that with more samples, as the histrogram approaches the smooth curve, the data would not be skewed. This would imply that police are as likely to divorce when they are younger as when they are older, if it is true that there is no tail in the theoretical police distribution. If this is true, it could be due to police officers retiring at much younger age than the average population and starting over, which would tend to counteract the lifespan asymmetry. Typical retirement ages in police pension plans vary from age 45 to age 55.
Except for the police case, the analysis confirms what most matrimonial practitioners already know, that most divorce cases occur in the mid 40s, and that people over 60 are more likely to get divorced than people under 30, and that fewer women have pensions than due men. The smooth curve fit for the police data implies that the distribution of divorce for police may be symmetric, as opposed to the general public.
More data and formal "goodness of fit" statistical analysis will be required to confirm this hypothesis.
Mark K. Altschuler, an actuary, is President of Pension Analysis Consultants, Inc., of Elkins Park, PA. He has prepared more than 7,000 pension valuations and has spoken about this and other topics at state and local bar associations and CLE workshops. An affiliated member of the American Society of Pension Actuaries, Mr. Altschuler writes a nationally distributed newsletter on pension issues in divorce (DIVTIPS) and is a contributing author to the book, Valuing Specific Assets in Divorce, published by Aspen Publishers (New York 2001). Counsel may contact Mr. Altschuler at (800) 288-3675.
The author expresses his appreciation to Jason Miller for his research assistance in writing this article.