You Say It’s Your Birthday

It’s my birthday, too.

Okay, not really, but it wouldn’t be surprising if it were. Today (September 17) is the fourth most common date on which to be born. Or at least it had been from 1973 to 1999 in the United States, according to an analysis by Amitabh Chandra, a professor of social policy at Harvard who compiled 28 years of birth date data.

Yesterday (September 16) was the overall most common birth date, and tomorrow (September 18) was the tenth most common.

In fact, all of the top ten most common birth dates occur in September, and even the lowest-ranking date of the month is still in the top third of the most common dates (September 5, #125). Apparently November and December are busy times for gettin’ busy.

It’s a busy birthday time elsewhere, too. Statistics New Zealand issued a warning about the excessive number of birthdays that will occur near the end of September.

How common is your birth date? Take a gander at the image below. Or, for an interactive experience, click around on Andy Kriebel’s version that uses Tableau.

This information (which is not new, btw — an initial analysis was completed by Chandra for a paper published in the Journal of Political Economy in 1999; a table with Chandra’s full data appeared in a column by David Leonhardt in 2006; and, various versions of the image above have been floating around the internet for several years) throws a wrench into the famous Birthday Problem, which asks, “At a party of n people, what is the probability that two of them share a birthday?”

The solution to the Birthday Problem assumes that birth dates are evenly distributed across the calendar. But clearly they’re not. How much does real data affect the solution, though? As it turns out, not much.

Using real birth date data from 1985-88 pulled from the CDC, Joe Rickert completed an analysis using Revelation R Enterprise 6. The results look like this:

On the other hand, the theoretical probability of two people sharing a birth date in a group of n people is given by the formula:

And if you try to graph that formula, you get something like the following:

Admittedly, I was lazy. The scatterplot above only includes n-values from 2‑10 and multiples of 5 from 15‑100. That’s because Excel can’t handle factorials larger than 170!, so I had to use a large number calculator online and enter the values into an Excel sheet by hand. Still, it gives a pretty good idea of the shape of the curve.

A typical benchmark for the Birthday Problem is n = 23, when the probability of a pair sharing the same birth date first exceeds 50%. Using real data, P(23) = 0.5087, and using the theoretical model, P(23) = 0.5073. Good enough for government work. By visual inspection, you can see that the other values match up rather well, too.

Finally, here are some math jokes about birthdays.

Statistics show that those who celebrate the most birthdays live longest.

I only drink twice a year: when it’s my birthday, and when it’s not.

Happy integer number of arbitrary units of time since the day of your birth!

You don’t need calculus to figure out your age.

They don’t make birthday cards for people who are 85 years old. So I almost bought you a card for an 80-year old and a 5-year old. But then I figured no one wants to do math on their birthday.