# Statistical significance of birth month of professional boxers

I looked at the birth dates of the top 100 ranked professional boxers of all time (67 of them to be exact). 40% of them were born during certain 3 month-long time periods. If birth date of boxers were totally random (as we'd expect) than in any 3 month period, about 25% of boxers should be born during any 3 month period, but I found a period where it was 40%. I found another 3 month period where only 12% of boxers were born. Question is--is that 40% statistically significant compared to the expected 25%? Same question for the 12% (is that statistically significant different from 25%)

I can't seem to figure out how to run the numbers. As mentioned n = 67.

(Actually, birth month is not perfectly distributed--somewhat more people are born in summer months than winter but I think this should not affect the total 'story' here). I very much appreciate any help.

• Please add the [self-study] tag & read its wiki. Dec 19, 2014 at 2:03
• Beware of data-dredging: finding such a large difference when you set out to compare Jan-Mar to Apr-Jun is more surprising than finding it when you set out to compare the quarters with highest & lowest birth rates across all three ways of dividing the year into quarters. Dec 19, 2014 at 11:11
• Dec 19, 2014 at 21:51

The results, as reported, are not statistically significant.

We can arrive at this conclusion (and better understand how it is meant to be interpreted) in steps. The first step is to take to heart Scortchi's comment,

Beware of data dredging.

This is the process of looking for "patterns" in data, finding one, and then applying a formal statistical test to determine its "significance." This would be an abuse of statistical testing, as has been amply explained and demonstrated in many places.

The second step is to ask whether the pattern found in these data is nevertheless so striking that it would be reasonable to take it as evidence of a meaningful variation in birth month. Some patterns are perfectly obvious, no matter what! Let's screen the results, using crude approximations and statistical models, to see how strong the results might be. Suppose that

1. The data could be conceived of as a random, representative sample of a well-defined population, such as "all champion professional boxers." Although this is obviously not a random sample, it is plausible to treat it as if it were, at least for these screening purposes.

2. Birth months are divided into four contiguous non-overlapping seasons (without reference to the data values).

3. As a null hypothesis (tentatively held, to be evaluated in light of the data), all variation observed in these seasonal totals is random.

With these assumptions, the count for any individual season has a Binomial$(67, 1/4)$ distribution. A Normal approximation to this distribution, which has a mean of $67/4\approx 17$ and standard deviation of $\sqrt{67(1/4)(1-1/4)}\approx 3.5$, suggests that values within a couple SDs of the mean should be expected as a result of sampling variation. This is the interval $[10, 24]$, having a width of $14$ (equal to $21\%$ of the total).

Although the quoted statistic of $40\% - 12\%$ = $28\%$ for the range, equal to $19$, is larger than $14$ (and therefore on the high side), it isn't that high. Variations in natural birth rates as well, variations in the lengths of quarters (which range from $90$ to $92$ days), and the fact there are $12$ (not just $4$) possible three-month sequences to look at, all suggest that $19$ might be on the margin of being statistically significant.

This takes us to the third step: let's try to reproduce the data evaluation that actually occurred. If one were exploring birth date data to look for patterns, the most powerful methods would look at individual dates. I will suppose, though, that this was not performed and that initially dates were summarized by month. One might then plot frequencies by month and look for patterns of highs and lows, much as described in the question. Plausibly, such a "pattern" would consist of some contiguous series of months with high average counts and some other contiguous series of months with low average counts.

We could generously characterize this search for patterns as a systematic statistical procedure. One way would be to look for statistically significant differences (at some desired level $\alpha$, such as $\alpha = 0.05 = 5\%$) among the individual months. If such differences did not appear, one would look for significant differences among windowed monthly sums using a two-month window, then a three-month window, and so on. (It is intuitively obvious that no more information is gained beyond a six-month window.)

The statistic for this procedure will be a vector $\mathbf t = (t_1, t_2, \ldots, t_6)$ giving the observed ranges of windowed means for windows of lengths $1, 2, \ldots, 6$ months. For, example, consider these simulated monthly counts (which occurred in the second of $1,000,000$ iterations of this experiment):

Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb
0   8   8  10   9   5   3   6   5   7   3   3


Their range is $t_1 = 10 - 0=10$. Their two-month sums (given by Mar+Apr, Apr+May, ..., Jan+Feb, Feb+Mar) are

  8   16   18   19   14    8    9   11   12    10     6     3


The range of those is $t_2 = 19-3=16$. Continuing like this through the six-month sums gives the vector of ranges

$$\mathbf t = (10,16,21,22,22,19).$$

Such a statistic will be considered "significant" if, as one scans through it, any of its components $t_k$ is in the critical region for a size-$\alpha$ test for windowed sums of width $k$. Because we are looking at ranges, the critical region of unusually high ranges for each $k$ can be described by a single number $c_k$. If any of the $t_k$ exceed $c_k$, one would have noticed a "pattern." 