Profs. Vicki McMillan and Rob Arnold posed the following question: When dragonflies mate, they fuse together into a unit and fly to a pond to deposit their eggs. The question is whether they prefer to settle close to other pairs as they lay their eggs or whether they don't care where other pairs are and land somewhere at random. To decide, McMillan and Arnold laid out plots one meter square along the edge of a pond and took periodic (hourly?) counts of how many mated pairs of dragonflies were in each of the plots. There were actually seven plots, but in four of them they never found any dragonfly pairs, so M&A concluded they weren't suitable egg-laying spots. That left three plots. Of the 17 counts in which there were only two pairs on the pond, 12 times they were in the same plot. That certainly suggests that dragonfly pairs prefer to be close to others while laying eggs, but is it significant? I advised M&A to apply a t-test, with the null hypothesis that a dragonfly pair, coming to the pond and seeing another pair already there, doesn't care which of the three plots it lands in, so it chooses the one where the other pair is with probability 1/3. Not too surprisingly, the 12 out of 17 is significant evidence against that hypothesis. When there were more than two dragonfly pairs on the pond, here are the numbers of dragonfly pairs were in each plot: Plot 3 pairs A 0 0 3 1 1 3 1 2 1 0 1 B 0 3 0 0 2 0 2 1 2 0 2 C 3 0 0 2 0 0 0 0 0 3 0 Plot 4 pairs A 4 0 3 4 0 0 B 0 0 1 0 4 2 C 0 4 0 0 0 2 Plot 5 pairs A 3 0 0 3 2 2 1 1 1 0 0 B 2 0 3 1 3 3 3 0 2 1 0 C 0 5 2 1 0 0 1 4 2 4 5 Plot 6 pairs A 5 1 0 B 0 2 4 C 1 3 2 Plot 7 pairs A 2 1 0 0 B 3 0 3 3 C 2 6 4 4 Plot 8 pairs A 5 B 2 C 1 Plot 9 pairs A 4 B 3 C 2 Plot 12 pairs A 5 B 5 C 2 M&A suggested doing some kind of chi-squared test with this data, but they didn't know what expected distribution to use. It seemed reasonable to me to make the same "We don't care which plot we land in" null hypothesis, but because there are more possibilities with more pairs, there are more possibilities for distribution of the pairs in the plots. Because M&A didn't care which plot the pairs were in, just whether they were sharing a plot or not, I suggested that they group the three-pair distributions according to the corresponding "partition" of the integer 3 into three nonnegative integers: 3,0,0 or 2,1,0 or 1,1,1. There were 6 counts of the first partition, 6 of the second and none of the third, for a total of 12. On the other hand, we can compute the probabilities of each partition on the basis of the null hypothesis: In order for the first partition to hold, the second dragonfly (respectively the third) dragonfly pair to arrive, having found one pair already there (respectively two pairs already in the same plot), choose the plot containing the earlier pair(s) with probability 1/3. So the probability of the partition 3,0,0 is 1/9. Similarly, the probability of the partition 1,1,1 is 4/9. There are two ways of getting to the partition 2,1,0, depending on whether the first two pairs were in the same plot or not, and we could compute its probability by adding the probabilities of each way; but it is simpler just to reason that the probability of this partition is 1 - (1/9) - (4/9) = 4/9. So M&A could do a chi-squared test comparing the observed distribution 6, 6, 0 with the expected distribution 12(1/9), 12(4/9), 12(4/9) . Again, not surprisingly the difference was significant, providing more evidence against the null hypothesis, i.e., that dragonfly prefer, for whatever reason, to be close to each other as they lay eggs. (That doesn't give evidence of why; there may have been predators in the other two plots when the researchers found the partition 3,0,0.) It might have been worthwhile to run the test again with 4 and 5 pairs, but after that it seems to me that there are too few in the sample to be worthwhile.