# Satterthwaite, variances, walruses and uteruses

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Statistics applies to everything. Today I was looking up examples of the Satterthwaite alternative to the pooled variance t-test.

In short, a t-test is used when one wants to answer the question, “Is the difference between these two groups greater than one would expect to find by chance?”

Any time you measure two groups, whether it is the number of walruses on two beaches or the number of live births in a sample of 400 women, you are not going to get the exact same number twice. Just by chance, one of the walruses could have swam out to sea, or one of the women could have given birth to triplets. Random events happen. A t-test compares the difference between two groups to the differences that could be attributable just random events.

The denominator used to calculate the t-test is based on the variance. If the population differs a lot, you wouldn’t be surprised to find a fair amount of difference between two groups. For example, let’s say you take two groups of 100 people each. You measure their average annual income and you find out one is \$300 more than the other. That wouldn’t seem too unlikely to you. On the other hand, if the average number of children in one group was 300 more than the other, that would be pretty amazing. Who has 300 children, anyway? (I’m not sure what the expected number of walruses would be for any given group of 100 people, but I am pretty certain it would be low.)

walrus

If the variance for two different groups is the same, then when you calculate the the t-test you use the variance for the two groups combined (also called the pooled variance). If the groups are different, say you are comparing number of walruses and you have a group comprised of statisticians (who tend, on the whole, to be rather short of walruses) and a second group comprised of zookeepers (who might be expected to have a walrus or two around), the variance in the two groups could be expected to be quite different. In this case, you could use Satterthwaite’s method which uses the individual group variances.

All of this brings me to the point that statistics applies to everything. Yesterday, I was preparing for a class and I wanted to give an example of a t-test using Satterthwaite’s method. I typed it into Google and the first two articles that came up were on

1. the effectiveness of assisted fertilization methods in women who had problems with their uterus and either did or did not have a particular diagnosis of a disease which I can neither pronounce nor spell, and
2. Accuracy of pinniped counts using individual or paired observers enumerating the Pacific walrus.

I thought this was a great example of what I always say about why statistics are wonderful. You can analyze the stars in the sky, atoms in a grain of sand, or as it turns out, the number of walruses on the beach or what is growing in your uterus.

And aren’t you glad I did not include a picture of a uterus?