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How I Introduce Statistics to Graduate Students
The math I use to understand what is observed in studies and experiments, and how those results differ from what is expected.
The classic example I use with biostatistics (statistics of biological data) students to kick off the course is this: A school of 1,000 students has a measles outbreak. 300 students come down with measles before we can order everyone to stay home for a few weeks. Of those 300, 150 are vaccinated and 150 are not. Is this proof that the vaccine doesn’t work, and that students have a 50/50 shot at getting measles if they are vaccinated?
From there, we start with the basics of statistics. While these are graduate students who I expect to have some background courses in statistics, I also understand that statistics are not intuitive for everyone. They weren’t for me. My statistics courses in college were a fever dream, with a professor who opted to tell us more about his exploits at the casino than the fundamentals behind how the numbers work. So, I took it easy for the first few lectures I gave.
“Suppose you flip a coin,” I tell my students. “You would be right to assume you’ll get a heads or tails. Those are the two most likely outcomes. They’re a binary outcome if the coin is fair and truly has a 50/50 chance of either side falling facing upward. So you see heads and are not impressed. Or you see tails, and you’re not impressed. But what about two heads in a row? Three heads? How many heads in a row do you see before you start thinking the coin is rigged?”
A Very Old Math
A long time ago, mathematicians calculated the probabilities of (or expected) outcomes of trials where there are two possible outcomes (heads or tails, sick or healthy, etc.). In our example, each trial is flipping a coin. For one trial, the probability of heads is 50%. For two? It’s 25%. Three in a row? 12.5%
If you’re noticing a pattern, that’s good. The probability of heads trial after trial goes down by half. What’s best is that it can be predicted with the binomial distribution probability function:
Here, P(X=k) represents the probability of getting exactly k successes in n independent trials. Since we…
