I realised today I’ve been advising my students… not wrong, exactly, but imprecisely. Capriciously. Unmathematically. Even through it was in statistics, where such things are usually tolerated, I felt it was worth putting it right.

It was in a scenario such as this:

The times an athlete takes to run 200m are assumed to follow a normal distribution with a mean of 22s and a standard deviation of 0.4s. What is the probability of the athlete running 200m in less than 21s?

The traditional way

In the olden days, before the Classwiz was pretty-much-mandatory, the method was always:

  • Find the z-score: here, 21s is 2.5 standard deviations below the mean, so $z = -2.5$
  • Look this up in the table: or rather, look up $z=2.5$ to find that $P(Z < 2.5) = 0.9938$
  • Think about what’s going on: the answer we want is definitely less than a half, so we want $P(Z < -2.5) = 0.0062$, and that’s the same as $P(X < 21)$.

That’s not too bad, but the calculator makes it better.

The Classwiz! (My original way)

My up-to-today approach on the Classwiz would have been:

  • Put it in Normal CD mode: (for me, that’s menu 7, option 2)
  • Fill out the statistics: Three of these are straightforward: Upper is 21, $\sigma$ is 0.4 and $\mu$ is 22. ((Aside: why have they put sigma before mu? That makes no sense at all.)) I usually say “put Lower as a big negative number, minus a billion or something.”
  • Press equals: We get 0.062 directly (perhaps in standard form.)

But that ‘a big negative number’ bugs me. What if it’s the wrong big number? Minus a billion will usually work, but what if mu is big and negative, or is sigma is large?

A slightly more reliable way

  • Fill out the statistics differently: Make Upper 22 - being the mean - and lower 21 (the observation).
  • Press equals: this give 0.4938.
  • Think about what’s going on: The remaining part of the left tail must be 0.0062 to make a total of 0.5

In practical terms, doing it ‘right’ will make no difference at all - the imprecisions in the model will usually dwarf the tiny difference it makes.

But, it removes a bit of arbitrariness that was bugging me. And while I’m hiding the fact that I’m using a calculator from the Mathematical Ninja, I’m sure they’d approve.