# The Classwiz and the Normal Distribution

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.