Yesterday’s post was entitled, “You cannot measure a mean”, which is both true and false depending—thanks to Bill Clinton for the never-ending stream of satire—on what the meaning of mean means.
The plot I used was a numerical average at each point. This implies that at each year there were several direct measures that were averaged together and then plotted. This numerical average is called, among other things, a mean.
In this sense of the word, a mean is obviously observable, and so yesterday’s title was false. You can see a mean, they do exist in the world, they are just (possibly weighted) functions of other observable data. We can obviously make predictions of average values, too.
However, there is another sense of the word mean that is used as a technical concept in statistics, and an unfortunate sense, one that leads to confusion. I was hoping some people would call me on this, and some of you did, which makes me very proud.
The technical sense of mean is as an expected value, which is a probabilistic concept, and is itself another poorly chosen term, for you often never expect, and cannot even see, an expected value. A stock example is a throw of a die, which has an expected value of 3.5.
Yesterday’s model B was this
y = a + b*t + OS
I now have to explain what I passed over yesterday, the
OS. Recall that
OS stood for “Other Stuff”; it consisted of mystery numbers we had to add to the straight line so that model B reproduced the observed data. We never know what
OS is in advance, so we call it random. Since we quantify our uncertainty in the unknown using probability, we assign a probability distribution to
For lots of reasons (not all of them creditable), the distribution is nearly always a normal (the bell-shaped curve), which itself has two unobservable parameters, typically labeled μ and σ^2. We set μ=0 and guess σ^2. Doing this implies—via some simple math which I’ll skip—that the unknown observed data is itself described by a normal distribution, with two parameters
μ = a + b*t and the same σ^2 that
Unfortunately, that μ parameter is often called “the mean“. It is, however, just a parameter, an unobservable index used for the normal distribution. As I stressed yesterday (as I always stress), this “mean” cannot be seen or measured or experienced. It is a mathematical crutch used to help in the real work of explaining what we really want to know: how to quantify our uncertainty in the observables.
You cannot forecast this “mean” either, and you don’t need any math to prove this. The parameter μ is just some fixed number, after all, so any “forecast” for it would just say what that value is. Like I said yesterday, even if you knew the exact value of μ you still do not know the value of future observables, because
OS is always unknown (or random).
We usually do not know the value of μ exactly. It is unknown—and here we depart the world of classical statistics where statements like I am about to make are taboo—or “random”, so we have to quantify our uncertainty in its value, which we do using a probability distribution. We take some data and modify this probability distribution to sharpen our knowledge of μ. We then present this sharpened information and consider ourselves done (these were the blue dashed lines on the plot yesterday).
The unfortunate thing is that the bulk of statistics was developed to make more and more precise statements about μ : how to avoid bias in its measurement, what happens (actually, what never can happen) when we take an infinite amount of data, how estimates of it are ruled by the central limit theorem, and on and on. All good, quality mathematics, but mostly besides the point. Why? Again, because even if we knew the value of μ we still do not know the value of future observables. And because people tend to confuse their certainty in μ with their certainty in the observables, which as we saw yesterday, usually leads to vast overconfidence.
From now on, I will not make the mistake of calling a parameter a “mean”, and you won’t either.