I am a guest here, having responded to a general invitation extended to the Cross Validated community, to possibly contribute answers whenever some question related to Statistics comes up in this site. I do not teach Mathematics, but I do occasionally teach Statistics. And one of the less obvious and most difficult aspects to teach, is the difference between Descriptive and Inferential Statistics.
To me of course, it seems pretty clear: Descriptive Statistics just summarize some characteristics of a specific data set. Inferential Statistics is our attempt to draw inferences about something "larger" than the data set available. How do we manage that? By making a whole new set of assumptions. And what do we do after? We use the exact same results we derived from Descriptive Statistics -but which now lead us to totally different conclusions in nature and in scope.
And here lies the problem: these additional assumptions are simply sewn alongside the tools of Descriptive Statistics. And students get uneasy: in the previous (Descriptive Statistics) class, this was just the "average of the data set", "a centrality measure". How on earth the exact same number, calculated in the exact same way, has now become "an estimate of the population mean" that moreover has been derived through the interaction of the data set with a function of random variables,(the estimator), a function that is (say), "unbiased and asymptotically consistent"?
The problem is not whether the concepts themselves need work and mental effort to understand. The problem is that this "switch of vision" of the same thing (the data set), from a "vector of numbers" to a "set of realizations of distinct random variables that belong to the same (statistical) population, forming a sample of this population" is so big, that, the consequent use of the exact same tools and results seems in total disharmony: surely, such a big step in the set up should lead to some brand new tools also... and it does, but mostly "later on", while the tools of Descriptive Statistics remain prominent, with maybe minor modifications (like bias-correction).
The bigger problem, is that the true problem takes time to show: students may play along, some may even like this new stochastic and probabilistic world, -but I keep getting the feeling that deep down, they feel that all the theoretical apparatus of Inferential Statistics is just an ingenious way to make something out of nothing (or too much out of too little), since after all, we keep adding the values in the data set and we divide by their number...
Since "too much out of too little" is demonstrably not the case (if the general public knew how many procedures they consider as deterministically controlled, are in reality driven by statistical algorithms, I suspect they would had a serious panic attack), I believe it is important to find ways to deal with this.
One way would be to start with Inferential Statistics, and get rid of the notion that "Descriptive Statistics are a good introduction and familiarization step" (I just argued that they are not).
My question(s)? 1) To those that teach Statistics, what are your experiences and how do you deal with the passage from Descriptive to Inferential Statistics?
And to everybody,
2) what are some other fields in Mathematics where such "structural breaks" happen, i.e. where objects and concepts already taught, acquire a totally different meaning by activating a new set of assumptions? And how do you teach that?