# Passage from Descriptive to Inferential Statistics - analogies with other Math-fields?

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?

• Welcome to MathEducators.SE! Aug 7, 2014 at 23:49
• @MarkFantini Thanks Mark. Aug 7, 2014 at 23:53

On the (false) distinction between descriptive and inferential statistics

In my view, it is rare that your only purpose is simply to describe the data you have. Even a simple graph is usually used to make a tentative hypothesis about the relationship between variables or the distribution of a single variable, and this is really the beginnings of inferential statistics. Also, presenting means in a research paper is usually to forward your argument about differences in populations. In that sense the mean is already a representation of the population and not technically only a representation of your sample.

On the other hand, if you really want to investigate the data you currently have, then summary stats are not usually fit for the purpose - you are often much more interested in identifying outliers and investigating them further. For example you might notice a negative skew in your class results, but then you would naturally be interested in those students in that left tail and how you might help them, so you end up looking at the individual data points. Even if you do calculate a summary, such as the pass rate, then your thoughts naturally turn to how you can make the pass rate higher in future students, and now you're thinking about a population beyond your current class.

Even the usual measures of central tendency are problematic. Many students I talk to express concern that they remove so much of the data they are trying to describe, and their natural instinct is instead to talk about clumps in the data or the percentage in a certain zone. It seems to me that the choice of the mean is actually guided by the estimation if a parameter, rather than actually being natural way for people to summarise data.

Finally, in practice, when a statistician does analyse data with the view to make conclusions about the population, they always draw graphs and calculate summaries. But we have a hard time convincing students that they should do this. I wonder if the dichotomy we set up between descriptive statistics and inferential statistics encourages students not to draw graphs. It seems they have learned that graphs only describe the data you have and they think they therefore cannot help in the process of making inferences.

Big ideas of variability, population and distribution

It distresses me that even though we know that stats is built on the big ideas of variability, population and distribution, we tend to forget about them when teaching certain parts of the course.

Even when teaching probability, we often focus on specific events, whereas in fact the only reason any one event has a probability is because it is in comparison to all the other events that could have been.

My experience teaching all stats as inference

To orient you, I am coordinator of a Maths Learning Centre at a university, so I talk to many students from many disciplines about their stats. I also do guest lectures on stats within courses (such as the med students during their research course).

Taking the previous discussion into account, I decided that when helping students understand statistics, I would try to remove the distinction between descriptive and inferential stats. I tell them that the main purpose of data collection and analysis is actually to make conclusions about populations, and that descriptive statistics are the essential first step.

I talk about distributions very early on, usually when they are studying descriptive stats. I introduce the big idea that any number you might measure or calculate has a distribution that describes all the possibilities and how likely they are. Your data is always a sample of the data you could have had, and the mean is just one mean that could have happened. Yet it has the tendency to be close to that central clump of the distribution, which is why it's pretty good as a measure of centre if your view is towards inference.

I find that recognising at the outset that almost all stats has a view to inference matches best with the reasons why they are forced to learn stats (generally it's a research methods course), and also helps to unify what they are learning in their stars course.

The concept of "all things have a distribution" helps to get them to understand why the calculations for test statistics need to be different, because we need to create something with a distribution we actually know. It also helps them to pull away from the data itself to imagine the bigger picture of the distribution/population it came from.

• (+1) and thanks for this rich answer. It is my first impression that you already do what I think should be done (the "start with Inferential Statistics" part at the end of my question, which essentially also dispenses with the dichotomy). Aug 8, 2014 at 0:39

I've encountered something similar in logic, where symbols can be interpreted either syntactically ($\vdash$) or semantically ($\models$). I've had the same feeling that logic students were only playing along about appreciating the difference between the two.

The statistical results connecting descriptive and inferential statistics look like:

The sample mean is an unbiased estimate of the population mean.

The sample variance times $N/(N-1)$ is an unbiased estimate of the population variance.

Fortunately, these results are easy to prove. The numerical factor helps show the difference clearly. There are lots of other nearby results to consider.

The logical results connecting syntax and semantics look like Godel's completeness theorem:

A set of sentences is consistent by these inference rules iff it has a model with these semantics.

These are harder to prove. I suspect that most logic teachers would have a hard time coming up with different hypotheses and nearby theorems. So it's harder to demonstrate understanding.

Perhaps this can reframe your pedagogical issues? $\vdash$ vs $\models$ might be even harder for students and teachers than $s^2$ vs $\sigma^2$ and $N$ vs $N-1$.

• Logic can be very hard, that's a fact. But to use your examples, my issue is more like: at one instance "$\bar X$ is a summary measure of the location tendency of the data set" and the next instance "$\bar X$ is an estimator of the population expected value". It is the exact same quantity/concept/tool that acquires a totally different meaning (while not stopping representing also its first meaning, whenever we "feel like it"). Is this analogous to what are you describing for the field of logic? Aug 8, 2014 at 1:36
• @AlecosPapadopoulos, I think it is analogous. We say $\phi \& \psi$, and interpret it either syntactically ($T \vdash \phi \& \psi \leftrightarrow T\vdash\phi\ \&\ T\vdash\psi$) or semantically ($M\models\phi \& \psi \leftrightarrow M\models\phi\ \&\ M\models\psi$). The exact same symbols acquire a totally different meaning.
– user173
Aug 8, 2014 at 1:44
• Yes, that's totally analogous. So how do you teach that? How the students react to the fact that it can be either way? Are there side rules that dictate the choice? Aug 8, 2014 at 1:49
• @AlecosPapadopoulos, I wish I knew how to teach it well, it would probably help me explain lots of other things. There are side rules of thumb: Disproofs are usually semantic, proofs often syntactic. Use of the word "true" suggests semantic reasoning, use of the phrase "first-order" suggests syntactic reasoning. But I'm not sure I would want to recommend those sociological observations to students.
– user173
Aug 8, 2014 at 2:06

You might find this article in the Journal of Mathematical Behavior of interest: Conceptual issues in understanding the inner logic of statistical inference: Insights from two teaching experiments, by Luis A. Saldanha, Patrick W. Thompson, pages 1-30, Volume 35, September, 2014.