# Teaching Statistics in High School

There is an interesting discussion about a topic where I know especially little: K-12 education. Within it, there is a narrow discussion about whether it makes sense to try to teach statistics to people who don’t know calculus. This is a clear question. However, it seems that the people who discuss it skip a much more basic question which is why mathematical statistics should be taught to high school students, and an even more basic question which is what do statisticians have to teach us.

Links and snippets.

Kareem Carr tweeted

Which causes me to ask how exactly calculus helps us develop valid statistical tools.

When mostly writing about other things, Matthew Yglesias wrote

“But one big problem with this idea, as Kareem Carr noted, is you can’t really teach statistics properly without calculus, which I think goes to underscore that this is not really a debate about the proper sequencing of math classes. It’s instead another manifestation of the dysfunctional tendency in some edu-left circles to stigmatize all efforts at measurement. If you sort kids into different math tracks based on their test scores, that might reveal that Black and Latino kids are doing worse than white and Asian ones. If you refuse to sort, you can pretend you’ve achieved equality. But will you provide useful education?”

Will you ? Can you teach statistics properly without calculus ? Is the effort to develop useful mathematical statistics at the point where one can “teach statistics properly” . I stress I think that the accomplishments of mathematical statisticians are immense and immensely important. However, I also think there is a risk of keeping things simple by making strong false assumptions and teaching students things which just aren’t true.

I agree in part (and disagree in part with Brad DeLong who wrote in response to Carr mostly

“Here is the fact: the rules of statistics

really arbitrary. The formulasarecome out of nowhere.doThe formulas come out of the brain of Carl Friedrich Gauss. The formulas for average (rather than median) and for least-squares linear fit make sense if the idea is to minimize the sum-of-squares of the retrodiction errors, which makes sense if the distribution of disturbances to some underlying linear true relationship is: [Gaussian]”

I suspect that high school (and college and graduate) students are taught formulas which are valid only under very strong assumptions or asymptotically (note the slogan of my defunct personal blog “asymptotically we will all be dead”).

But really the phrase “The formulas” refers to a whole lot of formulas only some of which follow from the assumption that some stochastic variable is normally distributed or the sample size is large enough (with no analysis of how large is large enough or how to tell if we have enough data).

High schools students certainly should not be taught that the right way to estimate the location of a population is to calculate the mean of a sample. As written without qualifications that is simply a false statement. They should also not be taught that the right way to estimate the mean mode and median of a normal distribution is to calculate a sample average — that is a true mathematical statement but too strongly suggests that this example is very often useful without giving any hints about how to find out when it is useful.

I am quite sure that one shouldn’t introduce statistics by teaching about what one would do if one were to know that stochastic variables are normally distributed. That is a worthwhile topic (or at least I hope so as I have spent a fair amount of time considering the distributions of estimators and test statistics under the assumption of normality). But it seems to me to be a very bad thing to present it as a useful first step.

Brad goes on

“We, today, rest [our confidence] on the Central Limit Theorem and the convergence-in-distribution of a sum of independent random variables to a Gaussian distribution if [one of many sufficient conditions is assumed].

In all this, knowing calculus—in the sense of having it in your intellectual panoply—is very useful for getting from [one of many sufficient assumptions] to the Central Limit Theorem. Knowing calculus is very useful in getting from the Gaussian distribution to the optimality of taking-averages and least-squares. [skip]

And here’s the crux: in the real world of using statistics, where your sample is finite, where one or a few of the disturbances are large relative to the total, where your sample is non-random, where your observations are not independent, the Central Limit and Gauss-Markov theorems are of little use. Yes, taking averages and least-squares fits are the first things you should do. But then you should not down tools, because of calculus! Then you should do other things and see if they agree with taking-averages and least-squares. And if they do not you should think hard about the problem.

Statistics is, IMHO, better taught as if you are teaching engineers rather than mathematicians. And I think that California is probably right in wanting to put engineering-focused statistics before calculus in the sequence.”

I absolutely agree with this. I think it is important.

Again and before going on I don’t think it is fair to statisticians to claim they rely on the central limit theorem and the Gauss-Markov theorem. Who is this “we” Brad ? I sure don’t rest my confidence on that. I don’t think there are many mathematical statisticians who rest their confidence (if any) on that.

But, while I think Brad conflates current research statistics and a possible bad introductory statistics course , I think he describes a very important problem. I recall Noah Smith (following others I am sure) denouncing economics 101 ism. This is an ideology based on the assertion that economists consider the very first models we teach to be useful approximtions to reality. It is actually unfair to really existing economics 101 courses which go on. It is more accurately described as “first two months of economics 101 ism”. But it is definitely a problem. In particular, there is a problem with teaching first about perfect competition (and rationality and symmetric information and static models or models with complete markets and … I shouldn’t have gotten myself started on that).

I will now try to get to an actual point. I have some thoughts about some things which everyone should be taught and they do not include calculus (trying to teach everyone calculus would be impractical anyway).

I think it is important to try to teach people about probability. I think it is possible, not easy, and useful to teach people Baye’s formula, present cases in which it is useful, and prove it is useful in those cases. People do not, in fact, think about probabilities and conditional probabilities in any way which could be rational, nor do they make choices under uncertainty which are arguably optimal. I am fairly confident that people can be convinced of this and convinced to guard against misleading heuristics.

I think it is useful to teach people about summary statistics (because they are presented all the time to the public) and the risks of relying on means and variances. The cases in which the median is a better estimate of the location of a random variable are not rare and should not be presented as if they are obscure special cases.

I think it is important to teach about causal inference and valid instruments. Here it seems that people have been taught that correlation is not causation. That is a good thing. Also post hoc is not propter hoc. However, it is also true some data sets described as natural experiments are properly described as natural experiments. I think the (not so few anymore) cases of valid causal inference based on non-experimental data are fairly easy to understand. I think a good rule is;

“If you can’t follow the data analysts argue for why his calculation is worthy of your interest, it probably isn’t.”

The lesson that mathematics can be used to cover up reliance on strong implausible assumptions is, I think, very useful. It is the opposite of the lesson that you can obtain a basic understanding of mathematical statistics by considering normally distributed variables.

I think it is important to teach people that they shouldn’t trust a model without checking whether it forecasts well out of sample. This is easy to teach with simple examples. It is important.

I think it is easy to present monte carlo simulations. Here the teacher says “I will make the computer generate some pseudo random data.” Notice that I have to tell it something very specific. Anything we learn may depend on that specific assumption. Now I tell it to calculate this summary statistic, point estimate, test statistic. Now we draw 10000 pseudo samples. Here’s what comes out.

That would give the students the impression that statisticians rely on a black box. That they are doing numerical experiments which has the fault of mathematics (all conclusions are based on assumptions) without the elegance. I think an advantage of giving students this impression is that I think it is 100% accurate.

I think it is very important to explain what the Neyman Pearson framework isn’t. What it means if a null hypothesis isn’t rejected. Here it is very easy to do numerical experiments. It is very easy to choose a data generating process, not tell the students what it is, and figure out things about it in front of them (with massive use of a pseudo random number generator). I think one should choose 10 data generating processes, have the computer pick one at pseudo random at the beginning of the lecture and figure out which one it picked in front of the students.

Most of all, I think the course should be focused on actual empirical problems — Questions which people in the field agree have been answered by analysing data and then the presentation of convincing data analysis.

Basically I agree with Brad.

__________________________________________________________________________________________________________________________

OK enough constructive discussion of possible curricula. I will now return to polemic. I claim that asymptotic theory is not useful and is not used. I have seen a lot of it presented and enjoyed the proofs greatly. I have also noticed that *after* the asymptotic analysis, mathematical statisticians consider a special case, generate 10000 (or now I guess many more) pseudo data sets and check the asymptotic analysis using monte carlo simulations. Why not cut out the middle man ? What was the point of the asymptotic analysis ? Is it like the dread DSGE used to go from an IS-LM idea to find some way to get a DSGE model to act that way, to an IS-LM explanation of what happened in the computer during the otherwise incomprehensible simulation ?

More generally, I would suggest abandoning the concept of infinity. I really think everyone should read “Avatars of the Tortoise” or at least the opening;

“There is a concept which corrupts and upsets all others. I refer not to Evil, whose limited realm is that of ethics; I refer to the infinite.” any claim in asymptotic theory is “for any positive number epsilon, there is a N so large such that this calculation is correct within epsilon if your sample size is greater than N”.

No asymptotic theory can be used to calculate that N. In my experience, mathematical statisticians have always (always) used numerical simulations to check if some N is large enough.

Applying the insight that there is some N large enough is like solving the halting problem. It is impossible. It is obviously impossible. The fact that it is so obviously impossible causes people to assume that they must have misunderstood something when they have undestood perfectly.

I hated calculus, but I loved statistics – used it all the time on my job. Then of course I lived in a world with computers and higher level programming languages including SAS. So, I did not need to do all that arithmetic, but just know how to correctly use the output.

That statistics career/application was in large computer system performance management and capacity planning. There was way way too much data to process to do it by hand. Also, SAS without SAS GRAPH (at least for statistics applications) is like Hamlet without the Prince. Plotting hundreds of data points or a smoothed curve from millions of data points complete with least squares median surrounded by the requested confidence interval of data points takes only a few statements in SAS GRAPH. It will even do predictions of the median and CIs forecast into an extended domain.

Far more important than calculus knowledge in applied statistics is thorough application knowledge. When I started in computer performance evaluation, then I had already worked four years in computer operations, seven years in computer application programming, and two years in computer system programming (OS care and feeding).

Thorough applications knowledge is the difference between success and failure. Statistics is a great tool for many application purposes when used rather than abused. Statistical analysis can be performed without any calculus knowledge by any analyst whose computer knows calculus so that (s)he does not need to.

All that aside, then this article was very well done within the bounds of its context. Other tools such as discrete event simulation are necessary for better answers to specific questions than normal stat since stats are rarely normal in distribution. The errors inherent in the assumptions of mathematicians become very apparent to the analyst within the context of their particular application.

Robert

Interesting post to this LSS Black Belt. One thing our teacher emphasized was reaching a conclusion only means you need to explore further.

I am not a statistician, but I was allowed to teach the subject to college students.

I was not a very good teacher, and they were not very good students. My mistake was to try to “explain” the math to them. Their mistake was to expect me to explain the math to them.

What they needed is a course that discusses in some depth the mis-uses of statistics.

There was a book by Pisani, Purves…and one other. I have forgotten the title. It did a pretty good job of discussing errors statisticians make. A good teacher could begin here and make an interesting …the important part…course in “statistical thinking.” One which depends not at all on calculus or other math…though a little math is always fun and hopeful as long as it is not used to make kids feel stupid.

I’m pretty sure some of the comments in this post are leaning it that direction, but they seem to be from statisticians who are starting from too high a level to have any hope of teaching anything useful to non statisticians, or the sort of journalists who say “it’s the math” without having ever done any mathematical thinking at all. [this includes people who use “math” every day in their professions.]

on the other hand, there are books about “lying with statistics” that may remain at too low a level (political dishonesty as opposed to insufficient analysis on the part of people who know the formulas) to be useful except as a club for partisans to hit each other with.

oh, to answer the original question: my experience teaching statistics to young people tells me that high school is too soon. some mathematically gifted might do well. but it seems to take some experience with both math and its applications befoe kids have the ….i call it synaptic weight… to really understand what it is all about.

makes me think of the school systems who think teaching second graders “the associative property of addition” adds anything useful to their understanding at all. or merely teaches them to learn by rote words that don’t mean anything to them.

i said “systems” instead of “teachers” because i had a bad experience once when i scabbed during a teachers strike and found the 12th graders hopelessly lost in their geometry classes taught out of a big book with pictues and colored ink and a teachers addition with instructions on how to teach this in a different colored ink. the approach was a combination of “new math” and dumbed down sesame street gain- their-interest cuteness. i decided to try teaching it the way i learned it.: here are the postulates, here are the axioms, here is the first proof. the kids got it, and i told the county inspector the kids got it. she told me to teach it “their” [her] way. well, i have never been much for that level of management. fortunately the strike ended before they found me out. i don’t know what happened to the kids. i did notice that a fellow scab seemed to be having pretty good luck teaching literature “his” way. probably smarter than me. didn’t tell on himself.

I think I know the geometry book you are talking about or at least one very similar to it. Having tutored high school kids in geometry, I’ve found that the textbook as written is useless. It’s full of distractions to the point that there is nothing there to learn. Most of the kids found a more theorem based approach a lot easier to learn. Proving theorems gave them something to hang all that other crap on.

A lot of memory work.

Not sure if the math part of statistical analysis makes sense prior to some exposure to calculus. I did not study statistics prior to calculus. But I do think that the work that is involved in picking meaningful hypotheses for problems that interest the student might be useful even if then some of the math involved in evaluating the data seems to come out of nowhere for students.

People who never go to college need to understand something about statistics just as they try to learn something about physics without calculus. It probably should be a very different course than what the above discussion is about.

You don’t need calculus to understand that if you repeat and experiment enough times, “unlikely” events will occur. You don’t need calculus to understand the value and difficulty of accomplishing random sampling.

Arne

I thought I mentioned that very different course.

I was brought up in a school that believed if you needed statistics you didn’t know what you were talking about. True enough for their subject, less true for others.

In my eventual job that i was good at, I found that my school was absolutely right…though my subject was very different from theirs. My colleagues relied on statistics. I did not. I got ten times more accurate results than they did. Sometimes that was important.

Arne:

I agree.

Exactly. Are we teaching statistics or statistical awareness? Everyone needs to be aware of basic statistical terms, what they mean, how they are used, and what information they can and cannot indicate. Everyone needs to know how statistics can misinform, and how to tell if the “statistics” really mean what someone is saying they mean. High school statistics is a lot like high school algebra. Knowing how to work out the math for word problems is more important in later life than actually solving the equations in class. Knowing how to evaluate what you are hearing is more important than being able to replicate the calculations.

Freedman, Pisani, Purves

I probably had the first edition. Can’t say anything about the fourth. I suppose I could put all four into a jar and pull out one or two and make an infomred guess about the other two.

here is a review i copied from google. another review found the book sexist because the examples “showed bias against women.” i suspect that lady did not understand the the book was showing bad statistics showing bias…

review

5.0 out of 5 stars This book is perhaps the best I’ve on statistics for several reasons

Reviewed in the United Kingdom on November 1, 2017

Verified Purchase

I am a practising engineer currently taking postgraduate course in Applied Statistics. This book is perhaps the best I’ve on statistics for several reasons:

1. The author arranged the concepts in a way which helps to build my understanding gradually. Important concepts and fundamentals get highlighted and are easy to refer.

2. The reading is effortless as the writing style is straightforward. You don’t even need to have science background to understand.

3. The examples given can be easily replicated, e.g., dice tossing and drawing cards from a box. For any reader who understands better from practical works, this is great.

Perhaps the 3 points above are common in many books, but I particularly appreciate them because my university is using a textbook from the U.S. which has too many jargons and felt like ideas are being forced down my throat. There aren’t much explanation on the fundamentals. I took this as a norm since many engineering textbooks are written this way as well. This book is so effortless that I can go back to it multiple times to re-cap some important basics.

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You can teach statistics without calculus, but it really helps to build it on top of probability. I took a good high school probability and statistics course before I took calculus. Then I took a college level probability and statistics course that required calculus. It was nice finding out where a lot of the stuff in the first course came from, but not necessary for a basic understanding.

Mathematicians can overthink things. Consider the problem of area as discussed on Twitter:

-> 8 yr old: “Dad, what is area?” Me, heart and mind racing, quietly to myself: “Take it slow, take it slow.”

-> “Well, first we have to define sigma algebras. They’re kinda like topologies, but closed under countable union and intersection instead of arbitrarily union and finite intersection. Anyway, a measure is a …”

Gee, I wish I’d learned that in second grade. It would have made graduate school so much easier.

Funny thing is I did know a girl in the eighth grade who did know all of that stuff. She went on to become an astrophysicist with an asteroid named after her. but when I tried to explain Social Security’s giant great big looming unfunded Many Trillion Dollar Deficit! to her, she said, “well, Social Security has aspects of a Ponzi scheme…”

balloon burst . . . knowing about hemorrhoids does not give the intellect about Social Security unfortunately. Got stuck on the word deficit probably.

that’s pretty much what i have learned after all these years.

you can be a genius but if you won’t look at the subject you won’t learn much about it.

[i was going to say “if you won’t look at the facts” but i realized the facts were yet to be determined, and contrary to Patrick Moynihan, you

canhave your own facts.]