Correlation for schoolchildren

A few comments on MEI‘s draft “Critical Maths” Curriculum. They list

Glossary of terms which students are expected to know and be able to use […]

Association: A tendency for two events to occur together.

Correlation: An association between two variables which is approximately linear.

This definition of correlation seems rather odd.  If $latex y = x^2$  aren’t $latex x$ and $latex y$  correlated?   What does “an association” mean here?  The suggested definition of association given above is for events, not “variables”.   Presumably the authors have in mind random variables.
There is a serious problem here in the use of language.  It needs to be made clear whether the notion being described is an intuitive one or a mathematical definition. I am not a statistician, but it seems to me that there are (at least) three common distinct types of usage of the word “correlation”,  none of which is captured by the “definition” proposed:
(1)  The vernacular usage. The  Merriam-Webster dictionary gives
  “a relation existing between phenomena or things or between mathematical or statistical variables which tend to vary, be associated, or occur together in a way not expected on the basis of chance alone”
which seems to me a reasonable description of the vernacular or intuitive non-mathematical meaning of the term.    This is clearly much broader than the meaning suggested above.
(2)  The intended meaning proposed seems to correspond closest to the use of the  (Pearson) correlation coefficient  in statistics, although even then it is not  accurate, since  the correlation coefficient is not always a  reliable indicator of the existence of a linear relationship.   This meaning is that which tends to be used by a large class of people who have had some minimal exposure to statistics.
(3)  More generally correlation can be used to indicate a variety of mathematical measures of probabilistic interdependence  (e.g. mutual information).
On a separate point the very heavy concentration on statistical reasoning to the exclusion of other mathematics (including perhaps more elementary logical reasoning such as manipulation of quantifiers and logical connectives) rather worries me, since it may encourage the idea that  almost the only practical applications of mathematics are statistical.
Another  serious danger in my opinion is that statistics at this level tends to be more  like cookery than mathematics and it would have to be extremely well taught by a gifted and highly educated teacher if  conceptual precision is not going to be completely lost.  The danger is partially raised by Gowers in Objection 5 listed in his blog (though he doesn’t mention cookery), but I think his own answer is rather optimistic.
Somewhat in this connection there is an interesting passage in Noam Chomsky on Where Artificial Intelligence Went Wrong where Noam Chomsky is interviewed on various topics concerning science, in particular AI and  cognitive science, and what he clearly regards as a modern deviation from the classical scientific method, which has been indirectly caused by the power of modern computers .  The article is quite long, but I found his example of “how to justify the abolition of physics departments” very nice;  it could  equally well used to justify closing down everything in mathematics departments except statistics.

One thought on “Correlation for schoolchildren

  1. I’m not familiar with this subject, to be honest. Here are my very biased comments.

    I do not like the way statistics is explained in the MEI‘s draft “Critical Maths” Curriculum.

    It appears from “Mathematical ideas which students should encounter through discussion of problems” (p. 8 of the document) that students are supposed to learn, understand, and apply the three basic limit theorems of probability theory/theoretical statistics:

    * the (strong) law of large numbers — “average in time converges to average in space”;

    * the central limit theorem — “all roads lead to Rome” (where “Rome” is the standard normal distribution);

    * empirical distribution converges to theoretical distribution (Glivenko-Cantelli theorem).

    I would prefer these three goals being presented in a more definite way, and then reformulated in practical terms, with a number of applications. At the moment, there is no any structure here.

    A few more comments.

    Page 9: Graph of a normal distribution is wrong: it must be strictly
    positive everywhere!

    “Very unlikely” does not mean “never happens”.

    By the way, why there is 2\sqrt{n}, and not 3\sqrt{n}?

Leave a Reply