I am in the middle of my highschool maths work, today's topic being variance
$var(X)$, so I am just commenting, sharing a thought/criticism:
Is MathWorld (MW) owned by Wolfram Mathematica (WM)? Then why doesn't WM link to the respective article on MW? @Eric's MW article is well-written in the sense that it tries to point out the potential of confusion regarding variance-related quantities, and fortunately it clarifies that WM's Variance[list]
is the so-called "bias-corrected sample variance" and does not produce the output of what textbooks teach as the square of the standard deviation,
$\sigma ^2=\text{var}(x)$. In order to get the value for
$\text{var}(x)$, the student must enter a rather complicated code line:
In[10]:= list = {1, 2, 3, 4, 5, 6}; (* playing dice *)
var = Expectation[(x - \[Mu])^2, x \[Distributed] list]
Out[10]= 35/12
In[11]:= Variance[list]
Out[11]= 7/2
His attached notebook is poorly curated and more than lacking. It is wishful thinking but imho the attached notebook should be full-fledged and nicely Mathematica-like formatted and basically continue where a "textbook page/subchapter" ends. My dream: First a student learns about a maths subtopic from his J.Wiley text, then he learns from the MW's attached notebook how the learned item (here: variance, from a stochastics intro course) is put into code in Mathematica. Basically I am criticizing that much much more effort should be put into the MW's attached notebooks! The notebook content should be really resourceful, informative, with a couple of simple/standard textbook examples, look attractive, maybe even include some mathematical proofs, and look like an impressively helpful page from an expensive colorful J.Wiley textbook. It would be also a great way of showing to the world, what the Mathematica software is capable of doing: creating professional-quality documents and doing all the maths which a student has learned from a printed textbook.
Sure, a student who wants to do maths with Mathematica could buy another printed book instead, like 'Probability and Statistics With Mathematica' from an unknown third-party author. But why another printed book? Mathematica is done on the PC, not on paper! A notebook nb-file is much more suitable to show and teach maths or, at least, to show how a Mathematica function like Probability[]
is put to use, how it relates to the things one has learned in class and from a book.
Am I saying that the current versions of notebook files on the MW website are nearly worthless? Well, for sure they are disappointing. And many articles don't even have one, like the one for Conditional Probability, which is another good example where MW and WM fail to make a connection. A related point of criticism is that the Wolfram Documentation examples should be more illustrative too, preceding a code line. I learned all about conditional probability for a full month, then I switch over to the WDC and cannot see the connection. I don't find the page helpful at all sorry.
Wolfram can't rely or hope that other authors around the world churn out books to document more clearly what Conditioned
does for the student. Wrong attitude. My attitude is, if I create a math function and want it to be useful for others, then it is my job to make clear, maybe in form of a long colorful nb document, how the code relates to what is seen and learned from textbooks and textbook examples and textbook problems.
The motto of the beautifully presenting notebook should be like "Let's do some maths with illustrative textbook-like examples showcasing the functionality and uses of the function .. in erh … maths!!".
Am not looking for a discussion. I am pretty sure that everyone who knows about MW and WM is aware of the lamentable situation, and I don't expect any changes. Funds and energy at the company are spent on other areas of interest (like artificial intelligence, neural networks, machine learning), not on spoon feeding the maths-learning high schoolers, I get it and accept.
I haven't ranted in a while. Lol. Okay, lemme get back to trying to figure out what the statistics functions do for me… Back to work, c u, bye!
EDIT 14 days of hard studying later: i am slowly getting the hang of (understanding) the basic Mma statistics functions/functionality (all very powerful, wow!) and how they relate to problems from my statistics introtexts. The Probability ref page is helping me a lot to get over the hurdle - great work there by the authors! The above rant from a statistics-first-learner's frustration was a bit unfair .. but there's some truth to it. Anyway, my apologies to anyone responsible at Mathworld or Wolfram who felt being criticized unfairly by it.