This blog post will focus on the intuition behind the divisor of the standard variance. No proofs, but lots of context and motivation.
And you’ll be pleased to know that this post is self contained. You don’t need to read either of my previous two posts on sample variance.
For a population the population variance is defined by
and for a sample the sample variance is defined by
And the question is “why doesn’t the sample variance have the same formula as the population variance?” i.e why isn’t the sample variance given by the direct analogue of the population variance, which is known as the “uncorrected sample variance”:
For the purpose of building intuition, there’s two facts that I think are extremely important to keep in mind.
Continue reading “Sample Variance Intuition”
This post is about an elementary result from mathematical logic (due to Stephen Yablo in the early 1990’s) that I think deserves to better known.
I’d always assumed that “Russell-like” logical paradoxes (e.g. the inability to assign either a true or a false value to the statement “this sentence is false”, or to the statement “the collection of collections that don’t contain themselves contains itself”) required circular self-reference. Apparently not, provided you use an infinite number of statements.
The construction is simple: consider an infinite sequence of sentences where is
is false for all
If you want to see the details of why this is problematic, it’s in the paper “Paradox Without Self Reference” available at http://www.mit.edu/~yablo/pwsr.pdf But I think it’s more fun to figure it out for yourself, and it’s not that hard. The difficulty was in realizing that the result was possible (I didn’t realize it, and I had more than enough mathematical training to do so before the paper was published), not in proving it.
In you really want to see an argument for why there’s a paradox, I’ll provide one here that’s slightly different in unimportant details only from the one in the paper. Continue reading “Paradox Without Self Reference”