Sample Variance Intuition

This blog post will focus on the intuition behind the n-1 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 X_1, X_2, \ldots , X_N the population variance is defined by

\sigma^2 = \frac{1}{N} \sum\limits_{i=1}^{N} (X_i - \mu)^2 where \mu = \frac{1}{N} \sum\limits_{i=1}^{N} X_i

and for a sample x_1, x_2, \ldots , x_n the sample variance is defined by

s^2 = \frac{1}{n-1} \sum\limits_{i=1}^{n} (x_i - \overline{x})^2 where \overline{x} = \frac{1}{n} \sum\limits_{i=1}^{n} x_i

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”:

{s_n}^2 = \frac{1}{n} \sum\limits_{i=1}^{n} (x_i - \overline{x})^2

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”

Rational and Irrational Values of Powers

Here’s a cute little existence argument that I was exposed to as an undergraduate and have never forgotten. It shows that there must be irrational positive reals \alpha and \beta for which \alpha^\beta is rational. Furthermore, it’s done by showing that one of two very specific pairs of numbers (either \alpha=\sqrt{2}, \beta=\sqrt{2} or \alpha={\sqrt{2}}^{\sqrt{2}}, \beta=\sqrt{2}) satisfies the condition, but without establishing which of the two pairs satisfies the condition.

The argument is elementary. First look at the case \alpha=\sqrt{2}, \beta=\sqrt{2} If \alpha^\beta = {\sqrt{2}}^{\sqrt{2}} is rational, we have an example, and we’re done. If not then {\sqrt{2}}^{\sqrt{2}} must be irrational, and so \alpha^\beta where \alpha = {\sqrt{2}}^{\sqrt{2}} (assumed to be irrational) and \beta=\sqrt{2} is an example of an irrational raised to an irrational. And it’s rational. In fact, it’s 2, because \alpha^\beta =({{\sqrt{2}}^{\sqrt{2}}})^{\sqrt{2}} = {{\sqrt{2}}^{{\sqrt{2}}{\sqrt{2}}}} = {\sqrt{2}}^2 = 2

It’s a great little elementary existence argument, and I think it’s worth being exposed to it. But, it’s also worth knowing that with (much) more advanced techniques we can actually say which of the two choices satisfies the condition Continue reading “Rational and Irrational Values of Powers”

Paradox Without Self Reference

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 S_1, S_2, S_3,\ldots where S_i is

S_j is false for all j \textgreater i

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”