Year Two

After a long break from my first module, I decided to start my next module in February 2014, half-way though the academic year.

My second module was to be Statistics. I was looking forward to this, and wasn't disappointed. The module covered a lot about survey results, and how to conduct surveys effectively and make sense of the data. It also had a biology experiment thrown in there for good measure, the results of which were meant to be used in an assignment.

One of the things that really bothered me though, was their description of statistical variance. To my mind, one measure of variance would be the sum of the difference between the population mean and the sample mean, all divided by the sample count.

In fact, that's not the case. It's not divided by the sample count, it's divided by the sample count minus one. This seemed totally counter-intuitive, and apparently the full explanation was outside of the course material!!! Frustrated I scoured the web looking for an explanation... Eventually I came across a decent youtube vid. Thanks youtube!

To summarise, this video proves that sample variation s^2 is an unbiased estimator of the population variation \sigma^2 as shown below:

    \[<span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="http://adrianbell.me/wp-content/ql-cache/quicklatex.com-8309d8e21d5c74b4a21588f648103d5a_l3.png" height="45" width="261" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{align*} E(s^2) =& E \left( \frac{ \sum_{i=1}^n( x_i - \bar{x} )^2 }{n-1}\right) = \sigma^2 \end{align*}" title="Rendered by QuickLaTeX.com"/>\]

See that pesky "n-1" divisor? Check out the full explanation here.

So after my study starting in February and ending in September, I felt I'd had a decent introduction to statistics.

Although, something that does surprise me is that looking ahead at my possible future module choices... statistics doesn't really crop up again for the rest of my maths degree. With the growing importance of statistics in modern society (let alone mathematics itself!) I would have expected a lot more of those modules on offer. Having said that, there is a separate BSc (hons) Mathematics and Statistics that the OU offers, but this appears to almost be purely statistics and doesn't give much variety. -certainly wouldn't be good for me.

In fact, there are  only two real interesting modules on this Maths and Stats degree course, one of which is the final year module "Mathematical Statistics" (M347) that introduces Markov Chain Monte Carlo, which is an area of interest. I suppose after my ten-year degree, if I'm still desperate for more, I can sign up for it as a stand-alone module. 🙂

 

Year One

The countdown to starting my latest module has begun!!! With about a month left to go before all my new material arrives I thought I'd sum up the years I've completed thus far.

My first year I spent studying a module the Open University call "Using Mathematics". It was essentially A-level-ish in terms of the content. Although I have never studied A-level in mathematics, I was thankful that a lot of it seemed like revision.

The module covered things like recurrence sequences, vectors and matrices, calculus, and some basic statistics in the form of probability work. Calculus being the most challenging area, as I'd not encountered it before.

This first module was the biggest jump I've experienced so far, as when I started I hadn't done any formal learning for about 12 years. Doing well in this first module acted as a massive confidence boost, acted as a confirmation that I actually knew what I was doing, and had a fighting chance of seeing the whole thing through!

Starting Late

I really should've started this site sooner, but I only considered how useful it would be to document mathematical learning just the other week.

I'd made post on the Open University forums, and one of the students (who is also a mathematics teacher) gave me a link to her site. It contained all kinds of observations on common mistakes and examples to circumvent them.  So good it was, it inspired me to make my own site!

As it stands I've studied three modules already. The first of which was more like an A-Level-like introduction to mathematics. As I don't have an A-Level in mathematics, this was actually rather useful.

Second year came a module on statistics, and the third year, followed directly on from the first year introductory module.

This most recent third year was particularly good, as it introduced complex numbers (of which I've never formally been taught), and group theory which included some brand new mathematical concepts to get my head around.

So with three years in the bag, I'm now looking ahead at another seven (at least). I'll be writing about my learning process, the great parts of learning and the bad. I'll be writing about mathematical techniques and even about various tools and books I discover to help me along.

Oh, and I'll also be writing lots of pretty maths:

    \[\pi\int_{0}^{\frac{\pi}{3}} \tan^{2}x\: dx - \pi\int_{-\frac{\pi}{6}}^{0} \tan^{2}x\: dx\]