I’ve been doing a lot of mathematics – mostly trigonometry and algebra – over the past year or so. In fact, I’ve been doing it for years. And the result is that I’ve become fairly proficient at it.
But today I strayed off the beaten track, and started doing some statistics. I got lots of statistics lectures at university, most of which I didn’t understand very well (or even at all). And it’s been slowly rusting ever since, because I’ve seldom had any data that needed any statistical analysis done on it.
But now that I’ve got 400 completed questionnaires from the ISIS study in front of me, for once I’ve got some data that does need some statistical analysis. So yesterday I dug out an old statistics textbook, and started reading it. Because, when the study gets published it might be helpful if I could say whether it was “statistically significant” at the 95% level, with Confidence Interval included.
A lot of the ISIS questions had 5 optional checkboxes with each question (e.g. “How much more/less do you go to pubs?”), so that people could answer “much more”, “more”, “same”, or “less”, or “much less”. And if I’ve got, say, 16o responses with “same” ticked, 200 with “less” ticked, and 40 with “much less” ticked, I wondered how I might convert words into numbers. And I decided that I’d call “same” 1, and “much less” 0, and “much more” 2, with “less” 0.5, and “more” 1.5. In fact “much less” shouldn’t really be 0, but perhaps 0.1. And to preserved symmetry, “much more” should be 1.9.
Once these numbers had been assigned to the words, it was easy to work out the mean value of all 400 responses as being u = 0.65. And the standard deviation – how much the numbers differed from the mean – as s = 0.32.
A bit more reading of my textbooks and online searching then turned up the formula to calculate the confidence interval = u ± z. s / √n. where z was a number pulled out of a normal distribution table to represent the confidence level. For 95% confidence, z was 1.96, and that gave a 95% confidence interval between 0.68 and 0.62. I think that means that 95% of samples from a larger population, assuming it had the same standard deviation, could be expected to fall within this range.
If the null hypothesis for the question ”How much more/less do you go to pubs?” is the “same” or 1, that would seem to indicate that a result with a 95% confidence interval between 0.68 and 0.62 does not straddle 1, and is therefore a statistically significant result. But I wouldn’t like to bet on it. Particularly since my stats textbook doesn’t seem to approach hypothesis testing quite this way. Maybe tomorrow I’ll try and figure that out.
Anyway today I worked out my first 95% confidence interval in over 40 years. I think I’ll celebrate with a few shots of Talisker later.
But the process of reading maths textbooks reminded me of one reason why I often found mathematics hard. And it wasn’t the logic or reasoning behind any of it, which is very often astoundingly simple, but the words that mathematicians use.
Like mathematics. That’s a pretty daunting word. And so also are addition and subtraction and multiplication and division. They’ve all got between 3 and 5 syllables in them. They’re big words. And they’re strange words. And mathematics is full of them. Like numerator and denominator. They’re like mountains, and doing mathematics often seems like climbing mountains, rising ever higher. Beyond denominator there’s common denominator and beyond that, obscured by cloud, lowest common denominator. And the branches of mathematics have similar menacing names. Algebra. Geometry. Trigonometry (on which one could do oneself a serious injury). Calculus. Matrices. Determinants. Probability. Statistics. And they’re all words which ring and reverberate with multiple hidden means. Matrices has a touch of matriarchy about it, calculus a vein of calcium, determinants a shot of determination.
And I would regularly get shipwrecked on these words. I’d get stuck on the slopes of lowest common denominator, suffering from vertigo, and running out of water. Or normal distribution. What on earth was “normal” about it? Was there an “abnormal” distribution? Or a “post-normal” one? Or right angles? What’s “right” about them? Are there “wrong angles”? And is a confidence interval the time it takes to build up sufficient courage to ask Mary Jane out to a dance?
Yet once I’d managed to slither my way round the word algebra, and absorbed the notion of an equation, I found algebra remarkably simple and straightforward. So also with geometry, because I already knew what a point and a line and an angle were. But I was in terror of the vast and inconceivable Theorem of Pythagoras, rising like Everest, with its deadly hypotenuses, which were a species of horse that roamed its slopes.
I think that if I was ever to try to teach mathematics, I’d want to throw away all these words, because they just cause problems. And they’re not necessary. I think I’d maybe call mathematics numbers, and addition summing, and subtraction taking, and multiply times, and division over. And then nobody – no child at least – would ever be terrified by some multi-syllabic Kilimanjaro mountain of a word.
Because we’re all mathematicians. We use it every time we buy something in a shop with money. And anyone who can figure out how much money is needed to buy 3 packets of toffees at 85 pence/packet is already a considerable mathematician. And higher mathematics ought to be as easy as shopping on the fifth floor of Debenhams.