Students starting A level maths from September 2017 will study both mechanics and statistics. The exam boards now require candidates to have a calculator that ‘gives access to’ certain probability distributions. In other words, statistical tables may not be enough to answer all questions completely. For example, the WJEC sample Unit 2 includes a question on the binomial distribution where n = 60, a value not found in most (any?) sets of tables.
This change means that it will no longer be possible for students to carry on using their GCSE calculators for the whole of A level. Schools will undoubtedly recommend a replacement, and may well organise a bulk purchase scheme, but here are my thoughts in case you want to go it alone.
All the Casio graphical calculators include a wide range of probability distributions. I have an fx-9750GII (inherited from a former pupil whose school made them all buy one as an alternative to tables), and it’s a great machine. I use it for stats, matrices and complex numbers…. But never for graphs!
The truth is that unless you buy a high end graphical calculator, graphs will always look better on a computer. I use Graph (http://www.padowan.dk/) which runs on any Windows device including tablets (and on Linux under Wine), or Desmos (https://www.desmos.com/calculator) which runs in a web browser, and even on a smart phone.
Graphical calculators are heavier and bulkier than standard scientific calculators. I never take my graphical calculator out of the house. They are also much more expensive. An fx-9750GII will cost between £50 and £80, and the ‘natural display’ fx-9860GII about £30 more. School students will want the fx-9860GII simply because it displays fractions as fractions (and they would be right!), so you could well be looking at £100 for a calculator. Ouch!
Scientific calculators with probability distributions
The Casio fx-991EX is a new model which does just enough for the new syllabus. It seems to be a successor to the fx-991ES (see below), and will be suitable for further maths A level as well. It has good reviews and is widely available, and I think it is likely to be the model recommended by most schools. It costs about £25 to £30. Just make sure you get an English language version!
The TI36X-PRO is Texas Instruments’ equivalent to the fx-991EX. It does much the same things and retails at about the same price, though your are less likely to find it in the shops in the UK. Be warned that TI calculators do not work in quite the same way as Casio models. Some people like this, others don’t – but either way, if you are used to a Casio then a TI36X is going to feel a bit strange to begin with.
Thirdly, if you are travelling to the States, the Canon F-792SGA sounds like it could be a worthy rival to the two machines mentioned above. Unfortunately, it doesn’t seem to be available in the UK.
The Casio fx-991ES PLUS
This has been widely used by A level further maths students for several years, mainly because it handles complex numbers and matrices sufficiently well (but not as well as the graphical fx-9750GII). It also performs numerical integration – very handing for checking! It is my everyday pencil case calculator, familiar enough to lend to GCSE students, powerful enough for almost all situations I am likely to encounter. And you can buy it for £15 to £20, so it’s not much pricier than a basic GCSE model. (I’m hoping that with the advent of the fx-991EX this one will be discounted, so I can pick up a bargain to keep as a spare!)
If you are about to buy a calculator for A level, you would be wise to go for the fx-911EX unless you really can’t afford the extra tenner. However, if you already have an fx-911ES (perhaps from an older sibling), then you can keep it for the new syllabus by using the sum and integral functions. This is what I plan to do, at least for the time being.
The rest of this post is a bit technical, and assumes you know what the probability distributions are and how they work, so if you are just starting A level it won’t make much sense. But if you are familiar with the binomial distribution and its friends, read on…
Finding cumulative probabilities with the fx-991ES
For these comparisons I used the WJEC statistical tables, the appropriate function on the Casio fx-9750 and the sum or integral described on the fx-991ES. For completeness, I have included integration as a way of dealing with the normal distribution, even though students will find it easier to standardize their distribution and use z tables in the usual manner.
a) Binomial distribution X~ B(20, 0.1)
- Test: P(X ≤ 7)
- Tables 0.9996
- fx-9750 0.99958436
- fx-991ES Using
p = 0.999584365
The binomial coefficient is entered using the nCr button (shift then divide).
b) Poisson distribution X~ Po(9.5)
- Test: P(X ≤ 8)
- Tables 0.3918
- fx-9750 0.39182348
- fx-991ES Using
p = 0.3918234825
This is easiest using natural fractions display inside the sum function.
c) Normal distribution X~ N(10, 22)
- Test: P(X < 12)
- Tables Using P(Z < 1) = 0.84134
- fx-9750 0.84134445
(calculator shows z limits of 1 and -5, i.e. 5 s.d. below the mean)
- fx-991ES Using
p = 0.8413444594
(Lower limit is 5 s.d. below mean).
As students will be expected to find single value probabilities by calculation for the discrete distributions, it is only a small step to learn how to sum these into cumulative values. The integral for cumulative normal probabilities is a bit of a pain to type and tables will always give a good approximation. However, these calculator methods do give a quick(ish) way of finding the probability that X lies between two values (e.g. P(2 < x < 15) ), which would otherwise require finding two probabilities and subtracting.