# Two Significant Figures

## Hardly any school leavers can work out percentages in their head. Does it matter?

07 February 2017

Can you work out four sevenths as a percentage, in your head - to the nearest percent?

If you can, you have a rare skill.

I recently ran an experiment in two good schools, where I asked the sixth form (i.e. 17 year old) mathematicians* to do various calculations in their heads.

The challenge included asking them to work out 4/7 to the nearest percent. This calculation stumped almost all of them. The majority knew that it was "a bit more than half", a few got 58% or 56%, but only 3 out of 60 (that's 5% of them) could mentally work out that four sevenths is 57% to two significant figures.

Does this matter? A lot of people seem to think so. I did a survey on Twitter. Of the 266 people who responded, 68% of them (that's nearly *five *sevenths) thought that we should expect "most" sixth form mathematicians to be able to mentally work out that 4/7 is 57%.

Only 10% of respondents thought it didn't matter as "this is not an important skill". These people pointed out: "Surely it's enough to know that 4/7 is a bit more than half...maybe about 60%".

And yes, I agree that in most circumstances, it's the ability to be able to come up with a ballpark figure that is the most vital skill. We need it to defend us against spurious statistics thrown out by politicians and salesmen. If we want more accuracy we can easily resort to a calculator or pen and paper. And yes the group I sampled were 'mathematicians' and not 'arithmeticians' (see my related blog on Arithematicians)

And yet...

Of all the maths learned at school, being able to work out and interpret percentages ranks right at the top in terms of its everyday use.

I just checked today's main news stories, and well over half of them contained a percentage. Here are just a few examples:

*"17% of mobile phone users will face mobile phone bill increases of over £100"*

*"...there has been a 5.2% surge in German factory orders"*

*"37% of the public believe it would be reasonable to charge for some NHS services"*

*"Alastair Cook won 41% of the Tests that he captained"*

Notice how in all the examples above, the percentage has been quoted to two significant figures. That's typical. Sometimes a percentage will be rounded to the nearest ten percent (one headline today read: "90% of hospitals are overcrowded") but generally the world deals in percentages to two S.F.**

So if you can work out a percentage to two significant figures, you have a tool that will serve you well as a journalist, a business analyst, a sports statistician, an exam marker - and also as a member of the public who has to engage with all these people. If you happen to have learned how to do it quickly and accurately in your head, then there will be no shortage of opportunities to apply this skill.

And this is not a hard skill that can only be mastered by the elite few. 4/7 as a percentage requires two steps of mental short division. Seven into 40 goes 5, remainder 5, seven into fifty goes 7, remainder 1....so that's 57% and a bit. Most primary school children can do it when they've learned about short division. But by the time they are sixth formers - when percentages have become a significant feature of daily life - they've forgotten the method.

How many of our 'mathematical' school-leavers should we expect to be able to do this? I think it would be excessive to expect it of all of them, or even half of them. This is certainly not an essential skill - though people who have mastered exact calculations have a strong foundation for doing ballpark calculations.

But remember that examiners expect those school-leavers who are the "best" at maths to be able to factorise a cubic equation and find the precise co-ordinates of the turning point of the graph. In contrast the education system seems to be (to borrow a phrase from Peter Mandelson) intensely relaxed that almost *no *school-leavers can mentally work out four sevenths as a decimal.

Dare I suggest we've got our priorities slightly wrong?

** I know sixth form maths is a far cry from arithmetic. But the sixth formers that I'd expect to be the most numerate are the ones studying maths - at least until 'Core maths' takes off. If it turns out A Level historians or geography students are better at their times tables than A level maths students, I'll be pleasantly surprised.*

** *two significant-figures is ok as a guide, though if one's being picky, sometimes a third significant figure is needed to give the same level of precision. Other things being equal, a number rounded to 84% is more precise than one that's been rounded to 11%.*

Related blogs: Three Sevenths; Arithmeticians