Errors, Accuracy, Precision.
Dad drove to Cambridge last weekend. He filled the petrol tank on the way home from work and he filled up again when he reached Cambridge. He says it is 320 miles to Cambridge. He says he put 32 litres in on Friday and another 30 in on Saturday afternoon. What do you think the car’s consumption is in miles per gallon?
i) Lower school methods: 320/30 is miles per litre and this answer *4.54 is m.p.g. The figure of 32 litres is irrelevant.
ii) Higher order thinking: 320 miles is 2 s.f. implied precision, so the distance is between 315 and 325 miles. When he fills the tank, what does ‘full’ mean? Is it the same ‘full’ every time? Well, probably not. Since he filled the tank with 32 litres on Friday, we might assume he gives the 30 litre figure to 2 s.f. too. We need to consider that the previous 32 litres’ ‘full’ is not the same as the 30 litres’ ‘full’. Experience suggests he might well have a variability of 10%, i.e. he used 27 to 33 litres. So the calculation has, as the lower bound, 315/33*4.54 and as the upper bound 325/27*4.54. This gives a range of answers and the true value is somewhere in between.
iii) So, when Dad starts sounding off about the car’s improved consumption on a long run, he needs to be challenged on his assumptions, doesn’t he? There is not enough evidence – unless and until he takes more care with the process of filling the tank: he needs to go from absolutely full and return to that state. He could go from empty to empty, but that produces hazardous driving!
Yes, I used gallons. Dad still thinks in those, not litres. When he says 32 litres, he doesn’t mean 35 - we assume he is giving us good information.
(ii) 320 miles could be 3s.f. but then Dad probably would have said “320 exactly”; he didn’t, so it is assumed to be “about 320”. Since Dad said 30 litres and 32 litres, we will assume he is using the same precision, because taking ‘thirty’ as between 25 and 35 is most unhelpful and makes Dad inconsistent with his numbers. Since you’re reading this, we will assume that your Dad is reasonably good with figures, too.