How wrong is wrong?
Tonight's idea
You measure a pencil and are off by 2 cm. A builder measures the length of a house and is off by 2 cm. The same mistake? In centimetres, yes. But the pencil is 16 cm long and the house is 1,000 cm long. Against the pencil, 2 cm is a big slice. Against the house you can hardly see it.
To compare fairly, measure the mistake against the size of the thing. First find the error: the gap between your value and the true value, counted as positive whichever way you missed. Then:
Percent error = error ÷ true value × 100.
Pencil: \(2 \div 16 = 0.125\), which is 12.5%. House: \(2 \div 1000 = 0.002\), which is 0.2%. With the same 2 cm, the builder did more than sixty times better.
Divide by the true value, not by your own value. The true value is the one thing everybody is aiming at. (For a reminder about percentages, see Percentages in your head.)
Watch one: beans in a jar
A jar holds 80 beans. You guess 60.
Error: \(80 - 60 = 20\). Percent error: \(20 \div 80 = 0.25\), which is 25%.
Your friend guesses 100. Error: 20 again, so 25% again. Too low or too high, the same miss gets the same score.
Now see why we divide by the true value. Divide by the guesses instead and you get \(20 \div 60\), about 33%, and \(20 \div 100 = 20\%\). Two scores for the same miss. That cannot be fair.
Your turn
- Find the percent error. (a) Measured 47, true value 50. (b) Estimated 230, true value 200. (c) Measured 9.8, true value 10.
- Mia guesses a dog weighs 22 kg. It weighs 20 kg. Leo guesses a car weighs 1,300 kg. It weighs 1,250 kg. (a) Who missed by more kilograms? (b) Find both percent errors. Who made the better guess?
- A kitchen scale promises a percent error of at most 2%. You put on a true 250 g. What is the lowest reading it could show? The highest?
- A ruler marked only in whole centimetres can be off by up to 0.5 cm on any measurement. (a) What is the largest percent error on a 4 cm leaf? (b) On a 50 cm shelf? (c) How long must something be for the percent error to be at most 0.1%?
Stretch: errors grow
A square tile has sides of 10 cm. You measure each side as 11 cm, a 10% error. You then work out the area from your measurement. What is the percent error in the area? Why is it more than 10%?
Answers
- (a) Error 3, and \(3 \div 50 = 0.06\): 6%. (b) Error 30, and \(30 \div 200 = 0.15\): 15%. (c) Error 0.2, and \(0.2 \div 10 = 0.02\): 2%.
- (a) Leo, by 50 kg against Mia's 2 kg. (b) Mia: \(2 \div 20 = 10\%\). Leo: \(50 \div 1250 = 4\%\). Leo made the better guess, even though he missed by 25 times as many kilograms.
- 2% of 250 g is \(0.02 \times 250 = 5\) g. The reading lies between 245 g and 255 g.
- (a) \(0.5 \div 4 = 0.125\): 12.5%. (b) \(0.5 \div 50 = 0.01\): 1%. (c) 0.1% is one thousandth, so 0.5 cm must be at most a thousandth of the length: at least \(0.5 \times 1000 = 500\) cm, which is 5 m.
Stretch. True area \(10 \times 10 = 100\) cm². Your area \(11 \times 11 = 121\) cm². Error 21, and \(21 \div 100 = 21\%\). The mistake is made on both sides, and multiplying stacks them: two strips of 10 cm² each, plus a 1 cm² corner where the two mistakes meet.