Can you can make a measurement that's very precise, but not very accurate? Can a number be accurate, but not very precise? Let's find out the difference between these two terms; you'll see that precision and accuracy are really two different things.
The precision of a measurement describes the units you used to measure something. For example, you might describe your height as 'about 6 feet'. That wouldn't be very precise. If however you said that you were '74 inches tall', that would be more precise, because inches are smaller than feet.
Of course, sometimes in real life we don't want to be precise.
For example, if someone driving by stops and asks you how far it is to the next town, you wouldn't reply 'It's about 3,270,132 centimetres'. Or if asked your age, you certainly wouldn't say '7776023 minutes'.
These answers are more precise than 'about 30 kilometres' and 'I'm fifteen'. But you use the less precise answers anyway ... probably because they're easier to remember.
The smaller the unit you use to measure with, the more precise the measurement is.
In mathematics and physics, it is often necessary to make measurements that are as precise as you can make them. This requires that you use measuring instruments with smaller units. Have a look at the pencil being measured below.
How long is the pencil? The best you can say is 'about 9 centimetres'. You might guess and say 'about 9.5 centimetres', but the decimal place is just a guess. Because the smallest unit on the ruler you are using is one centimetre, the precision of your measurement is to the nearest centimetre.
Now look at the picture below, where we are using a different ruler to measure the pencil.
How long is the pencil? The best you can say is 'about 9.5 centimetres'. Again, you might guess and say 'about 9.51 centimetres, but the second decimal place is just a guess. Because the smallest unit on the ruler you are now using is one millimetre (one tenth of a centimetre), the precision of your measurement is to the nearest millimetre, or tenth of a centimetre.
This second measurement is more precise, because you used a smaller unit to measure with.
O.K., so how long is the ruler really?
Strangely enough, there is no answer to that question.
It is impossible to make a perfectly precise measurement.
Suppose for example, you purchased a very expensive measuring device, with units of one thousandth of a centimetre, and measured the pencil again. You might find its length to be 9.503 cm. This is certainly a more precise measurement, and you now have a more precise value for the length of the pencil.
But this isn't the real length either ... an even more expensive device might give a value of 9.50312 cm.
Since there is no limit to the tiny size of the unit you can measure with (as long as you can afford the instrument), there is no limit to how precisely you can measure the pencil.
Currently, scientists use instruments which can measure lengths as precise as the width of a subatomic particle (millionths of a centimetre).
The accuracy of a measurement describes how close it is to the 'real' value. This real value need not be very precise; it just needs to be the 'accepted correct value'.
For example, suppose your math textbook tells you that the value of Pi is 3.14. You do a careful measurement by drawing a circle and measuring the circumference and diameter, and then you divide the circumference by the diameter to get a value for Pi of 3.16.
The accuracy of your answer is how much it differs from the accepted value.
In this case, the accuracy is 3.16 - 3.14 = 0.02.
The precision with which the accepted value has been measured is not important. All that matters is how different your measurement is from that value.
Here's another example. The mass of the moon is often given in textbooks as 7.3 x 1022 kg.
This isn't a very precise figure, with only one decimal place.
However, that doesn't matter. The accepted value we are going to use is 7.3 x 1022 kg.
If we were to do our own calculation and work out the mass of the moon to be 6.9 x 1022 kg, the accuracy of our calculation could be measured this way:
Accuracy is the difference between our own calculation and the accepted real value.
So now we can answer our original questions:
Can you can make a measurement that's very precise, but not very accurate?
Yes. Suppose you measured the voltage of a 9-volt battery and get an answer of 17.1453 volts. You must have used a device that allowed 4 decimal places of precision ... but your answer was wrong. It was inaccurate by over 8 volts!Can a number be accurate, but not very precise?
Yes. Suppose you have a can of soup that really does contain 451 ml of soup, as it says on the side. You pour it out and measure it and find there are 450 ml. Neither value is very precise, both being measured to the nearest ml.