Significant figures

Many people, even many scientists, find statistics and error analysis to be boring. But it is a necessary evil, both to get high grades and to know whether we can confidently say something is true.

From A-level we should remember that a result of 25 has two significiant figures, and a result of 25.0 has three significant figures, 0.000025 has two significiant figures, while 0.0000205 has three.

It is useful to write results in exponential form. You are likely to make mistakes when writing out multiple zeros. But more importantly it prevents ambiguity in significiant figures. Consider the number 92500, it could represent:

9.25 x 10⁴ (three s.f.)

9.250 x 10⁴ (four s.f.)

9.2500 x 10⁴ (five s.f.)

If we multiply or divide numbers, the answer is limited by whichever number has the least significant figures:

If we add or subtract numbers, it is OK to get answer with a different amount of significant figures. 

But the sum is limited by whichever number has the least decimal places:

The answer above should be rounded to the nearest significant digit, which in this case is 121.795.

To add numbers with different exponents, first convert them to the same exponential form. Then use the same rules as before.

In the above example, the answer is rounded to 11.51 x 10⁵ because the number 9.84 x 10⁵ limits the answer to two decimal places.

To remember the difference, think "multiplication = significant figures" and "addition = decimal places".