Measurement uncertainty and significant figures

TL;DR:

• Every measurement has an uncertainty. For a ruler, a good estimate is half the smallest division.

• Significant figures show how precisely a number is known. Multiply and divide: keep the fewest sig figs. Add and subtract: keep the fewest decimal places.

• Relative uncertainties add when you multiply measurements: 0.05/17.6 + 0.05/0.70.

You measure a pencil with a ruler and write down 17.6 cm. Your friend gets 17.5 cm. Neither of you made a mistake. Measuring never gives a perfect number, so a careful measurement always comes with a statement of how far off it might be.

Reading a scale and stating the uncertainty

Say your ruler is marked in millimetres. The smallest division is 1 mm, which is 0.1 cm. You can read to the nearest mark, and often estimate between marks by eye. A common, simple rule is to take the uncertainty as half the smallest division: 0.5 mm = 0.05 cm.

So the length is written 17.6 ± 0.05 cm. This says the true value probably lies between 17.55 and 17.65 cm. Your friend's 17.5 is a little outside, which is a hint that the pencil end or the ruler zero was lined up differently.

The last digit you write should be the first uncertain one. Writing 17.62 cm on a mm ruler claims a precision you do not have. Writing 17 cm throws away information you do have.

Significant figures

Significant figures are the digits that carry information. The rules are short.

• All non-zero digits count: 17.6 has 3.

• Zeros between digits count: 1005 has 4.

• Leading zeros do not count: 0.0072 has 2.

• Trailing zeros after a decimal point count: 0.70 has 2, and 12.00 has 4.

• In a plain whole number like 1500, trailing zeros are unclear. Scientific notation settles it: 1.5 × 10³ has 2, 1.500 × 10³ has 4.

For calculations, two working rules. When you multiply or divide, the answer has as many sig figs as the least precise input. When you add or subtract, the answer has as many decimal places as the least precise input.

Worked example: the side area of a pencil

Treat the pencil as a rectangle seen from the side. Length L = 17.6 ± 0.05 cm. Width, measured across the body, w = 0.70 ± 0.05 cm. Find the area with its uncertainty.

1. Area: 17.6 × 0.70 = 12.32 cm².

2. Relative uncertainty of length: 0.05 / 17.6 = 0.0028.

3. Relative uncertainty of width: 0.05 / 0.70 = 0.071.

4. When multiplying, add the relative uncertainties: 0.0028 + 0.071 = 0.074.

5. Absolute uncertainty of area: 12.32 × 0.074 = 0.91 cm², which we round to 0.9.

6. Quote the result: the uncertainty sits in the units place, so round the area to match: 12 ± 1 cm².

Check the sig-fig rule agrees: 0.70 has two sig figs, so the product should have two, giving 12 cm². That matches the uncertainty-based answer. Check the worst case directly: 17.65 × 0.75 = 13.24 and 17.55 × 0.65 = 11.41, so the area really could be anywhere from about 11.4 to 13.2, a spread of roughly ±0.9 about 12.3.

Notice what dominates. The length is known to a tiny fraction (0.0028), but the width to about 7 parts in 100 (0.071). The area is only as good as the weakest measurement. To improve it, measure the width better, for example with calipers, rather than the length.

One more habit helps: take repeated readings. Measure the pencil three times, say 17.6, 17.5 and 17.6 cm, and the average is (17.6 + 17.5 + 17.6) ÷ 3 = 17.57, which you would still write as 17.6 cm. The spread between readings, here 0.1 cm, is a useful guide to how big your uncertainty really is, and it is a good reason not to quote more digits than the readings agree on.

Absolute versus relative uncertainty

The absolute uncertainty has the same units as the measurement (±0.05 cm). The relative uncertainty is the absolute one divided by the value, with no units (0.05 / 17.6). Relative uncertainty lets you compare measurements of different sizes: ±0.05 cm is tiny for a pencil and huge for a grain of rice.

Do not mix up precision and accuracy. Precise means repeated readings agree. Accurate means they are close to the true value. A ruler with a worn-off end can give very precise but inaccurate readings.

How to check your answer

• Reasonableness. The uncertainty should be small compared with the value, unless the measurement is poor.

• Worst-case test. Calculate the largest and smallest results using the extremes of each input.

• Consistent digits. The last digit of the value should sit in the same place as the uncertainty.

• Do not round early. Keep an extra digit in the middle steps and round at the end.

Mistakes that cost marks

• Quoting every digit your calculator shows.

• Counting leading zeros as significant.

• Dropping a trailing zero like 0.70, which loses a significant figure.

• Adding absolute uncertainties when multiplying, instead of relative ones.

• Rounding in the middle of a calculation and passing the error on.

Where you will use it next

Every lab report asks for uncertainties, and the units you carry through are covered in units and dimensional analysis. The same habit helps when you test numbers like those in orbits.

Frequently asked questions

Are exact numbers significant?

Exact numbers, such as 12 eggs in a dozen or the 2 in 2πr, have unlimited significant figures and never limit your answer.

Does taking more readings help?

Averaging repeated readings reduces random scatter, but it cannot fix a systematic error such as a miscalibrated ruler.