Uncertainty and significant figures

Two students read the same graduated cylinder. One writes “36 mL”, the other “36.4 mL”. Both are “right”, but they promise different things: the first says “somewhere between 35 and 37”, the second “between 36.2 and 36.6”. The number of digits you write down is itself a statement of uncertainty. This lesson is about writing, reading and calculating those digits honestly.

• Accuracy is how close a result (or mean) is to the true value. It suffers from systematic error, which pushes every reading the same way: an unzeroed balance, an uncalibrated instrument, a biased method.
• Precision is how close repeated results are to one another. It suffers from random error, which is sometimes up and sometimes down: judging the last estimated digit, small temperature fluctuations.

Random error shrinks when you average repeated measurements and is measured by the standard deviation $s$; systematic error is found only by comparison with a reference standard.

Write down every digit you can count from the marks, then estimate one more — to about a tenth of the smallest division. A 100 mL cylinder is marked every 1 mL, so read to 0.1 mL (e.g. 36.4 mL); a 50 mL burette is marked every 0.1 mL, so read to 0.01 mL (e.g. 23.62 mL). The last digit is an estimate that two careful readers may disagree on by 1 or 2 — that is the reading uncertainty.

For aqueous solutions read the bottom of the meniscus with your eye level with it. On a cylinder, looking down reads high and looking up reads low; a burette is numbered from the top down, so there the effects reverse. In the magnifier below you can change your eye height and see how large parallax is.

1. Every non-zero digit is significant: $4.23$ has 3.
2. Zeros between non-zero digits are significant: $4005$ has 4.
3. Leading zeros only locate the decimal point: the $0.00$ in $0.00340$ does not count.
4. With a decimal point, trailing zeros are significant: $0.00340$ has 3 and $100.0$ has 4.
5. Without a decimal point, trailing zeros of a whole number are ambiguous: $1200$ may have 2, 3 or 4. Writing $1.20\times10^{3}$ settles it.
6. Exact numbers (counted items, definitions such as 1 in = 2.54 cm, and $N_\mathrm{A}$ since 2019) have unlimited significant figures and never limit a result.

• Multiply and divide: keep as many significant figures as the input with the fewest. $2.5 \times 3.42 = 8.55 \to 8.6$ (two).
• Add and subtract: keep digits only down to the leftmost last place — the input with the fewest decimal places. $12.11 + 18.0 = 30.11 \to 30.1$.
• In multi-step work carry one or two extra digits and round only at the end, so rounding errors do not pile up.

Subtraction can destroy significant figures: $1.0 - 0.98 = 0.02 \to 0.0$ — two numbers with two significant figures each leave almost nothing reliable. That is why weighing by difference needs a suitably sized sample.