Uncertainty and Significant Figures

Every measurement carries some doubt. A ruler marked in millimeters cannot tell you a length to the nearest nanometer. Pretending otherwise misrepresents your data. Significant figures are the bookkeeping system chemists use. They communicate exactly how much a number can be trusted. This ensures a calculated result never claims more precision than the instruments that produced it. This idea builds directly on Metric Prefixes and Scientific Notation. Scientific notation is the cleanest way to display significant figures without ambiguity. Mastering this bookkeeping is essential before tackling Dimensional Analysis. It is also essential before the calculations behind Density, where multiple measured quantities combine in a single formula.

Key takeaways

Precision describes how reproducible measurements are; accuracy describes how close they are to the true value — the two are independent.
Significant figures are the digits in a measurement known with certainty, plus one final estimated digit.
Counting rules: all nonzero digits count; zeros count only in specific positions (captive, trailing-with-decimal); leading zeros never count.
In multiplication/division, the answer is limited by the factor with the fewest significant figures.
In addition/subtraction, the answer is limited by the term with the least precise decimal place.
Key idea: a calculated number can never be more certain than the least certain measurement that went into it.

The core insight

Significant figures track certainty, not just digit count — the last digit written is always a reasonable estimate, never noise to be trimmed or padded away.

The physical picture

Imagine reading a graduated cylinder marked every 1 mL. You can see the liquid level sits between the 36 and 37 mL marks, closer to 36, so you report 36.3 mL. The "36" is certain — anyone reading the same cylinder agrees on it. The final "3" is your best estimate of the fraction between marks; a different person might read 36.2 or 36.4. That estimated digit is still meaningful, but it is uncertain. Reporting "36.30 mL" would be dishonest, because no one can resolve a hundredth of a milliliter on this cylinder — it invents certainty that doesn't exist. Reporting just "36 mL" throws away real information, since the observer clearly could distinguish more than whole milliliters. This is the core tension significant figures resolve: report all the digits known with confidence, plus exactly one uncertain digit — no more, no less. A digital balance reading 14.702 g is telling you the same story with more digits, because its instrument resolves finer differences; the last "2" is its estimated digit. Precision and accuracy are separate ideas layered on top of this. Precision asks whether repeated measurements agree with each other — tight clustering, regardless of where that cluster sits. Accuracy asks whether the measurements agree with the true, accepted value. A poorly calibrated balance can give very precise but inaccurate readings (consistently off by the same offset), while a rough, badly-maintained instrument can occasionally land near the truth by chance without being precise at all.

Figure

Four dart-board patterns illustrating precision versus accuracy

The core relations

\text{Significant figures} = \text{all certain digits} + \text{one estimated digit}

where:

certain digits are read directly off the instrument's marked scale,
the estimated digit is the observer's interpolation between the finest marked divisions.

\text{(product or quotient): sig figs in answer} = \min(\text{sig figs of each factor})

where:

the minimum is taken over all measured factors entering a multiplication or division,
exact numbers (like a defined conversion factor or a counted integer) do not limit the count.

\text{(sum or difference): decimal places in answer} = \min(\text{decimal places of each term})

where:

the limiting term is whichever addend is known to the fewest decimal places, not the fewest sig figs,
the final answer is then rounded to that decimal place, after which its sig-fig count is whatever it is.

Derivation AWhy multiplication/division and addition/subtraction use different rules

Goal. Show that the "fewest sig figs" rule for products and the "fewest decimal places" rule for sums both fall out of tracking the absolute uncertainty in each measurement honestly.

Step 1. Model uncertainty as a range. Suppose a measured length $a$ has an uncertainty $\delta a$ in its last digit, so the true value lies in $a \pm \delta a$ . Likewise, $b$ lies in $b \pm \delta b$ . Every physical measurement in this course is written this way implicitly — the last digit is the one that carries $\delta$ .

Step 2. Consider a product $P = a \times b$ . Expanding the extremes,

P_{\max} \approx ab + a\,\delta b + b\,\delta a

so the absolute uncertainty in $P$ is approximately

\delta P \approx a\,\delta b + b\,\delta a

Dividing through by $P = ab$ gives the relative uncertainty,

\frac{\delta P}{P} \approx \frac{\delta a}{a} + \frac{\delta b}{b}

Relative uncertainties add. A quantity known to 2 sig figs has a relative uncertainty around 1 part in 10–100; a quantity known to 5 sig figs has a relative uncertainty around 1 part in 10,000–100,000. When these are summed, the larger relative uncertainty (fewer sig figs) completely dominates the total. This is exactly why the product's sig-fig count collapses to match the least-precise factor — extra digits in the more precise factor are swamped and meaningless.

Step 3. Consider a sum $S = a + b$ . Here the absolute uncertainties simply add,

\delta S = \delta a + \delta b

with no rescaling by $a$ or $b$ . What matters now is not relative precision but where the uncertain digit sits — i.e., which decimal place is the last trustworthy one. If $a$ is known only to the tenths place ( $\delta a \sim 0.1$ ) and $b$ is known to the thousandths place ( $\delta b \sim 0.001$ ), then

\delta S \approx 0.1 + 0.001 = 0.101 \approx 0.1

The sum's uncertainty is dominated by the coarser decimal place, not by sig-fig counts. This is why addition and subtraction are governed by decimal places rather than total digit counts.

Step 4. Limiting case. If $a$ and $b$ are known to the same decimal place (equal precision), both rules reduce to simply keeping that shared decimal place — no information is lost or invented, confirming the rule behaves sensibly when there is no asymmetry between the inputs.

Worked examples

Example 1 — multiplication/division with mixed precision. A rectangular sample of metal has measured dimensions length $= 4.5$ cm (2 sig figs), width $= 2.36$ cm (3 sig figs), thickness $= 0.71$ cm (2 sig figs). Find its volume with correct significant figures.

V = 4.5 \text{ cm} \times 2.36 \text{ cm} \times 0.71 \text{ cm}

First multiply the raw numbers without rounding intermediate steps:

4.5 \times 2.36 = 10.62

10.62 \times 0.71 = 7.5402 \text{ cm}^3

The sig-fig rule says the answer is limited by the least precise factor. Here two factors (4.5 and 0.71) each carry only 2 sig figs, so the answer must be rounded to 2 sig figs:

V = 7.5 \text{ cm}^3

Limiting-case check: if all three dimensions had been given to 4 sig figs instead (e.g., 4.500, 2.360, 0.7100 cm), the rule would allow the full 4 sig figs in the product, $7.540\ \text{cm}^3$ — confirming that precision is preserved rather than arbitrarily discarded when every input actually supports it.

Example 2 — addition/subtraction with mixed decimal places. A chemist weighs an empty flask at $54.3$ g (1 decimal place), adds a liquid, and the filled flask reads $67.851$ g (3 decimal places). Find the mass of liquid added.

m_{\text{liquid}} = 67.851 \text{ g} - 54.3 \text{ g} = 13.551 \text{ g}

The rule for subtraction looks at decimal places, not sig figs. The term $54.3$ g is known only to the tenths place, so the answer cannot be trusted past the tenths place either:

m_{\text{liquid}} = 13.6 \text{ g}

Notice that the raw subtraction $13.551$ g has 5 sig figs, but only 3 are meaningful once rounded — this is precisely why addition/subtraction is not governed by the multiplication rule. As a limiting check: if the flask masses had instead both been recorded to 3 decimal places (say $54.300$ g and $67.851$ g), the difference $13.551$ g would stand with all three decimal places intact, since no coarser measurement is dragging the precision down.

Limiting cases & common traps

Situation	Correct treatment	Common mistake
Leading zeros (e.g., 0.0042)	Never significant — they only locate the decimal point	Counting them as significant, inflating sig-fig count
Trailing zeros with no decimal point (e.g., 1500)	Ambiguous; write as $1.5\times10^{3}$ or $1.500\times10^{3}$ to clarify	Assuming trailing zeros are always significant or always not
Captive zeros (e.g., 405)	Always significant	Ignoring zeros sandwiched between nonzero digits
Exact/counted numbers (e.g., 12 eggs, conversion 100 cm = 1 m)	Infinite sig figs; never limits a calculation	Treating a defined constant as if it had few sig figs
Mixed add/subtract then multiply	Round after each type of operation, tracking rules separately	Applying the multiplication rule to an intermediate sum, or vice versa

The chemist behind it

Significant-figure conventions are not the work of one person; they crystallized gradually through the 19th and 20th centuries as chemistry became a quantitative science demanding reproducible measurement. One towering figure who insisted on rigorous quantitative precision, setting the tone for how chemists report and trust numbers, is worth knowing.

Origins · 1 minAntoine-Laurent de Lavoisier and how this was discovered

Antoine-Laurent de Lavoisier

1743–1794 · Founder of quantitative chemistry, Paris

Lavoisier transformed chemistry from a qualitative, often mystical craft into a quantitative science built on careful mass measurements. His insistence on weighing reactants and products before and after reactions — famously demonstrating the conservation of mass in combustion — required exactly the kind of disciplined attention to instrument precision that significant figures later formalized. Without his push toward exact, reproducible measurement, the entire tradition of reporting uncertainty honestly in chemical data would have far shakier roots. He was executed during the French Revolution, cutting short a career that had already redefined what counted as rigorous chemical evidence.

Ask the tutor: If I multiply a measured value by an exact defined constant and then add a separately measured quantity, in what order should I apply the sig-fig and decimal-place rules to get the correct final answer?

Why do trailing zeros in a number like 1500 cause ambiguity?

Without additional notation, it's unclear whether the zeros were actually measured or are just placeholders locating the decimal point. Scientific notation resolves this instantly: $1.5\times10^{3}$ signals 2 sig figs, while $1.500\times10^{3}$ signals 4. This is exactly why pairing significant-figure rules with Metric Prefixes and Scientific Notation removes the ambiguity that plain decimal notation leaves open.

Do significant figure rules apply to exact numbers, like the '2' in a formula such as circumference = 2πr?

No. Exact numbers — defined constants, conversion factors, or counted quantities (like 12 items in a dozen) — carry infinite implied precision and never limit a calculation's significant figures. Only measured quantities, which carry genuine instrumental uncertainty, restrict the final answer.

How does rounding differ from truncating, and why does it matter here?

Rounding adjusts the last retained digit based on the digit that follows (round up at 5 or above, keep as is below 5, with the common convention of rounding to even on an exact half), while truncating simply chops off extra digits without adjustment. Truncating systematically biases results low, while proper rounding keeps errors balanced across many calculations — important once significant-figure results feed into further work like Dimensional Analysis.

Can a measurement have too many significant figures, and does that make it 'wrong'?

A number itself isn't wrong for having many digits, but reporting more sig figs than an instrument can support misrepresents the data's actual certainty. The number becomes misleading, not mathematically incorrect — which is why matching sig figs to instrument resolution is a matter of honest scientific communication as much as arithmetic.