Chemistry: Atoms First 2e · Essential Ideas
Measurement Uncertainty, Accuracy, and Precision
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In 30 seconds
Every measurement is imperfect: the ruler has finer divisions than your eye can resolve, the balance drifts, the thermometer lags. Uncertainty The unavoidable limit on how exactly a measurement is known. Full entry → is the honest acknowledgment that a measured value is never exact — it is an estimate with a limited number of reliable digits, called Significant figures The meaningful digits in a measurement: certain digits plus one estimated digit. Full entry →, which tell the reader how much confidence to place in a number.
This topic covers the rules for counting significant figures in measured values and calculation results; the distinction between Accuracy How close a measurement is to the true/accepted value. Full entry → (closeness to the true value) and Precision How close repeated measurements are to each other. Full entry → (reproducibility) — two ideas students constantly mix up; and percent error, the standard way to express how far a measurement sits from an accepted value.
Why this matters
- Honest data reporting. "3.0 g" vs. "3.00 g" says whether your balance reads to tenths or hundredths of a gram — significant figures state a measurement's uncertainty.
- Calculators lie. They happily report 12.4567832 from numbers measured to different precisions. Sig-fig rules separate meaningful digits from noise.
- Lab quality and patient safety. A lab test must be both accurate (close to the true concentration) and precise (repeatable). A precise-but-inaccurate instrument gives confident-looking wrong answers — the most dangerous kind.
- Exam value: sig-fig counting, sig-fig arithmetic, and accuracy-vs-precision scenarios are classic test questions.
The college version
Core Concepts
Uncertainty and significant figures
A significant figure is any digit that carries real information about a measurement — all digits known for certain plus one estimated. Reading a centimeter ruler, you can report 4.35 cm: the "4" and "3" are certain, the "5" is your best estimate between marks.
Counting rules:
- Non-zero digits are always significant (452 has 3).
- Zeros between non-zero digits are significant (4005 has 4).
- Leading zeros are never significant — they only position the decimal point (0.0045 has 2).
- Trailing zeros are significant only with a decimal point (4500 has 2; 4500. has 4; 4.500 has 4).
- Scientific notation a × 10n form that makes significant figures explicit. Full entry → removes ambiguity: 4.50 × 103 has 3 sig figs; the exponent never counts.
Exact numbers — counted values (12 eggs) and defined relationships (1 km = 1000 m) — have infinite sig figs and never limit calculations.
Arithmetic with significant figures
A result cannot be more precise than its least precise input.
- Multiplication and division: the answer matches the factor with the fewest sig figs — 2.5 × 3.14159 is limited by 2.5 (2 sig figs).
- Addition and subtraction: the answer matches the term with the fewest decimal places — 12.11 + 18.0 + 1.013 is limited by 18.0 (one decimal place).
Rounding: carry extra digits through multi-step work and round only at the end (round up on 5 or greater; some courses use "round half to even").
Accuracy vs. precision
- Accuracy is closeness to the true (accepted) value; inaccuracy comes from Systematic error Error that shifts all measurements the same way (miscalibration). Full entry → — a flaw shifting every measurement the same way (a scale reading 0.5 g high every time).
- Precision is how close repeated measurements are to each other; imprecision comes from random error — fluctuations scattering measurements (air currents, reaction time).
The classic analogy is a target. Shots clustered tightly are precise; if that spot is the bullseye, they are also accurate. Tight clusters off-center = precise but not accurate (systematic error); scattered shots near the bullseye = accurate on average but not precise (random error). The ideal is a tight cluster on the bullseye — precise and accurate. Good strategy reduces both: calibrate instruments (systematic) and repeat and average (random).
Percent error
Percent error compares a measured value to the accepted (true) value:
% error = |measured - accepted|accepted × 100%
The absolute value makes the error positive regardless of direction; a small percent error means high accuracy. Percent error measures accuracy only.
Common Confusions
| Do Not Confuse | With | Difference |
|---|---|---|
| Accuracy | Precision | Accuracy = closeness to the true value (systematic error); precision = reproducibility (random error). Precise can still be inaccurate. |
| Trailing zeros with decimal | Trailing zeros without decimal | 4500 has 2 sig figs (ambiguous); 4500. has 4; 4.500 has 4. A decimal point locks in trailing zeros. |
| Leading zeros | Captive zeros | Leading zeros (0.0045) never count; captive zeros (4005) always count. |
| Sig figs in add/subtract | Sig figs in multiply/divide | Add/subtract: fewest decimal places; multiply/divide: fewest sig figs. |
| Rounding mid-calculation | Rounding at the end | Rounding intermediate steps propagates error; round once at the end. |
| Percent error | Precision | Percent error measures accuracy only — one lucky reading can give a tiny error; consistent precise misses give a large one. |
| Counted values | Measured values | 3 beakers is exact; "3 g" is measured (1 sig fig). |
Eli explains
The same idea, in plain words
Explain it like I’m 10
When you measure something, your answer can never be perfectly exact — there is always a little guessing in the last digit. Significant figures are the "honest digits" you are sure about, plus one you had to estimate. Accuracy is how close your answer is to the real answer; precision is whether your repeated tries give nearly the same numbers. If a dart player always hits the same spot on the wall but misses the board entirely, that's precise but not accurate — practice the same spot, but aim at the bullseye!
Worked example
A student weighs a coin three times on the same balance: 12.4 g, 12.3 g, and 12.4 g. The coin's accepted mass (from the mint's specification, measured by a certified lab) is 12.48 g.
Precision: the three readings (12.4, 12.3, 12.4) spread over only 0.1 g.
Accuracy: compare the average — (12.4 + 12.3 + 12.4)/3 = 12.37 g — with the accepted value. Percent error:
% error = |measured - accepted|accepted × 100%
% error = |12.37 - 12.48|12.48 × 100% = 0.1112.48 × 100% = 0.88%
An error under 1% means the measurements are quite accurate — the balance reads about 0.1 g low.
Now apply sig-fig arithmetic to a typical calculation. A rectangular block measures length 2.35 cm (3 sig figs), width 1.4 cm (2 sig figs), height 0.52 cm (2 sig figs). Volume = length × width × height:
V = 2.35 × 1.4 × 0.52 = 1.7108 cm3
The calculator says 1.7108, but width and height carry only 2 sig figs, so the answer may show only 2: 1.7 cm³.
The clinical trap: a thermometer consistently reading 0.5 °C high (systematic error) gives precise but inaccurate readings. Calibration — comparing against a standard and correcting — fixes it. Precision alone is not enough.
Key takeaways
- Significant figures = certain digits + one estimated digit. Non-zero digits count; leading zeros never count; captive zeros count; trailing zeros count only with a decimal point.
- Scientific notation removes zero ambiguity: 4.50 × 103 = 3 sig figs; the exponent never counts.
- Exact numbers (counts, defined conversions) have infinite sig figs.
- Multiply/divide: result has the fewest sig figs of any factor. Add/subtract: result has the fewest decimal places of any term.
- Round only at the end of a multi-step calculation.
- Accuracy = closeness to true value (systematic error); precision = reproducibility (random error). They are independent — precise ≠ accurate.
- Percent error = |measured - accepted|/accepted × 100% — a measure of accuracy only.
- Target analogy: tight cluster on the bullseye = precise and accurate; tight cluster off-target = precise but not accurate.
Check yourself
6 review questions from the chapter. Try each one, then open the answer.
How many significant figures are in each: (a) 0.00450, (b) 1200, (c) 1200., (d) 4.50 × 103?
Show answer
(a) 3 — leading zeros do not count, but the trailing 0 after the decimal does; (b) 2; (c) 4 — the decimal makes trailing zeros count; (d) 3 — the exponent never counts.
A length is measured as 4.35 cm on a centimeter ruler. Which digits are certain, and which is estimated?
Show answer
The 4 and the 3 are certain; the 5 is estimated between the centimeter marks.
Calculate 3.20 × 4.1 and report the answer with the correct number of significant figures.
Show answer
3.20 × 4.1 = 13.12, but 4.1 has 2 sig figs, so the answer is 13.
Calculate 12.11 + 18.0 + 1.013 and report with the correct number of decimal places.
Show answer
12.11 + 18.0 + 1.013 = 31.123, limited by 18.0 (one decimal place), so the answer is 31.1.
Three trials give 9.8, 10.1, and 9.9 m/s² for the acceleration of gravity (accepted value 9.81 m/s²). Are the trials precise? Accurate? Compute the percent error of the average.
Show answer
The trials cluster within 0.3 of each other — precise. Average = (9.8 + 10.1 + 9.9)/3 = 9.93 m/s². Percent error = |9.93 − 9.81|/9.81 × 100% ≈ 1.2% — accurate as well.
A scale is miscalibrated to read 0.5 g high on every weighing. Is the resulting error systematic or random? Will repeating measurements fix it?
Show answer
Systematic error: every reading shifts high by the same amount. Repeating will not fix it — the same offset appears every time; only calibration corrects it.
Study tools & related lessonsKey vocabulary · Related
Key vocabulary
- Uncertainty
- The unavoidable limit on how exactly a measurement is known.
- Significant figures
- The meaningful digits in a measurement: certain digits plus one estimated digit.
- Exact number
- A counted or defined value with no uncertainty (12 eggs, 1000 m/km).
- Accuracy
- How close a measurement is to the true/accepted value.
- Precision
- How close repeated measurements are to each other.
- Systematic error
- Error that shifts all measurements the same way (miscalibration).
- Scientific notation
- a × 10n form that makes significant figures explicit.
Sources & references
This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.
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