Chemistry 2e · Essential Ideas

Measurement Uncertainty, Accuracy, and Precision

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On this page 9 sections

In 30 seconds
Why this matters
The college version
Eli explains
Worked example
Key takeaway
Check yourself
Study tools
Sources & references

In 30 seconds

No measurement is perfect. Every reading carries — the last digit you write down is an estimate — so a value is meaningful only if you know how trustworthy it is. Scientists communicate trustworthiness with (the digits that carry real information) and with two quality words students constantly mix up: (closeness to the true value) and (closeness of repeated measurements to each other). This topic gives the counting rules for significant figures, the arithmetic rules for carrying them through calculations, and the accuracy-versus-precision framework for judging data quality.

Why this matters

Reporting "2.0 g" versus "2.00 g" makes different claims: the first says the mass is known to one decimal place, the second to two. In a pharmacy, that precision difference can matter for potent drugs. Asking is the data accurate, precise, both, or neither? tells you whether a method is reliable and whether an instrument is biased. On exams, significant-figure arithmetic is guaranteed points if you follow the rules.

The college version

Core Concepts

Uncertainty is built into every measurement

A measurement consists of all the digits you are certain of, plus one digit you estimate. If a graduated cylinder is marked in 1 mL divisions, you read to the nearest 0.1 mL by estimating between marks — "25.3 mL" means 25 is certain and 0.3 is your best estimate. Digital instruments show their uncertainty in the last digit.

Accuracy versus precision

Accuracy describes how close a measured value is to the accepted (true) value. Precision describes how close repeated measurements are to one another — the reproducibility of the method. The archery-target analogy gives four cases: a tight cluster at the bullseye is accurate and precise (ideal); a tight cluster off-center is precise but not accurate (, e.g., an uncalibrated balance); a scattered ring around the center is accurate on average but imprecise (random errors); a scattered cluster far from the center is neither. Systematic errors shift every result the same way; random errors scatter results unpredictably.

Significant figures: the counting rules

Significant figures are the digits that carry meaning. The rules: (1) all nonzero digits are significant (123 has 3); (2) zeros between nonzero digits are significant (1005 has 4); (3) leading zeros are not significant — they only place the decimal (0.0012 has 2); (4) trailing zeros are significant only if the number contains a decimal point (1200 has 2, but 1200. has 4 and 1.200 has 4); (5) exact numbers — counted objects and defined conversion factors (5 coins, 1000 mL = 1 L) — have unlimited significant figures; (6) removes ambiguity: 1.2 × 10³ has 2 significant figures, 1.200 × 10³ has 4.

Rounding

Carry all digits through the calculation and round only at the end. Standard rounding: if the first digit to be dropped is 5 or greater, round up; if it is 4 or less, round down. For example, 81.51 rounds to 82 with two significant figures.

Significant figures in calculations

The rules differ by operation. For multiplication and division, the answer has the same number of significant figures as the factor with the fewest. For addition and subtraction, the answer has the same number of decimal places as the term with the fewest decimal places. Mixed problems apply each rule at the corresponding step. Reason: multiplication uncertainty is relative, while addition uncertainty is absolute.

Quantifying accuracy: percent error

Percent error compares a measured value with the accepted value:

$% error = | measured - accepted | accepted \times 100 %$

A small percent error means high accuracy; a large one flags a systematic problem or a blunder.

How It Works / Step-by-Step Process

Record the measurement with all certain digits plus one estimated digit.
Count significant figures using the five counting rules; use scientific notation if trailing zeros are ambiguous.
Perform the calculation, keeping extra digits until the end.
Apply the operation rule: fewest significant figures for ×/÷, fewest decimal places for +/−.
Round the final answer and, if an accepted value exists, compute percent error.

Common Confusions

Do Not Confuse	With	Difference
Precision	Accuracy	Precision is repeatability; accuracy is closeness to truth. Tight but off-center results are precise, not accurate.
Multiplication rule	Addition rule	×/÷ uses fewest significant figures; +/− uses fewest decimal places — applying the wrong rule is a top exam error.
Exact numbers in sig-fig limits	Measured numbers	Exact numbers (5 coins, 1000 mL = 1 L) have unlimited significant figures and never limit an answer.

Eli explains

The same idea, in plain words

Explain it like I’m 10

When you measure anything, the last number you write down is a careful guess — "2.0 g" means you're sure to one decimal place, and "2.00 g" means you're sure to two. Accuracy is how close your answer is to the real value; precision is how close your repeated tries are to each other. Think darts: hitting near the bullseye is accurate, and landing in a tight bunch is precise — you can be precise without being accurate if you always aim at the wrong spot. Significant-figure rules tell you how many digits your answer honestly deserves.

Worked examples

A rectangle measures 24.7 cm by 3.3 cm. Compute its area. First, the formula and substitution:

$area = 24.7 cm \times 3.3 cm = 81.51 cm 2$

24.7 has three significant figures; 3.3 has two. For multiplication, the answer takes the fewest significant figures, so the result rounds to two: 82 cm². Reporting 81.51 cm² would overstate the precision, because 3.3 cm does not justify four digits.

Three masses are added: 103.42 g + 0.0041 g + 12.5 g. First, the sum:

$103.42 + 0.0041 + 12.5 = 115.9241 g$

For addition, the answer takes the fewest decimal places. The terms have 2, 4, and 1 decimal places, so the answer has 1 decimal place: 115.9 g — the 12.5 g term is the least precisely known.

A student measures the density of a liquid three times: 1.28, 1.29, and 1.27 g/mL; the accepted value is 1.250 g/mL. The results agree closely (spread of 0.02 g/mL), so the method is precise. The mean is 1.28 g/mL, above the accepted value, so the results are systematically high — accurate only to a limited degree. Quantify the accuracy of the mean:

$% error = |1.28 - 1.250| 1.250 \times 100 % = 0.030 1.250 \times 100 % = 2.4 %$

The 2.4% error suggests a small systematic bias (perhaps an uncalibrated balance), not random scatter — a precise-but-not-quite-accurate data set.

Key takeaways

Every measurement has uncertainty; the last digit is estimated.
Accuracy = closeness to the accepted value; precision = agreement among repeats. Precise ≠ accurate (systematic error).
Count significant figures: nonzero digits always; interior zeros yes; leading zeros no; trailing zeros only with a decimal point.
Exact numbers (counted objects, defined conversions) have unlimited significant figures.
Multiplication/division: fewest significant figures. Addition/subtraction: fewest decimal places.
Round only at the end of a calculation.
Scientific notation removes trailing-zero ambiguity: 1.2 × 10³ vs 1.200 × 10³.
Percent error quantifies accuracy: |measured - accepted|accepted × 100%.

Check yourself

6 review questions from the chapter. Try each one, then open the answer.

How many significant figures are in each: 0.00450, 1005, 1000, 1.00 × 10³?
Show answer
0.00450 has 3 (leading zeros not significant, trailing zero after the decimal is). 1005 has 4 (interior zeros significant). 1000 has 1 (trailing zeros, no decimal). 1.00 × 10³ has 3.
Compute 3.20 × 4.1 and report the answer with the correct number of significant figures.
Show answer
3.20 × 4.1 = 13.12, rounded to 2 significant figures (4.1 has 2): 13.
Compute 25.46 + 3.7 and report the answer with the correct number of significant figures.
Show answer
25.46 + 3.7 = 29.16, rounded to 1 decimal place (3.7 has 1): 29.2.
A balance reads 12.3456 g, and three weighings of the same object give 12.35, 12.36, and 12.35 g; the accepted mass is 12.50 g. Is the balance accurate? Is it precise? Compute the percent error of the mean.
Show answer
Precise (values agree within 0.01 g) but not accurate (all well above 12.50 g — systematic bias). Mean = 12.3533 g; percent error = |12.3533 - 12.50|12.50 × 100% = 1.2%.
Why is a counted value such as "5 coins" treated as having unlimited significant figures?
Show answer
Counting is exact: there are exactly 5 coins, with no measurement uncertainty, so it cannot limit the significant figures of a calculated result.
A 2.50 L sample of water has a mass of 2.50 kg. Report its density in g/mL with the correct significant figures (1 L = 1000 mL, 1 kg = 1000 g).
Show answer
ρ= 2500 g2500 mL = 1.00 g/mL. Both inputs have 3 significant figures (2.50 L = 2.50 × 10³ mL; 2.50 kg = 2.50 × 10³ g), so the answer keeps 3: 1.00 g/mL.

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Study tools & related lessonsKey vocabulary · Related

Key vocabulary

uncertainty: The inherent doubt in a measurement, shown by its last estimated digit.
accuracy: Closeness of a measurement to the accepted (true) value.
precision: Closeness of repeated measurements to each other.
significant figures: The digits in a measurement that carry real information.
exact number: A counted or defined value with unlimited significant figures.
systematic error: An error that shifts all measurements the same way.
random error: An error that scatters measurements unpredictably.
scientific notation: Writing numbers as a × 10ⁿ with one digit before the decimal.

Sources & references

openstax.org — Chemistry 2e

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