Significant Figures and Error Propagation: Worked Examples
By ChemistryIQ Team · April 26, 2026
Direct Answer: The Three Rules You Need
Three rules cover almost every general chemistry significant-figures question. (1) Counting sig figs: all non-zero digits count. Zeros between non-zero digits count. Trailing zeros count if there is a decimal point. Leading zeros do NOT count. Scientific notation removes ambiguity (3.20 × 10^2 has 3 sig figs, unambiguous). (2) Addition/subtraction rule: the answer keeps the SAME NUMBER OF DECIMAL PLACES as the input with the FEWEST decimal places. (3) Multiplication/division rule: the answer keeps the SAME NUMBER OF SIGNIFICANT FIGURES as the input with the FEWEST sig figs. For mixed operations, apply each rule at its appropriate step but carry extra digits through the calculation; round only at the end.
Counting Sig Figs: Worked Example 1
How many sig figs in each number?
• 0.00420 → 3 sig figs (the leading zeros don't count, but the trailing zero does because of the decimal point) • 5,400 → ambiguous (could be 2, 3, or 4 sig figs); write as 5.40 × 10^3 to specify 3 sig figs • 5,400. → 4 sig figs (the decimal point makes the trailing zeros count) • 1.0089 → 5 sig figs (all digits including the interior zeros) • 6.022 × 10^23 → 4 sig figs (only the coefficient digits matter; the exponent is exact) • 100.0 → 4 sig figs (the decimal point makes all digits count)
The ambiguity in numbers like 5,400 is exactly why scientific notation is preferred in scientific writing — there is no way to misread 5.40 × 10^3.
Addition and Subtraction: Worked Example 2
Add: 23.456 + 1.2 + 0.083 = ?
The input with the fewest decimal places is 1.2 (one decimal place). The answer must have one decimal place.
Sum: 24.739 Rounded: 24.7
Note that this rule is about DECIMAL PLACES, not sig figs. The answer 24.7 has 3 sig figs, but the constraint is the one decimal place from 1.2. If you had added 23,456 + 1,200,000 + 83, the answer would round to the nearest thousand (the precision of 1,200,000 expressed with 2 sig figs gives a precision of 100,000, but the absolute precision being matched is the lowest decimal place, which here means the nearest ten thousand or hundred thousand depending on interpretation — this is why scientific notation matters).
Multiplication and Division: Worked Example 3
Multiply: 4.567 × 2.3 = ?
The input with the fewest sig figs is 2.3 (2 sig figs). The answer must have 2 sig figs.
Product: 10.5041 Rounded: 11 (2 sig figs)
Divide: 12.456 / 3.0 = ?
The input with the fewest sig figs is 3.0 (2 sig figs). The answer must have 2 sig figs.
Quotient: 4.152 Rounded: 4.2 (2 sig figs)
Watch the rounding: 4.152 rounds to 4.2 (the next digit after the second sig fig is 5, but we look at the digit AFTER the rounding spot — here it's actually a 5 followed by 2, which means round up).
IMPORTANT: Carry one or two extra digits through multi-step calculations and round only at the END. Rounding intermediate results compounds errors.
Mixed Operations: Worked Example 4
Calculate the molarity of a solution made by dissolving 4.567 g of NaCl (molar mass 58.44 g/mol) in 250.0 mL of water.
Step 1: Moles = mass / molar mass = 4.567 / 58.44 Step 1 has 4 sig figs / 4 sig figs = 4 sig figs allowed at this step Intermediate: 0.07815 mol (carry 4 sig figs into step 2)
Step 2: Volume in L = 250.0 mL × (1 L / 1000 mL) The 1000 conversion is EXACT (defined), so 250.0 mL = 0.2500 L (4 sig figs)
Step 3: Molarity = moles / volume = 0.07815 / 0.2500 Limiting factor: 4 sig figs (both inputs) Result: 0.3126 M
Final answer: 0.3126 M (4 sig figs).
Key insight: exact conversions (1 L = 1000 mL, 1 mol = 6.022 × 10^23 molecules — well, 6.022 has 4 sig figs but is treated as more by convention because Avogadro's number is known to higher precision than experimental data) don't affect sig fig counts. Only measured quantities limit sig figs.
Log and Exponential Functions: Worked Example 5
When taking log of a number, the rule changes: the number of decimal places in the LOG equals the number of sig figs in the original number.
Example 1: pH calculation. [H+] = 1.5 × 10^-4 M (2 sig figs) pH = -log(1.5 × 10^-4) = 3.82
The pH has 2 DECIMAL PLACES because the [H+] had 2 sig figs. This is the source of the 'pH 7.00' confusion — pH 7.0 has 2 sig figs in the original concentration; pH 7.00 has 3 sig figs.
Example 2: pKa from Ka. Ka = 6.5 × 10^-5 (2 sig figs) pKa = -log(6.5 × 10^-5) = 4.19
pKa has 2 decimal places.
Reverse direction: if pH = 3.82 (2 decimal places), then [H+] = 10^-3.82 = 1.5 × 10^-4 M (2 sig figs). The exponential 'inverts' the rule: 2 decimal places in pH → 2 sig figs in [H+].
Error Propagation: Addition and Subtraction
When adding or subtracting measurements with uncertainties, the ABSOLUTE uncertainties add in quadrature (assuming the errors are independent and random).
Formula: σ(A ± B) = √(σ_A^2 + σ_B^2)
Worked Example: A balance reads (12.45 ± 0.02) g. A second balance reads (3.18 ± 0.01) g. Sum?
Sum: 12.45 + 3.18 = 15.63 g Uncertainty: √(0.02^2 + 0.01^2) = √(0.0004 + 0.0001) = √0.0005 ≈ 0.022 ≈ 0.02 g
Final: (15.63 ± 0.02) g
If the errors were correlated (same balance, systematic), they would simply add: 0.02 + 0.01 = 0.03. Random errors propagate in quadrature; systematic errors propagate linearly. Most lab measurements assume random unless explicitly noted.
Error Propagation: Multiplication and Division
When multiplying or dividing, the RELATIVE (fractional) uncertainties add in quadrature.
Formula: σ(A×B)/(A×B) = √((σ_A/A)^2 + (σ_B/B)^2)
Worked Example: Calculate the density of a solid with mass (4.55 ± 0.02) g and volume (1.85 ± 0.05) cm³.
Density: 4.55 / 1.85 = 2.459 g/cm³ Relative uncertainties: σ_m/m = 0.02/4.55 = 0.0044 = 0.44%; σ_V/V = 0.05/1.85 = 0.027 = 2.7% Combined relative uncertainty: √(0.0044^2 + 0.027^2) = √(0.0000194 + 0.000729) ≈ √0.000748 ≈ 0.0274 ≈ 2.7%
Absolute uncertainty in density: 2.459 × 0.0274 ≈ 0.067 g/cm³
Final: (2.46 ± 0.07) g/cm³
Observe that the volume uncertainty (2.7%) dominates the mass uncertainty (0.4%) in the combined result. The biggest relative uncertainty drives the propagation. To improve the density measurement, focus on improving volume measurement, not mass measurement.
Common Mistakes Students Make
(1) Treating 'sig figs' and 'decimal places' as the same — they aren't. Addition uses decimal places; multiplication uses sig figs. (2) Rounding intermediate results in multi-step calculations — always carry extra digits and round at the end. (3) Treating exact constants (1 L = 1000 mL, ratios from chemical formulas like '2 mol H per 1 mol H2O') as limiting sig figs — they are exact and don't limit. (4) Forgetting the log rule — pH should match the sig figs of [H+] in DECIMAL PLACES, not sig figs. (5) Adding uncertainties linearly when they should be added in quadrature (only for independent random errors). ChemistryIQ walks through each rule with practice problems graded for the right number of sig figs and the correct uncertainty propagation, which is the fastest way to internalize these rules for lab reports and exams.
FAQs
Common questions about significant figures and error propagation
Because sig figs communicate the precision of the measurement. An answer of 4 g and an answer of 4.0000 g are not equivalent — the second claims precision to 0.0001 g, which is a much stronger claim. In a lab report, claiming more precision than your instruments support is dishonest data presentation; claiming less precision wastes information. Both directions matter.
Without additional notation, 5400 is ambiguous — it could be 2, 3, or 4 sig figs. The convention is that trailing zeros in numbers WITHOUT a decimal point are not counted as significant, so 5400 is treated as 2 sig figs by default. To specify 3 sig figs, write 5.40 × 10^3. To specify 4 sig figs, write 5400. (with a trailing decimal point) or 5.400 × 10^3. Scientific notation eliminates the ambiguity entirely.
Logarithms transform the structure of numbers — the integer part of the log corresponds to the order of magnitude (the exponent in scientific notation), and the decimal part corresponds to the mantissa (the coefficient). Only the mantissa carries information about precision. So 2 sig figs in the original number translates to 2 decimal places in the log. This is why pH values are usually reported to 2 decimal places — typical pH meter precision corresponds to 2 sig figs in [H+].
Use quadrature for INDEPENDENT, RANDOM errors — measurements where the error in one quantity has no relationship to the error in another. Use linear addition for CORRELATED or SYSTEMATIC errors — for example, if both measurements use the same balance with a systematic bias, errors don't cancel and add linearly. Most introductory lab courses assume independent random errors and use quadrature unless explicitly stated otherwise.
No. Defined exact conversions (1 L = 1000 mL, 1 inch = 2.54 cm exactly, 1 mole = 6.02214076 × 10^23 by definition) have unlimited sig figs and don't limit your answer. Only measured quantities (with finite precision) limit sig figs. This is why writing '0.2500 L' from '250.0 mL' is correct — you didn't lose precision in the conversion.
Yes. Provide your calculation steps with each measured quantity and its uncertainty, and ChemistryIQ walks through the sig fig analysis at each step, applies the appropriate rule (decimal places for addition/subtraction, sig figs for multiplication/division, decimal places for logs), and propagates uncertainty correctly using quadrature for independent random errors. Especially useful for lab reports where reviewers grade on both calculation correctness and proper precision reporting.