Accuracy, Precision, and Significant Figures

Key Ideas

• Accuracy = nearness to the true value (systematic error pulls results consistently off-target).
• Precision = agreement among repeated measurements (random error scatters results).
• Exact numbers (counted objects, defined conversions like 100 cm = 1 m) have unlimited significant figures and never limit rounding.
• Measured numbers are limited by instrument resolution; the last digit is estimated (uncertain).
• Significant-figure rules for zeros:
  • Leading zeros are never significant (0.0032 → 2 sig figs).
  • Captive (interior) zeros are always significant (2008 → 4 sig figs).
  • Trailing zeros are significant only if there is a decimal point (25.00 → 4; 2500 → 2, ambiguous).
• Scientific notation removes ambiguity: 2.5 × 10³ (2 sig figs) vs. 2.500 × 10³ (4 sig figs).

Equations and Variables

This topic has rules, not equations, but two formulas frame them:

Rounding for multiplication/division: report the answer with the same number of significant figures as the factor having the fewest significant figures.

Rounding for addition/subtraction: report the answer with the same number of decimal places as the term having the fewest decimal places.

Intermediate (non-final) steps should be carried with at least one extra digit and rounded only at the end.

How It Works or Problem-Solving Method

Counting significant figures:

1. If the number has a decimal point, start at the first nonzero digit and count everything to the right of it (0.05040 → 4 sig figs).
2. If there is no decimal point, count from the first nonzero digit and stop at the last nonzero digit, treating trailing zeros as ambiguous (1500 → 2, or 3, or 4 — rewrite as 1.5 × 10³ to be clear).

Rounding rules:

• If the digit after the last kept digit is < 5, round down; if > 5, round up.
• If it is exactly 5 (or 5 followed only by zeros), round to the nearest even kept digit (banker's rounding), e.g., 2.35 → 2.4 and 2.45 → 2.4, to avoid systematic bias.

Choosing the rule: identify the operation first — multiply/divide → count sig figs; add/subtract → count decimal places. Mixed problems are done stepwise, rounding only at the very end.

Worked Example

Problem 1 (multiplication/division): Compute the density of a sample with m = 5.32 g and V = 2.1 mL.

d = m/V = 5.32 / 2.1 = 2.5333… g/mL → round to 2 significant figures (2.1 has the fewest): 2.5 g/mL.

Problem 2 (addition/subtraction): Add 12.11 g + 0.3 g + 1.234 g.

Raw sum = 13.644 g. The term 0.3 g is known only to the tenths place, so the answer must stop at the tenths place: 13.6 g.

Problem 3 (sig-fig counting): Count sig figs in (a) 0.00450, (b) 1080, (c) 1.080 × 10³.

(a) Leading zeros don't count; "450" → 3 sig figs (trailing zero counts because a decimal point is present). (b) "1080" with no decimal → the trailing zero is ambiguous; treat as 3 sig figs unless written in scientific notation. (c) 1.080 × 10³ → 4 sig figs (the trailing zero after the decimal point is significant).

Problem 4 (exact number): How many sig figs does "3 beakers" contribute to a calculation? None — a counted number is exact and has infinite sig figs, so it never limits the answer.

Common Confusions

• "0.00450 has five significant figures." Wrong — leading zeros are placeholders and never count; it has three (4, 5, and the trailing 0).
• "Precise data is automatically accurate." Wrong — a miscalibrated instrument can give a tight, reproducible but consistently wrong cluster (high precision, low accuracy).
• "For addition I round to the fewest significant figures." Wrong — addition/subtraction follows the fewest decimal places, not sig figs (13.644 g → 13.6 g, not 14 g).
• "2500 has four significant figures." Wrong — without a decimal point the trailing zeros are ambiguous; write 2.500 × 10³ if you truly know all four digits.