1.5 Measurement Uncertainty, Accuracy, and Precision

Counting as Measurement

Only measurement free of uncertainty. Example: 12 eggs counted exactly.

Sources of Measurement Uncertainty

Instrument limitations, human interpretation (e.g., reading a meniscus).

Defined Quantities

Exact numbers with no uncertainty. Examples: 1 foot = 12 inches, 1 inch = 2.54 cm, 1 g = 0.001 kg.

Accuracy

How close a measured value is to the true or accepted value. Example: Hitting the bullseye.

Precision

How close repeated measurements are to each other, regardless of accuracy. Example: Hitting the same spot repeatedly, even if far from bullseye.

Accuracy vs Precision

A measurement can be accurate but not precise, precise but not accurate, both, or neither.

Significant Figures (Sig Figs)

All digits known with certainty plus one estimated digit. Example: 21.6 mL → 3 sig figs.

Sig Fig Rule: Non-zero Digits

Always significant. Example: 45.2 → 3 sig figs.

Sig Fig Rule: Captive Zeros

Always significant. Example: 1003 → 4 sig figs.

Sig Fig Rule: Leading Zeros

Never significant; indicate decimal placement. Example: 0.00456 → 3 sig figs.

Sig Fig Rule: Trailing Zeros with Decimal

Always significant. Example: 2.300 → 4 sig figs.

Sig Fig Rule: Trailing Zeros without Decimal

Ambiguous; use scientific notation to clarify. Example: 1.3×10³ → 2 sig figs, 1.30×10³ → 3 sig figs.

Reporting Measurement Uncertainty

Measurement must reflect device precision. Example: 6.72 g → ±0.01 g; 6.723 g → ±0.001 g.

Meniscus Reading

Read at bottom of meniscus; estimate one digit beyond smallest marking. Example: 21.6 mL (3 sig figs).

Sig Figs in Multiplication & Division

Result has same number of sig figs as measurement with fewest sig figs. Example: 4.56 × 1.4 = 6.4 (2 sig figs).

Sig Figs in Addition & Subtraction

Result rounded to same decimal place as measurement with least precision. Example: 12.11 + 18.0 + 1.013 = 31.1.

Rounding Rules

Exact Number

Known with complete certainty, no estimation. Examples: Counting objects (12 eggs), defined conversions (1 inch = 2.54 cm).

Uncertain Number

Measurement with inherent uncertainty. Examples: 21.6 mL, 6.72 g.

Real-World Example of Uncertainty

U.S. Census 2014: 317,297,725 is not exact due to ongoing births/deaths; report as 317 million or 3.17×10⁸.

Key Takeaway: Accuracy vs Precision

Accuracy = closeness to true value, Precision = consistency of measurements.

Key Takeaway: Sig Figs

Express measurement precision; scientific notation avoids ambiguity.

Key Takeaway: Calculations

Follow sig fig rules for multiplication/division, addition/subtraction, and rounding.