1.5 Measurement Uncertainty, Accuracy, and Precision

Learning Objectives

Define accuracy and precision
Distinguish exact and uncertain numbers
Represent uncertainty using significant figures
Apply proper rounding rules to computed quantities

Core Concepts & Principles (De-Verbosed)

Every scientific measurement carries a degree of uncertainty due to the physical limits of measuring tools and human estimation. Managing this uncertainty correctly is essential for reliable calculations and data interpretation.

Exact vs. Uncertain Numbers

Exact Numbers: Values known with complete certainty, free from measurement error. These include counted objects (e.g., exactly 12 eggs in a carton) and defined conversion factors (e.g., $1 \text{ foot} = 12 \text{ inches}$ ).
Uncertain Numbers: Values obtained from physical instruments (mass, volume, length). Every physical measurement involves estimating one digit at the instrument's finest scale division.

The Rule of Measurement Readings

Always record all digits known with certainty plus one final estimated digit. This last uncertain digit represents the limit of precision for that specific tool.

Significant Figures

All digits in a measurement—including the uncertain last digit—are called significant figures (or significant digits).

Rules for Counting Significant Figures:

Nonzero digits are always significant.
Captive zeros (between nonzero digits) are always significant (e.g., $40.7 \text{ g}$ has 3 sig figs).
Leading zeros (before the first nonzero digit) are never significant; they only indicate decimal position (e.g., $0.0042 \text{ m}$ has 2 sig figs).
Trailing zeros (at the end of a number) are significant if they follow a decimal point ( $12.00 \text{ mL}$ ). If they sit to the left of an unwritten decimal point ( $1,300 \text{ g}$ ), they are ambiguous and should be written in scientific notation to clarify (e.g., $1.3 \times 10^3$ vs. $1.300 \times 10^3$ ).

Accuracy vs. Precision

Precision: How closely repeated measurements agree with one another.
Accuracy: How closely a measured value agrees with the true or accepted value.

Accuracy vs. Precision Mental Model

Think of an archery target: Accuracy means hitting the bullseye. Precision means clustering your arrows tightly together, even if they miss the bullseye entirely.

Problem-Solving Routines & Methods

Performing Calculations with Significant Figures

1
Complete the mathematical operation on your calculator to obtain the raw, unrounded result.
2
Identify whether the operation is addition/subtraction or multiplication/division.
3
Apply the specific rounding rule and report the final value with proper sig figs or decimal places.

Pro-Tip: Never round numbers mid-calculation; keep all calculator digits until the final step!

Sig Figs in Addition/Subtraction

\text{Addition & Subtraction Rule: Match fewest decimal places}

Round the final result to match the same number of decimal places as the least certain input value.

Variables: \text{e.g., } 486 - 421.23 = 64.77 \rightarrow 65 \text{ (rounded to ones place)}

Sig Figs in Multiplication/Division

\text{Multiplication & Division Rule: Match fewest sig figs}

Round the final result to match the total number of significant figures in the input value with the fewest sig figs.

Variables: \text{e.g., } 0.6238 \times 6.6 = 4.11708 \rightarrow 4.1 \text{ (2 sig figs)}

Practice & Concept Checks

Concept Check

A student measures the mass of a sample three times and gets: 4.12 g, 4.13 g, and 4.12 g. The true mass is 5.50 g. Are these measurements accurate, precise, both, or neither?

Concept Check

How many significant figures are in the measurement 0.005040 L?

Key Terms & Vocabulary

exact numberMeasurements

A value known with complete certainty, obtained through counting or definition.

Example: 1 foot = 12 inches

uncertaintyMeasurements

A quantitative estimate of how much a measured value deviates from the actual result.

Example: 6.72 g ± 0.01 g

significant figuresData Analysis

All digits in a measurement known with certainty plus one final estimated digit.

Example: 21.6 mL has 3 significant figures

significant digitsData Analysis

Alternative terminology for significant figures reflecting measurement certainty.

Example: 4.50 has 3 significant digits

roundingMath Operations

The procedure of reducing digits in a number while keeping its value close, using standard tie-breaking rules.

Example: 0.02867 rounds to 0.0287

precisionExperimental Quality

The degree of agreement among repeated measurements of the same quantity.

Example: Getting 10.1 mL, 10.2 mL, and 10.1 mL on repeated trials

accuracyExperimental Quality

The degree of closeness between a measured value and the true or accepted reference value.

Example: A measured density of 7.8 g/cm³ compared to a true value of 7.9 g/cm³