Lecture Notes: Accuracy, Precision, and Significant Figures

Introduction

• Understanding accuracy and precision is essential in measurement.
• Exact numbers come from counting or defined quantities.
• Uncertain numbers come from measurements and have varying degrees of uncertainty.

Definitions

• Accuracy: How close a measurement is to the true value.
• Precision: How repeatable a measurement is.

Exact vs. Uncertain Numbers

• Exact Numbers: Results of counting (e.g., 1 egg, 2 eggs) or defined quantities (e.g., 12 inches in a foot).
• Uncertain Numbers: Derived from measurements, always contain some level of uncertainty.

Representing Uncertainty

• Measurements should indicate uncertainty; estimate one uncertain digit.
• Significant Figures: All measured digits in a number, including uncertain ones.

Significant Figures

• Non-zero digits, captive zeros, and trailing zeros to the right of the decimal are significant.
• Leading zeros and trailing zeros to the left of the decimal are not significant.

Examples

• 0.00802 has three significant figures (8, 0 (captive), 2).
• 3090 has three significant figures (3, 0 (captive), 9).

Mathematical Operations and Significant Figures

• Addition/Subtraction: Round result to the least number of decimal places.
• Multiplication/Division: Round result to the least number of significant figures.
• Rounding Rules:
  • <5: Round down.

  • 5: Round up.

  • =5: Round to achieve even retained digit.

Rounding Examples

• 2.676 -> 2.67 (round down if digit <5)
• 18.35 -> 18.3 (round down if digit <5)
• 6.875 -> 6.88 (round up if digit is 5 to make even)
• 92.85 -> 92.8 (round down if digit is 5 to make even)

Accuracy vs. Precision

• Accuracy: Closeness to true value.
• Precision: Consistency of results in repeated measurements.

Visualization

• Accurate & Precise: Close to true value and repeatable.
• Precise Only: Consistently hitting the same spot, not near true value.
• Neither: Results are scattered, not repeatable or close to true value.

Conclusion

• Understanding and applying these concepts is crucial in scientific measurements to ensure validity and reliability of data.