1.3 Accuracy, Precision, and Significant Figures

Measurements in physics require precision and accuracy. Significant figures, uncertainty, and error analysis are crucial tools for ensuring reliable results. These concepts help scientists communicate the level of confidence in their measurements and calculations.

Understanding the difference between accuracy and precision is key. Accuracy measures how close a value is to the true value, while precision refers to the consistency of repeated measurements. Both are essential for drawing valid conclusions from experiments and advancing our understanding of the physical world.

Measurement and Uncertainty

Significant figures in calculations

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• Significant figures represent the number of digits in a measured value that are known with certainty plus one estimated digit
  • Non-zero digits are always significant (1, 2, 3, etc.)
  • Zeros between non-zero digits are significant (101, 1002, etc.)
  • Leading zeros are not significant (0.01, 0.0023, etc.)
  • Trailing zeros are significant only if the decimal point is present (1.0, 2.300, etc.)
• Addition and subtraction follow the decimal place rule
  • The result should have the same number of decimal places as the measurement with the least number of decimal places (1.2 + 3.45 = 4.7)
• Multiplication and division follow the significant figure rule
  • The result should have the same number of significant figures as the measurement with the least number of significant figures (2.3 × 1.2345 = 2.8)
• Scientific notation is often used to express very large or very small numbers while maintaining significant figures

Percent uncertainty in experiments

• Absolute uncertainty represents the smallest unit of measurement that can be reliably measured by the measuring instrument (ruler with mm markings has an absolute uncertainty of 1 mm)
• Relative uncertainty is the ratio of the absolute uncertainty to the measured value
  • Expressed as a fraction or percentage (absolute uncertainty of 1 mm for a measured length of 50 mm gives a relative uncertainty of 1/50 or 0.02)
  • Calculate relative uncertainty using the formula: $\text{Relative uncertainty} = \frac{\text{Absolute uncertainty}}{\text{Measured value}}$
• Percent uncertainty is the relative uncertainty expressed as a percentage
  • Calculate percent uncertainty using the formula: $\text{Percent uncertainty} = \text{Relative uncertainty} \times 100\%$ (relative uncertainty of 0.02 gives a percent uncertainty of 2%)
• Apply percent uncertainty to experimental results by expressing the final result with the appropriate number of significant figures based on the percent uncertainty (a measured value of 10.5 cm with a percent uncertainty of 2% should be expressed as 10.5 ± 0.2 cm)
• The least count of an instrument is the smallest measurement it can reliably make, which contributes to measurement uncertainty

Accuracy vs precision in measurements

• Accuracy refers to how close a measured value is to the true or accepted value
  • High accuracy means the measured value is very close to the true value (measuring a known 100 g mass and obtaining a result of 99.8 g)
  • Instrument calibration is crucial for maintaining accuracy in measurements
• Precision refers to how close multiple measurements of the same quantity are to each other
  • High precision means the measurements are very consistent and have little variation (multiple measurements of the same object yielding 10.1 cm, 10.2 cm, and 10.1 cm)
  • Repeatability (consistency of measurements by the same person) and reproducibility (consistency of measurements by different people) are aspects of precision
• Importance in physics:
  • Accurate measurements are essential for verifying scientific theories and laws (confirming the predicted value of the gravitational acceleration, $g$)
  • Precise measurements are necessary for reproducibility and reliability of experimental results (consistent values of the speed of light, $c$, obtained by different researchers)
  • Both accuracy and precision are crucial for drawing valid conclusions from scientific experiments and advancing our understanding of the physical world (precise and accurate measurements of atomic spectra leading to the development of quantum mechanics)

Error Analysis and Propagation

• Measurement uncertainty is inherent in all physical measurements and must be accounted for
• Error propagation involves calculating how uncertainties in individual measurements affect the uncertainty of a final calculated result
• Understanding error propagation is crucial for determining the reliability of experimental conclusions