1.6 Significant Figures

Measurements in physics aren't perfect. We use significant figures to show how precise our numbers are. This helps us understand the limits of our data and calculations.

Accuracy, precision, and uncertainty are key concepts in measurements. They help us gauge how reliable our results are and how to interpret them in experiments and real-world applications.

Significant Figures and Measurement Uncertainty

Significant figures in calculations

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Significant figures represent the meaningful digits in a measurement or calculation All non-zero digits are significant (1, 2, 3, 4, 5, 6, 7, 8, 9) Zeros between non-zero digits are significant (1.02 has three significant figures) Leading zeros are not significant (0.0012 has two significant figures) Trailing zeros are significant only if the decimal point is present (1.00 has three significant figures, but 100 has only one) When performing calculations, the result should have the same number of significant figures as the input with the least number of significant figures For addition and subtraction, the result should have the same number of decimal places as the input with the least number of decimal places (2.1 + 3.42 = 5.5) For multiplication and division, the result should have the same number of significant figures as the input with the least number of significant figures (2.1 × 3.42 = 7.2) • The position of the decimal point in a number does not affect its significant digits

Accuracy vs precision vs uncertainty

Accuracy measures how close a measured value is to the true value High accuracy means the measured value is very close to the true value (measuring a 10 cm object as 9.9 cm) Low accuracy means the measured value is far from the true value (measuring a 10 cm object as 15 cm) Precision measures how close multiple measurements are to each other, regardless of their accuracy High precision means multiple measurements are very close to each other (measuring a 10 cm object as 9.1 cm, 9.2 cm, and 9.1 cm) Low precision means multiple measurements are spread out (measuring a 10 cm object as 8 cm, 10 cm, and 12 cm) Uncertainty represents the range of values within which the true value is expected to lie Uncertainty is typically expressed as a ± value (10.0 ± 0.1 cm) Lower uncertainty indicates a more precise measurement (10.0 ± 0.1 cm has lower uncertainty than 10 ± 1 cm) • Experimental error refers to the difference between a measured value and the true value

Percent uncertainty calculation

Percent uncertainty expresses the uncertainty of a measurement relative to the measured value To calculate percent uncertainty:

1. Determine the absolute uncertainty (the ± value)
2. Divide the absolute uncertainty by the measured value
3. Multiply by 100 to express as a percentage Formula for percent uncertainty: $\text{Percent Uncertainty} = \frac{\text{Absolute Uncertainty}}{\text{Measured Value}} \times 100\%$ Example: If a mass is measured as 50 ± 1 g, the percent uncertainty is $\frac{1}{50} \times 100\% = 2\%$

Propagation of uncertainty rules

When performing calculations with measured values, the uncertainties in the input values propagate to the final result For addition and subtraction: $\text{Uncertainty in Result} = \sqrt{(\text{Uncertainty 1})^2 + (\text{Uncertainty 2})^2 + \ldots}$ Example: If $L = 10.0 \pm 0.1$ cm and $W = 5.0 \pm 0.2$ cm, then for $L + W$, the uncertainty is $\sqrt{(0.1)^2 + (0.2)^2}$ cm $= 0.22$ cm For multiplication and division: $\text{Percent Uncertainty in Result} = \sqrt{(\text{Percent Uncertainty 1})^2 + (\text{Percent Uncertainty 2})^2 + \ldots}$ Example: If $L = 10.0 \pm 0.1$ cm (1% uncertainty) and $W = 5.0 \pm 0.2$ cm (4% uncertainty), then for $L \times W$, the percent uncertainty is $\sqrt{(1\%)^2 + (4\%)^2} = 4.1\%$

Measurement and Significant Digits

• A measurement is a quantitative observation of a physical property • Significant digits (or significant figures) are all the digits in a measurement that are known with certainty, plus one estimated digit • The number of significant digits in a measurement reflects its precision