Citation:
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 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 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 expresses the uncertainty of a measurement relative to the measured value To calculate percent uncertainty:
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 $\sqrt{(1%)^2 + (4%)^2} = 4.1%$
• 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