Significant figures and measurement uncertainty help students report experimental data honestly and clearly. This topic explains how precise a measurement is, how many digits should be kept, and how uncertainty affects calculations. Students need this cheat sheet because physics answers are not complete unless the units, precision, and uncertainty make sense.
It is especially useful for labs, graphing, and problem solving with measured values.
The core ideas are that every measured value has limited precision and should be written with the correct number of significant figures. Addition and subtraction are rounded by decimal place, while multiplication and division are rounded by the number of significant figures. Absolute uncertainty uses units, while percent uncertainty compares uncertainty to the measured value using .
When measurements are combined, uncertainties must be propagated so the final answer does not look more precise than the data allow.
Key Facts
- All nonzero digits are significant, so has significant figures.
- Zeros between nonzero digits are significant, so has significant figures.
- Leading zeros are not significant, so has significant figures.
- Trailing zeros after a decimal point are significant, so has significant figures.
- For addition and subtraction, round the final answer to the least number of decimal places, such as .
- For multiplication and division, round the final answer to the least number of significant figures, such as .
- Percent uncertainty is calculated with , where is the measured value and is the absolute uncertainty.
- For powers, relative uncertainty is multiplied by the power, so if , then .
Vocabulary
- Significant Figures
- The meaningful digits in a measured or calculated value, including all certain digits and one estimated digit.
- Precision
- How closely repeated measurements agree with one another or how finely a measuring tool can measure.
- Accuracy
- How close a measured value is to the accepted or true value.
- Absolute Uncertainty
- The size of the possible error in a measurement, written with units as .
- Percent Uncertainty
- The uncertainty compared with the measured value, calculated as .
- Uncertainty Propagation
- The process of determining how measurement uncertainties affect a calculated result.
Common Mistakes to Avoid
- Counting leading zeros as significant figures is wrong because zeros before the first nonzero digit only locate the decimal point, such as in .
- Rounding too early is wrong because intermediate rounding can change the final result, so keep extra digits until the last step.
- Using the multiplication rule for addition is wrong because addition and subtraction are rounded by decimal places, not by total significant figures.
- Writing an answer with too many digits is wrong because it suggests more precision than the measuring tools provide.
- Forgetting units on uncertainty is wrong because absolute uncertainty is a measured quantity, such as .
Practice Questions
- 1 How many significant figures are in ?
- 2 Calculate and round using the correct significant figure rule.
- 3 A student measures a length as . What is the percent uncertainty?
- 4 A measurement is precise but not accurate. Explain what this means in terms of repeated trials and the accepted value.