Engineering measurements are never exact, so engineers use significant figures and error estimates to communicate how much confidence a value deserves. A digital caliper reading such as 12.36 mm does not mean the object is perfectly known, only that the measurement is limited by the tool and method. Significant figures keep calculations honest by preventing a final answer from looking more precise than the data used to produce it.
This matters in design, manufacturing, testing, and safety decisions where small differences can affect whether a part fits or fails.
Uncertainty is described using absolute error, relative error, percent error, and error propagation rules. When measured quantities are combined in formulas, their uncertainties combine too, often in predictable ways. For example, if a cylinder diameter and length are measured with a caliper, the uncertainty in its calculated volume depends on both measurements and on how the formula uses them.
A clear engineering report gives the value, the uncertainty, the units, and a sensible number of significant figures.
Key Facts
- Significant figures are the meaningful digits in a measured or calculated value, including all certain digits plus one estimated digit.
- Absolute error: absolute error = |measured value - accepted value|.
- Relative error: relative error = absolute error / |accepted value|.
- Percent error: percent error = (absolute error / |accepted value|) x 100%.
- For addition and subtraction, round the final result to the least precise decimal place among the inputs.
- For multiplication and division, round the final result to the same number of significant figures as the input with the fewest significant figures.
Vocabulary
- Significant figures
- The digits in a number that communicate the precision of a measurement or calculation.
- Precision
- How closely repeated measurements agree with each other.
- Accuracy
- How close a measured value is to the true or accepted value.
- Uncertainty
- A numerical estimate of the possible range around a measured value.
- Error propagation
- The process of determining how measurement uncertainties affect the uncertainty of a calculated result.
Common Mistakes to Avoid
- Reporting too many digits in the final answer, which makes the result look more precise than the measurements allow.
- Confusing precision with accuracy, because repeated measurements can be close to each other but still far from the true value.
- Rounding intermediate steps too early, which can increase rounding error and change the final result.
- Ignoring units when calculating error, because absolute error has the same units as the measurement while relative error and percent error do not.
Practice Questions
- 1 A digital caliper measures a cylinder diameter as 18.42 mm with an uncertainty of plus or minus 0.02 mm. What is the relative uncertainty and percent uncertainty in the diameter?
- 2 A metal rod has measured length 12.4 cm and width 3.16 cm. Calculate the area and round the answer to the correct number of significant figures.
- 3 Two teams measure the same part. Team A gets 10.02 mm, 10.03 mm, and 10.02 mm, while Team B gets 9.91 mm, 10.10 mm, and 10.00 mm. If the accepted value is 10.00 mm, compare the precision and accuracy of the two teams.