Learning outcomes
- Use accuracy and precision correctly.
- Distinguish random, systematic and measurement errors.
- Explain repeatability and reproducibility.
- Estimate and reduce percentage uncertainty.
- Choose significant figures consistent with raw data.
5.1 Accuracy and precision
An accurate result is close to the true value. Precise repeated results are close to one another. Measurements can be precise but inaccurate when a systematic error shifts every reading by a similar amount.
In an examination, the true value may not be known. You can still discuss precision from the scatter of repeats, but you should not claim accuracy merely because readings agree.
5.2 Repeatability and reproducibility
Repeatability means obtaining similar results under the same conditions, with the same method and apparatus. Reproducibility means obtaining similar results with changed conditions, a different method, apparatus or observer.
Repeating a reading tests repeatability and helps reveal random variation. It does not automatically reveal a calibration error because every repeated value may be shifted together.
5.3 Random and systematic error
Random errors produce unpredictable scatter. They arise from reaction time, fluctuating conditions or difficult judgment. Repeats and averaging reduce their influence on the final mean.
Systematic errors shift results consistently, such as a zero error, an incorrectly calibrated thermometer or heat loss ignored in every trial. Repetition does not remove systematic error; the cause must be corrected, calibrated or modelled.
5.4 Uncertainty and percentage effect
A scale with 1 mm divisions may be read to about 0.5 mm. When a length is found from two ruler readings, uncertainty enters at both ends. Percentage uncertainty is approximately absolute uncertainty divided by measured value multiplied by 100%.
Increase the measured quantity where possible. Timing 20 oscillations gives a much smaller percentage reaction-time uncertainty than timing one. Measuring the thickness of 50 sheets and dividing by 50 is better than measuring one sheet.
5.5 Significant figures
A calculated answer should normally have the same number of significant figures as the least precise raw data used. Keep extra digits during working and round only the final result.
Decimal places describe position relative to the decimal point; significant figures describe meaningful digits. A time of 12.0 s has three significant figures and one decimal place. Zeroes between non-zero digits are significant.
Worked examples
Percentage uncertainty
A diameter is 20.0 ± 0.5 mm. Percentage uncertainty ≈ (0.5/20.0) × 100% = 2.5%.
Why a mean helps
Three times are 9.8 s, 10.1 s and 10.0 s. Mean = 9.97 s, reported as 10.0 s to match 0.1 s resolution. The mean reduces random scatter but not a stopwatch calibration error.
Practical focus
Investigation or training activity
Measure one quantity repeatedly, calculate the mean and range, then deliberately introduce a zero offset. Compare how random scatter and systematic shift appear in the data.
Examination guidance
- Agreement between repeats shows precision, not necessarily accuracy.
- Repeats reduce random error, not systematic error.
- State how an improvement reduces the named uncertainty.
- Do not overstate significant figures.
- Use the term anomaly for a result outside the general pattern, not simply any imperfect point.
Check your understanding
- Can results be precise but inaccurate?
- What type of error is reduced by averaging?
- Why does timing many oscillations help?
- How many significant figures are in 0.0405?
Answers
- Yes, if a systematic error shifts a tight cluster.
- Random error.
- The same reaction-time uncertainty is a smaller fraction of the longer total time.
- Three.
