Accuracy, Precision, and Measurement Uncertainty
Accuracy describes how close a measurement is to the true value, precision describes how close repeated measurements are to each other, and measurement uncertainty is the range within which the true value is expected to lie. In brief: a measurement can be precise but not accurate (a consistent systematic error), accurate but not precise (large random scatter), both, or neither. A result is written as value ± uncertainty, for example (1.55 ± 0.03) s, and the percentage uncertainty is (uncertainty ÷ value) × 100.
No measurement is perfect, so every result should say how reliable it is. This guide explains accuracy, precision and uncertainty, shows how to estimate uncertainty from repeated readings, compares the concepts in a table, works through an example with three students and a pendulum, and corrects common mistakes.
What is accuracy?
Accuracy is the closeness of a measured value to the true (or accepted) value. The difference between them is the error. The percentage error is |measured − true| ÷ true × 100. Accuracy is lowered by systematic errors, which shift all results in the same direction: a zero error on an instrument, a scale that is not calibrated, or a clock that runs fast. Repeating the measurement does not remove a systematic error. It can be found by comparing with a standard and corrected.
What is precision?
Precision describes the agreement between repeated measurements of the same quantity under the same conditions. High precision means small scatter. It is limited by the resolution of the instrument and by random errors, which change from one reading to the next and can be positive or negative: small differences in reaction time, in reading a scale, or in the environment. Averaging many readings reduces the effect of random errors.
An analogy is a target: precise shots are tightly grouped, accurate shots are centered on the bullseye. A tight group away from the center is precise but not accurate.
What is measurement uncertainty?
The uncertainty gives an interval around the measured value in which the true value is likely to be. Common ways to estimate it in school physics are:
- Single reading, analog scale: half of the smallest division (for example ±0.5 mm on a millimeter ruler). Some courses use the full division when it is hard to read.
- Single reading, digital instrument: ±1 in the last digit displayed (or as stated by the manufacturer).
- Repeated readings: the mean value, with the uncertainty taken as half of the range: (maximum − minimum) ÷ 2.
The absolute uncertainty has the same unit as the measurement (±0.03 s). The fractional or percentage uncertainty compares it with the value (0.03 ÷ 1.55 = 0.019, or 1.9 percent). The percentage uncertainty is the best measure for comparing the quality of different measurements. A result should be given with the uncertainty to the same decimal place as the value, usually with one significant figure for the uncertainty.
Accuracy, precision and uncertainty compared
| Term | Meaning | Affected by | How to improve |
|---|---|---|---|
| Accuracy | Closeness to the true value | Systematic errors | Calibrate, correct zero error |
| Precision | Closeness of repeated results | Random errors, resolution | Repeat and average, better instrument |
| Uncertainty | Interval for the true value | Both, and instrument limits | Reduce errors, measure larger quantities |
Worked example: comparing three sets of measurements
The true length of a metal rod is 10.00 cm. Three students each measure it three times.
- Student A: 10.02, 10.01, 10.03 cm. Mean = 10.02 cm; range = 0.02 cm. Percentage error = 0.02 ÷ 10.00 × 100 = 0.2 percent. The results are close to each other and to the true value: accurate and precise.
- Student B: 9.80, 9.82, 9.81 cm. Mean = 9.81 cm; range = 0.02 cm. Percentage error = 0.19 ÷ 10.00 × 100 = 1.9 percent. The results are close to each other, but far from the true value, which suggests a systematic error such as a zero error: precise but not accurate.
- Student C: 9.5, 10.4, 10.1 cm. Mean = 30.0 ÷ 3 = 10.0 cm; range = 0.9 cm. The mean agrees with the true value, but the readings are scattered: accurate on average, but not precise.
Uncertainty from repeated readings: the times for one swing of a pendulum are measured five times as 1.52, 1.58, 1.55, 1.57 and 1.53 s.
- Mean = (1.52 + 1.58 + 1.55 + 1.57 + 1.53) ÷ 5 = 7.75 ÷ 5 = 1.55 s.
- Half range = (1.58 − 1.52) ÷ 2 = 0.03 s.
- Result: (1.55 ± 0.03) s.
- Percentage uncertainty = 0.03 ÷ 1.55 × 100 = 1.9 percent.
How did the idea of uncertainty develop?
Carl Friedrich Gauss developed the theory of random errors and the normal distribution in the early 1800s, when he studied astronomical observations. The method of least squares, used to fit lines to data, was published by Legendre in 1805. Today, the Guide to the Expression of Uncertainty in Measurement (GUM), first published in 1993 by international standards bodies, gives the standard procedures for laboratories.
Which mistakes are common?
Mistake 1: Using accuracy and precision as synonyms
Accuracy is about the closeness to the true value, and precision is about the closeness of repeated values.
Mistake 2: Thinking that repeating a measurement removes all error
Averaging reduces random errors, but not systematic ones.
Mistake 3: Writing the uncertainty with too many digits
Use one significant figure (or two at most) for the uncertainty and match the decimal place of the value, for example (1.55 ± 0.03) s and not (1.55 ± 0.0312) s.
Mistake 4: Confusing error and uncertainty
An error is the difference from the true value, which is usually unknown. The uncertainty is an estimate of the possible size of that difference.