Experimental Uncertainty and Percent Error
Experimental uncertainty is the range within which the true value of a measurement is expected to lie, and percent error expresses how far an experimental result is from the accepted value as a percentage: percent error = |experimental − accepted| ÷ accepted × 100%. In brief: if you measure the density of aluminum as 2.65 g/cm3 and the accepted value is 2.70 g/cm3, the percent error is 0.05 ÷ 2.70 × 100% = 1.9%.
No measurement is perfect. A scientific result is only meaningful if it comes with an estimate of how reliable it is. This article explains absolute and relative uncertainty, how to estimate uncertainty from the instrument and from repeated trials, how to combine uncertainties in calculations, how to calculate percent error and percent difference, with worked examples, and how to describe sources of error in a lab report.
What is experimental uncertainty?
Uncertainty states the doubt about a measurement, and it is written with a plus-or-minus sign: 25.0 ± 0.1 mL means that the true volume is probably between 24.9 and 25.1 mL. The absolute uncertainty (±0.1 mL) has the same unit as the measurement. The relative (percent) uncertainty compares the uncertainty with the size of the measurement: 0.1 ÷ 25.0 × 100% = 0.4%.
A measurement with a small relative uncertainty is more reliable. The same absolute uncertainty of 0.1 mL on a 2.5 mL measurement would be 4%, ten times larger in relative terms.
How do you estimate the uncertainty of a measurement?
| Situation | Estimate of uncertainty | Example |
|---|---|---|
| Digital instrument | ± one unit of the last digit (or the manufacturer’s value) | 12.34 g → ±0.01 g |
| Analog scale (ruler, thermometer) | ± half of the smallest division | 1 °C marks → ±0.5 °C |
| Marked glassware | Stated tolerance on the glassware | 25 mL volumetric pipette → about ±0.03 mL |
| Repeated trials | Half the range, or the standard deviation | 2.52, 2.55, 2.49 → mean 2.52, standard deviation 0.03 |
How do uncertainties combine in calculations?
- Addition or subtraction: add the absolute uncertainties. (A ± a) + (B ± b) = (A + B) ± (a + b).
- Multiplication or division: add the relative (percent) uncertainties. The relative uncertainty of the result is the sum of the relative uncertainties of the factors.
- Powers: multiply the relative uncertainty by the exponent. For r2, the relative uncertainty doubles.
These are simple, conservative rules suitable for introductory work. Statistical methods that add uncertainties in quadrature give slightly smaller combined values.
How do you calculate percent error?
Percent error compares an experimental value with an accepted (theoretical or literature) value:
percent error = |experimental value − accepted value| ÷ accepted value × 100%
The absolute value bars give a positive number. Some courses keep the sign to show whether the result is too high or too low, and then the formula is (experimental − accepted) ÷ accepted × 100%. When there is no accepted value and two experimental values are compared, percent difference is used: |A − B| ÷ the mean of A and B × 100%.
Worked examples
- Percent error of a density. Experimental 2.65 g/cm3, accepted 2.70 g/cm3. |2.65 − 2.70| = 0.05. 0.05 ÷ 2.70 × 100% = 1.9%.
- Boiling point. A student measures 98.5 °C for the boiling point of water at sea level. Percent error = |98.5 − 100.0| ÷ 100.0 × 100% = 1.5%. The result is slightly low, which may come from impurities, a thermometer error, or a pressure lower than standard.
- Signed error. A student’s result for the molar mass of a gas is 29.5 g/mol, and the accepted value is 28.0 g/mol. (29.5 − 28.0) ÷ 28.0 × 100% = +5.4%, so the result is too high.
- Subtracting measurements. A mass of 22.46 ± 0.02 g is measured for a flask with solid, and 10.12 ± 0.02 g for the empty flask. The mass of the solid is 22.46 − 10.12 = 12.34 g, and the uncertainty is 0.02 + 0.02 = 0.04 g, so the result is 12.34 ± 0.04 g. The relative uncertainty is 0.04 ÷ 12.34 × 100% = 0.32%.
- Area of a rectangle. Length 5.00 ± 0.02 cm (0.40%) and width 3.00 ± 0.02 cm (0.67%). The area is 15.0 cm2 with a relative uncertainty of 0.40 + 0.67 = 1.07%, so the absolute uncertainty is 15.0 × 0.0107 = 0.16 cm2. The result is 15.0 ± 0.2 cm2.
- Is the result consistent with the accepted value? A density of 2.65 ± 0.06 g/cm3 and an accepted value of 2.70 g/cm3 overlap, since the range is 2.59 to 2.71, and so the data agree with the accepted value within the uncertainty.
Which sources of error should you discuss?
A good lab report names specific sources of error and explains how each one would change the result. Avoid vague statements such as “human error”.
| Source | Type | Effect on result (example: density) |
|---|---|---|
| Balance not zeroed, reads 0.20 g high | Systematic | Mass too high, so density too high |
| Air bubbles on a submerged solid | Systematic | Volume too high, so density too low |
| Meniscus read from above | Systematic | Volume reading is off in the same direction each time |
| Drafts moving the balance pan | Random | Scatter in the mass readings |
| Judging the end point of a titration | Random | Scatter in volume |
| Splashing liquid out of the beaker | Procedural mistake | Lost mass; repeat the trial |
Where does error analysis come from?
The study of errors in measurement began in astronomy and surveying. In the early 1800s, Gauss and Laplace developed a theory in which errors have a bell-shaped (normal) distribution, which provides the basis for standard deviation and confidence intervals. In 1993 the international Guide to the Expression of Uncertainty in Measurement (GUM) established common rules for reporting uncertainty in science and industry.
Which mistakes are common?
Mistake 1: Calling a blunder an uncertainty
Spilled liquid or a misread scale is a mistake, not part of the uncertainty. Repeat the trial.
Mistake 2: Using percent error without an accepted value
If no accepted value exists, compare the trials by percent difference or standard deviation.
Mistake 3: Quoting an uncertainty with too many digits
Report the uncertainty to one significant figure (or two at most) and match the decimal places of the measurement: 12.34 ± 0.04 g.
Mistake 4: Adding relative uncertainties for sums
For addition and subtraction, add absolute uncertainties. For multiplication and division, add relative uncertainties.