Every measurement has an uncertainty, and every lab report is marked partly on how well you handle it. This guide covers random and systematic error, instrument uncertainty, significant figures, propagation rules, percent error and how to write an error discussion that earns marks.
A measurement without an uncertainty is incomplete. Writing "g = 9.6 m/s²" tells the reader nothing about whether your result agrees with the accepted 9.81 m/s². Writing "g = 9.6 ± 0.3 m/s²" does: the accepted value lies inside your range, so your experiment is consistent with it. Writing "g = 9.60 ± 0.05 m/s²" tells a different story: the accepted value lies well outside your range, which points to a systematic error worth investigating.
That is why lab rubrics typically reward three things: stating uncertainties correctly, carrying them through calculations, and using them to judge your result. This guide covers all three.
| Random error | Systematic error | |
|---|---|---|
| What it is | Unpredictable scatter in repeated measurements | A consistent offset in one direction |
| Examples | Reaction time with a stopwatch, fluctuating readings, judging a meniscus | Uncalibrated scale, zero error on a meter, heat loss to surroundings, parallax read the same way each time |
| Effect | Reduces precision | Reduces accuracy |
| Reduced by | Repeating measurements and averaging | Better calibration, improved method, correcting for the bias |
| Visible in data as | Spread around the mean | Results consistently too high or too low; a non-zero intercept where zero is expected |
Accuracy vs precision: accuracy is how close a result is to the true value; precision is how close repeated results are to each other. You can be precise but inaccurate (tight cluster, wrong place) if a systematic error is present.
"Human error" is not a type of error and should not appear in your report. Name the specific source instead: reaction time, parallax, misreading a scale.
For a single reading, the uncertainty comes from the instrument's resolution. Common conventions, which your lab manual may adjust, are:
| Instrument type | Typical convention | Example |
|---|---|---|
| Analogue scale (ruler, thermometer, burette) | ± half the smallest division | Ruler marked in mm: ± 0.5 mm |
| Digital display | ± 1 in the last displayed digit | Balance reading 12.47 g: ± 0.01 g |
| Manufacturer specification | Use the stated tolerance | Volumetric pipette: ± 0.05 mL (as printed) |
| Stopwatch timing by hand | Dominated by reaction time, often estimated at around 0.1 to 0.3 s | Time 10 oscillations and divide to reduce it |
Measuring a length with a ruler involves two readings (each end), so some courses double the reading uncertainty. Follow your instructor's convention and state it in the method.
When you repeat a measurement, the spread tells you the random uncertainty. Three approaches are common, from simplest to most rigorous:
| Trial | Period (s) |
|---|---|
| 1 | 1.42 |
| 2 | 1.45 |
| 3 | 1.40 |
| 4 | 1.44 |
| 5 | 1.43 |
| Mean | 1.428 |
Half the range = (1.45 − 1.40) ÷ 2 = 0.025 s, so T = 1.43 ± 0.03 s. Using the standard deviation (about 0.019 s) and n = 5, the standard error is about 0.009 s, giving T = 1.428 ± 0.009 s. Use whichever method your course specifies.
The uncertainty determines how many digits your result deserves:
| Calculator output | Correctly reported |
|---|---|
| 9.7342 ± 0.2817 m/s² | 9.7 ± 0.3 m/s² |
| 0.0045672 ± 0.00012 m | (4.57 ± 0.12) × 10⁻³ m |
| 1523.8 ± 46 J | (1.52 ± 0.05) × 10³ J |
Send your data and lab manual. We help with propagation, graphs and a discussion that matches your rubric.
When you calculate a result from measured quantities, their uncertainties combine. The rules below assume the measurements are independent. Introductory courses often use the simpler "add" versions; more advanced courses use addition in quadrature (square root of the sum of squares).
| Operation | Simple (worst case) | Quadrature (independent errors) |
|---|---|---|
| q = a + b or a − b | Δq = Δa + Δb | Δq = √(Δa² + Δb²) |
| q = a × b or a ÷ b | Δq/q = Δa/a + Δb/b | Δq/q = √((Δa/a)² + (Δb/b)²) |
| q = aⁿ | Δq/q = |n| × Δa/a | |
| q = k × a (k exact) | Δq = |k| × Δa | |
A block has mass m = 52.4 ± 0.1 g and volume V = 19.6 ± 0.4 cm³. Density ρ = m/V = 2.673 g/cm³.
Notice the volume dominates the uncertainty. Pointing this out in your discussion, and suggesting a better way to measure volume, is exactly the kind of insight markers reward.
For a pendulum, g = 4π²L/T². With T = 1.43 ± 0.03 s (2.1%), T² carries 2 × 2.1% = 4.2%. Squaring doubles the relative uncertainty, which is why careful timing matters more than careful length measurement in this experiment.
Percent error measures accuracy:
percent error = |experimental − accepted| ÷ accepted × 100%
But percent error alone does not tell you whether the difference is significant. The better test is whether the accepted value falls within your uncertainty range:
| Result | Accepted | Interpretation |
|---|---|---|
| 9.7 ± 0.3 m/s² | 9.81 m/s² | Agrees within uncertainty; no evidence of systematic error |
| 9.52 ± 0.06 m/s² | 9.81 m/s² | Disagrees; difference is about five times the uncertainty, suggesting a systematic error |
Some courses use percent difference instead, when comparing two experimental values with no accepted reference.
The error discussion is where many reports lose marks by listing generic sources. A strong discussion:
Weak: "Errors may have occurred due to human error and faulty equipment."
Strong: "The largest contribution came from hand timing (4.2% after squaring). Reaction time would produce random scatter rather than bias, consistent with the spread in Table 1. Using a light gate, or timing 20 oscillations instead of 10, would reduce this uncertainty."
For the full report structure, from abstract to conclusion, see our lab report guide. Presenting your data well matters too; our guide to scientific graphs covers error bars and best-fit lines.
Usually one. Many courses allow two when the first digit is 1 (for example ± 0.14). The measured value is then rounded to the same decimal place as the uncertainty.
Error is the difference between a measured value and the true value, which is usually unknown. Uncertainty is your estimate of the range within which the true value probably lies. Lab reports mainly quantify uncertainty and discuss likely sources of error.
Standard deviation describes the spread of individual measurements; standard error (s ÷ √n) describes the uncertainty of the mean. If you report a mean, the standard error is usually the appropriate uncertainty, unless your course specifies otherwise.
Avoid it. It is too vague to earn credit. Name the specific mechanism, such as reaction time in hand timing or parallax when reading a scale, and explain its effect.