Lab Math & Error Analysis

Every measurement you make in the laboratory is a little bit wrong. That is not a failure — it is a property of measurement, and a scientist's job is to know how wrong, which way, and why. This part is the working math for that: reading instruments honestly, combining uncertain numbers, and reporting a result with the confidence it actually deserves.

Use it alongside the Laboratory Program, the Study Toolkit error-analysis section, and the Problem-Solving Playbook.

Precision, accuracy, and the two kinds of error

Accuracy is how close a measurement is to the true value. Precision is how close repeated measurements are to each other. They are independent: you can be precise and wrong (every reading 2.10 g when the truth is 1.80 g), or accurate on average but scattered.

The two show up because there are two kinds of error:

• Random error scatters results above and below the true value unpredictably — the last estimated digit on a buret, small temperature drifts, judgment about when a color changes. It limits precision. You beat it down by measuring more times and averaging.
• Systematic error pushes every result the same direction by roughly the same amount — an unzeroed balance, a miscalibrated pipet, reading a meniscus from above, a thermometer that reads 1 °C high. It limits accuracy, and averaging does not remove it. You find it by calibrating against a standard and by checking your technique.

A tight cluster of results that misses the accepted value is the signature of systematic error. A wide spread centered on the right value is random error.

Reading an instrument: the last digit is estimated

Record every digit you know for certain, plus one estimated digit — the one you read between the smallest marked divisions. That last digit is uncertain, and it is the last significant figure.

Rules of thumb:

• Read a meniscus at eye level, at the bottom of the curve for water in glass. Reading from above makes every volume read low; from below, high — a systematic error.
• On a digital instrument, the uncertainty is usually ± 1 in the last displayed place unless the manual says otherwise. Do not add estimated digits a digital readout does not show.
• A buret is read twice (start and end); the volume delivered is the difference, and both readings carry the ± 0.01 mL uncertainty.

Absolute uncertainty, relative uncertainty, percent error

Absolute uncertainty is the ± value in the same units as the measurement: 24.37 ± 0.02 mL. Estimate it as about half the smallest division for analog scales, or the manufacturer's tolerance, or — better — the standard deviation of your own repeated trials.

Relative uncertainty is the absolute uncertainty divided by the measurement, usually as a percent:

relative uncertainty = (absolute uncertainty / measured value) × 100%

For 24.37 ± 0.02 mL: 0.02 / 24.37 × 100% = 0.08%. Relative uncertainty is what you compare across measurements — a ± 0.02 mL on a 2 mL delivery (1%) is a far worse measurement than the same ± 0.02 mL on 24 mL (0.08%). This is why you titrate to use most of the buret, and why you weigh by difference on the analytical balance.

Percent error compares your result to an accepted value:

percent error = |experimental − accepted| / accepted × 100%

It is a measure of accuracy after the fact, not an uncertainty. Report it with a sign or a direction when the direction is informative ("2.3% low").

Propagating uncertainty through a calculation

When measured numbers are combined, their uncertainties combine too. The two working rules for this course:

Addition and subtraction → add absolute uncertainties

If z = x + y or z = x − y, then the absolute uncertainty in z is the sum of the absolute uncertainties in x and y (a simple, slightly conservative rule; the statistical rule adds them in quadrature).

Buret: final 38.41 ± 0.02 mL, initial 2.05 ± 0.02 mL. Volume delivered = 36.36 mL, uncertainty = 0.02 + 0.02 = 0.04 mL. Report 36.36 ± 0.04 mL (relative uncertainty 0.11%).

Note that subtracting two close numbers is dangerous: 10.02 − 9.98 = 0.04, but the uncertainty is still ± 0.04, so the relative uncertainty is ~100%. Avoid experiment designs that hinge on a small difference of large measurements.

Multiplication and division → add relative uncertainties

If z = x · y or z = x / y, then the relative uncertainty in z is the sum of the relative uncertainties in x and y.

Density: mass 4.51 ± 0.01 g, volume 1.7 ± 0.1 mL. d = 4.51 / 1.7 = 2.653 g/mL. Relative uncertainties: mass 0.01/4.51 = 0.22%; volume 0.1/1.7 = 5.9%. Sum = 6.1%. Absolute = 0.061 × 2.653 = 0.16 g/mL. Report 2.65 ± 0.16 g/mL, or roughly 2.7 ± 0.2 g/mL.

The volume measurement dominates completely — it contributes 5.9% of the 6.1%. That tells you exactly where to improve the experiment: measure the volume better (a pipet or a volumetric flask instead of a rough graduated cylinder), and do not bother chasing a fourth decimal place on the mass.

Multiplying by an exact number

A pure count or a defined factor (the 2 in 2πr, Avogadro's number, a stoichiometric coefficient, "per 100 g") has no uncertainty and does not add to the total.

Repeated trials: mean, deviation, standard deviation

When you run a measurement several times, report the mean as your best value and a spread as its uncertainty.

• Mean: x̄ = (Σ xᵢ) / n.
• Deviation of one trial: dᵢ = xᵢ − x̄.
• Average deviation: Σ|dᵢ| / n — a quick, honest spread for three or four trials.
• Standard deviation (sample): s = √[ Σ(xᵢ − x̄)² / (n − 1) ]. Use this when you have enough trials; most calculators and spreadsheets compute it directly (STDEV / s, not the population σ).
• Standard deviation of the mean (standard error): s / √n. This is the uncertainty in x̄ itself, and it shrinks as you add trials — the quantitative reason more trials help.

Molarity from four titrations: 0.1042, 0.1038, 0.1051, 0.1045 M. Mean = 0.10440 M. s = 0.00055 M. Standard error = 0.00055/√4 = 0.00028 M. Report 0.1044 ± 0.0003 M (about 0.3%).

Rejecting an outlier

Do not discard a data point just because you dislike it. Discard it only if (a) you have a documented reason — you knocked the buret, overshot the endpoint, misread a digit — or (b) a defensible statistical test flags it (the Q-test at your chosen confidence level). Note every rejection and its reason in the notebook. Three good trials beat four with a silently deleted one.

Significant figures are shorthand for uncertainty

Significant-figure rules are a quick approximation of full propagation, and when the two disagree, the propagated uncertainty wins for the final reported value.

• Carry extra digits through every intermediate step; round only at the end.
• The final answer's last significant digit should sit in the same decimal place as the uncertainty. Report 2.65 ± 0.16 g/mL, not 2.6534 ± 0.16.
• Round the uncertainty to one significant figure (two if the first is 1), then match the value to it.
• log and pH: only the digits after the decimal point are significant. pH = 3.47 has two significant figures (the 3 just locates the power of ten).

Worked lab problems

1. Percent error on a density determination

You measure an aluminum cylinder: mass 13.62 ± 0.01 g, volume by displacement 5.1 ± 0.1 mL. Accepted density of aluminum is 2.70 g/mL.

• d = 13.62 / 5.1 = 2.671 g/mL.
• Relative uncertainty: 0.01/13.62 (0.07%) + 0.1/5.1 (2.0%) = 2.1%. Absolute: 0.021 × 2.671 = 0.055, so 2.67 ± 0.06 g/mL.
• Percent error: |2.671 − 2.70| / 2.70 × 100% = 1.1%.
• Interpretation: the 1.1% error is within the 2.1% measurement uncertainty, so the result is consistent with aluminum — you cannot claim a real discrepancy. The displacement volume is the limiting measurement.

2. Which measurement to improve

A molar-mass-by-freezing-point experiment gives molar mass with contributions: mass of solute ± 0.3%, mass of solvent ± 0.1%, freezing-point depression ΔT_f = 1.8 ± 0.2 °C (± 11%).

Total relative uncertainty ≈ 0.3 + 0.1 + 11 = 11%. The freezing-point depression dominates. Improving the balance does nothing useful; using more solute (a larger, easier-to-measure ΔT_f) or a better thermometer is the only change that matters.

3. Endpoint uncertainty in a titration

Four trials give the volume of titrant as 18.42, 18.55, 18.39, 18.47 mL.

• Mean = 18.458 mL, s = 0.070 mL, standard error = 0.035 mL.
• The buret's own reading uncertainty per delivery is ~± 0.03 mL — comparable to the scatter, so both the instrument and endpoint judgment contribute.
• Report the volume as 18.46 ± 0.04 mL (~0.2%) and carry that 0.2% forward. If the mass of your primary standard is known to 0.1%, the titrant molarity inherits about 0.2 + 0.1 = 0.3%.

4. Spotting a systematic error from the data

Five trials of a percent-water-in-hydrate experiment: 19.1, 19.3, 19.0, 19.2, 19.1 %. Accepted value 20.9 %.

• Precision is excellent: s = 0.11%, so random error is tiny.
• Every trial is ~1.8% (absolute) low, well outside the random scatter. That one-directional miss is a systematic error — most likely incomplete heating (some water never driven off) or reweighing before the sample fully cooled.
• The fix is procedural, not statistical: heat to constant mass, cool in a desiccator, repeat. More trials of the flawed procedure would just give a tighter wrong answer.

Reporting a result

A complete experimental result has four parts:

1. The value, to the right number of significant figures.
2. The uncertainty, with units, one or two significant figures.
3. The comparison, where an accepted value exists — percent error and whether the difference is inside or outside the uncertainty.
4. The dominant source, named — which single measurement contributed most of the uncertainty, and what would reduce it.

"Density = 2.67 ± 0.06 g/mL; 1.1% from the accepted value for aluminum, within the measurement uncertainty; the displacement volume (± 2%) dominates and a volumetric measurement would tighten it" is a full result. "Density ≈ 2.7, percent error 1.1%" is not.