Error Analysis for Physics Labs — Uncertainty Done Right

Error is not a mistake

In physics, error (or uncertainty) does not mean you messed up — it means no measurement is exact. A ruler's millimeter marks, your reaction time on a stopwatch, and the flicker of a digital meter all inject spread into results.

A mistake is reading 32 cm as 23 cm. An uncertainty is knowing the length is between 31.9 and 32.1 cm. Error analysis is the discipline of quantifying the second kind — and refusing to pretend measurements are more precise than they are.

Absolute vs relative uncertainty

Write a measurement as $x = (25.0 \pm 0.5)\ \mathrm{cm}$. The $\pm 0.5$ cm is the absolute uncertainty $\Delta x$ — it has units. Divide by the value to get the relative uncertainty $\Delta x / x = 0.5/25 = 0.02$, and multiply by 100 for percent uncertainty: 2%.

Absolute uncertainty answers 'how many units could I be off?' Relative answers 'how big is that compared to what I measured?' A ±1 mm error is superb on a football field and terrible on a coin — relative uncertainty captures that judgment.

Random vs systematic errors

Random errors scatter results around the true value — sometimes high, sometimes low. Timing a swing with a stopwatch gives slightly different readings each trial; averaging beats them down.

Systematic errors push every reading the same way: a scale that reads 5 g heavy, a ruler with its first millimeter worn off, parallax from always viewing at an angle. No amount of averaging fixes these — they must be found and corrected.

Quick test: repeat the measurement. If the spread shrinks with more trials but the center stays off, you are looking at a systematic error hiding behind small random ones.

Significant figures carry the uncertainty

Writing $12.3456$ s from a stopwatch readable to $0.1$ s is dishonest — the extra digits are fiction. The rule: the last reported digit is the first uncertain one, so $12.3 \pm 0.1$ s keeps three significant figures.

When you compute with measured values, the result cannot be more precise than its shakiest input. (The full rules live in the significant figures article — uncertainty is why those rules exist.)

Propagation: how errors travel through math

Measurements get added, multiplied, and raised to powers — and their uncertainties ride along:

• Add/subtract: add the absolute uncertainties: $\Delta z = \Delta x + \Delta y$.
• Multiply/divide: add the relative uncertainties: $\Delta z/z = \Delta x/x + \Delta y/y$.
• Powers: multiply the relative uncertainty by the exponent: for $z = x^2$, $\Delta z/z = 2\,\Delta x/x$.

Worked: a rectangle $12.0 \pm 0.2$ cm by $8.0 \pm 0.2$ cm has area $96\ \mathrm{cm^2}$ with relative uncertainty $0.2/12 + 0.2/8 = 0.0417$, so $\Delta A = 96 \times 0.0417 \approx 4\ \mathrm{cm^2}$. Report $(96 \pm 4)\ \mathrm{cm^2}$ — not $96.0$, not $96.00$.

Repeated measurements and the mean

Measure the same thing $N$ times and take the mean $\bar{x}$ as your best estimate — random errors average toward zero. For a quick uncertainty, use half the range: $\Delta x \approx (\max - \min)/2$.

Five stopwatch timings of a fall — 1.42, 1.51, 1.47, 1.44, 1.49 s — give mean $1.466$ s and half-range $0.045$ s, so report $(1.47 \pm 0.05)$ s. More trials shrink the uncertainty roughly as $1/\sqrt{N}$: quadrupling the trials halves the random error.

Reporting results honestly

The standard format packs everything into one line: $x = (\text{value} \pm \text{uncertainty})\ \text{unit}$, with the uncertainty quoted to one (or at most two) significant figures and the value rounded to match.

If an accepted value exists, compute percent error $= |\text{measured} - \text{accepted}| / \text{accepted} \times 100\%$. But remember: percent error judges agreement with the book, while your uncertainty judges your own measurement. A result can agree with the book by luck while its uncertainty is huge — report both.

Variables and units

Common mistakes

Real-world applications

• Every physics lab report uses uncertainty to decide whether results agree with theory.

• Manufacturers quote tolerances (±0.1 mm) that are just absolute uncertainties on parts.

• Medical dosing and analytical chemistry live or die by propagated uncertainties.

• Instrument makers specify accuracy (systematic) and precision (random) separately on datasheets.