Physics for ME · Module 15 of 16 · Experimental thinking
Measurement, Uncertainty, and Experimental Physics
An experiment is an argument with evidence. This module teaches the rules of evidence: instruments, errors, and the honest propagation of doubt.
Readiness check
From Module 1 and Math Modules 6 and 15. Tick only what you can do closed-notes.
- Hold the significant-figures discipline.
- Compute mean and standard deviation of a small sample (Math Module 15).
- Use the total differential for sensitivities (Math Module 6).
- Distinguish random scatter from a fixed offset.
- Convert relative and absolute uncertainties.
The core idea
Every measurement is value plus doubt. Derived quantities inherit doubt through their formulas.
urel(xⁿ) = n·urel(x)uf = √(Σ(∂f/∂x·ux)²)For products and powers, relative uncertainties combine in quadrature, with exponents as multipliers (the d² in an area doubles d's contribution). Random error shrinks with repeats; systematic error (bias) only yields to calibration.
The skills, taught in order
An experiment is an argument, and uncertainty is its honesty. Six skills take you from reading an instrument to reporting a defensible value with its doubt and improving the measurement that produced it.
15.1 Instruments and resolution
An instrument's resolution is the smallest step it can show, but a fine display is not the same as a true reading. Resolution sets a floor on precision; accuracy is a separate question answered only by calibration.
15.2 Random versus systematic error
Random error is scatter: readings dance around the true value and averaging pulls them in. Systematic error is bias: a fixed offset that every reading shares, which averaging can never remove. They are two different diseases with two different cures.
15.3 Repeats and statistics
Repeat a measurement and the scatter itself becomes data. The mean estimates the value, the standard deviation s measures single-reading scatter, and the standard error of the mean, s/√n, says how well the mean is known. Because of that √n, more repeats help with diminishing returns. Worked Example 2 works this through.
Micro-example. Averaging four independent readings halves the random error of the result (√4 = 2); reaching a quarter takes sixteen.
15.4 Propagation of uncertainty
A derived quantity inherits doubt from its inputs. Sums and differences combine absolute uncertainties in quadrature; products and quotients combine relative uncertainties in quadrature; and a power multiplies the relative uncertainty by its exponent.
| Operation | How uncertainties combine |
|---|---|
| q = a + b or a − b | absolute: uq = √(ua² + ub²) |
| q = a·b or a/b | relative: uq/q = √((ua/a)² + (ub/b)²) |
| q = xⁿ | relative: uq/q = |n|·(ux/x) |
| mean of n readings | SEM = s/√n |
Micro-example. A 1% uncertainty in a radius becomes a 2% uncertainty in the area, because area goes as r².
15.5 Reporting results
State a result as value ± uncertainty, with units, rounding so the uncertainty carries one or two significant figures and the value matches its last place. "7.85 ± 0.05 g/cm³" says exactly how much to trust the 7.85.
15.6 Designing better measurements
Once the uncertainty budget is built, improve the dominant term and ignore the rest. A better instrument for the input that contributes most (often the one raised to a power) buys far more than polishing the small contributors.
Engineering connection: instrumentation, lab courses, quality control, and reliability; the twin of Math Module 15. Bridge reference Taylor, An Introduction to Error Analysis.
Worked example 1: what metal is this cylinder?
A cylinder is measured: m = 155.0 ± 0.5 g, diameter d = 25.04 ± 0.05 mm, length L = 40.1 ± 0.1 mm. Find its density with uncertainty and identify the material.
- ProblemFind ρ and uρ for the cylinder in Figure 1 and name the material.
- Given / findm, d, L with uncertainties above. Find ρ ± u.
- AssumptionsIndependent uncertainties; a true cylinder; instruments calibrated (bias addressed separately).
- Modelρ = 4m/(πd²L). Relative uncertainties add in quadrature with d's doubled by its exponent.
- Equationsurel(ρ) = √(um² + (2ud)² + uL²)
- SolveV = π × 25.04² × 40.1/4 = 19 748 mm³ = 19.75 cm³. ρ = 155.0/19.75 = 7.849 g/cm³. Relative doubts: m 0.32%, d 0.20% (doubled: 0.40%), L 0.25%. Combined: √(0.32² + 0.40² + 0.25²) = 0.57%, so u = 0.045. ρ = 7.85 ± 0.05 g/cm³.
- CheckSteel sits at 7.85; aluminium (2.70), brass (8.5), and copper (8.96) all fall far outside the band. The dominant doubt is the diameter, exactly because it enters squared.
- ConclusionThe material verdict is defensible because the uncertainty band excludes the alternatives. To tighten the result, attack d first (a micrometer instead of calipers): designing the next measurement from the budget is the whole craft.
Worked example 2: how many repeats?
A diameter is measured five times with calipers: 25.01, 25.06, 24.98, 25.10, 25.00 mm. Find the mean, the standard deviation, and the standard error of the mean, then work out how many readings would halve that error.
- ProblemFrom the five caliper readings, find the mean, the standard deviation, and the standard error of the mean, then the number of readings that would halve the error.
- Given / findn = 5 readings: 25.01, 25.06, 24.98, 25.10, 25.00 mm. Find the mean, the sample standard deviation s, the standard error of the mean (SEM), and the n for half the SEM.
- AssumptionsIndependent readings of a stable quantity with random scatter only (the instrument already calibrated, so no bias).
- ModelThe mean estimates the value; the sample standard deviation measures single-reading scatter; the standard error of the mean, s/√n, measures how well the mean is known and falls as 1/√n.
- Equationsmean = (Σ di)/n s = √(Σ(di − mean)² / (n − 1)) SEM = s/√n
- SolveMean = 125.15/5 = 25.03 mm. The squared deviations sum to 0.0096, so s = √(0.0096/4) = 0.049 mm. SEM = 0.049/√5 = 0.022 mm, giving d = 25.03 ± 0.02 mm. Halving the SEM needs four times the readings (the 1/√n rule), so 20 measurements.
- CheckThe SEM (0.02 mm) is smaller than the ±0.05 mm caliper-resolution guess used in Example 1, so repeated readings can beat a single-reading estimate. But the return diminishes: going from 5 to 20 readings only halves the error.
- ConclusionScatter across repeats is not failure; it is the measurement reporting its own reliability. Averaging tames random error on a √n budget but never bias, which is why this module's two tools, statistics and calibration, are always needed together.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Resolution mistaken for accuracy | Five digits trusted from an uncalibrated meter | "Fine display, but is the reading true?" | Resolution is the step size; accuracy needs calibration against a standard. |
| Exponents ignored in propagation | d's doubt counted once under a square | "What power does this input carry?" | Relative uncertainty multiplies by the exponent: the d² doubled it here. |
| Uncertainties added linearly | Doubt budgets pessimistically fat | "Are these error sources independent?" | Independent doubts combine in quadrature (the root-sum-square), not by addition. |
| One reading treated as truth | No repeats, no scatter estimate | "What would a second measurement show?" | Repeat at least a few times; scatter is data about your own measurement. |
Practice ladder
A voltage is 12.06 ± 0.05 V and a current 2.31 ± 0.02 A. Find the power and its uncertainty.
Show answer
P = 27.86 W. Relative: √(0.41%² + 0.87%²) = 0.96%, so u = 0.27. P = 27.9 ± 0.3 W.
The pendulum formula g = 4π²L/T² is used with L = 1.000 ± 0.002 m and T = 2.007 ± 0.005 s. Which input dominates the doubt in g, and what is g ± u?
Show answer
g = 4π² × 1.000/4.028 = 9.80 m/s². Relative: L gives 0.2%; T, squared, gives 2 × 0.25% = 0.5%: T dominates. Combined 0.54%: g = 9.80 ± 0.05 m/s². Time the swing better before measuring the string better.
The Module 11 kettle experiment gave c = 4500 J/kg·K versus the true 4186. Scatter across repeats was only ±100. Diagnose: random or systematic, and name the likely bias.
Show answer
The 314 J/kg·K excess is three times the scatter: systematic. Likely bias: heat lost to the kettle body and air is charged to the water, inflating c. Averaging more runs would never fix it; insulating or accounting for the kettle would.
Design and run an indirect measurement of your own (density of a coin, g from a phone-timed drop, spring constant from masses and a ruler). Produce the full uncertainty budget and state the dominant term and your next improvement.
What good work looks like
The formula with all inputs and doubts, exponents handled, quadrature shown, the result with the band, and a concrete next-instrument recommendation aimed at the dominant term.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Write the cylinder-density experiment (or your transfer task design) as a one-page lab report in the professional format: aim, apparatus with resolutions, data, propagation, result with band, comparison, and the next-improvement recommendation.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. Resolution, precision, accuracy: define and separate.
Resolution: smallest displayed step. Precision: repeatability (scatter). Accuracy: closeness to truth (bias). All three are independent properties.
2. Which error type does averaging cure, and which does it not?
Averaging shrinks random error by √n; it never touches systematic bias.
3. How do relative uncertainties combine for products and powers?
In quadrature, each multiplied by its exponent: urel = √(Σ(nᵢ·urel,i)²).
4. What is the correct way to report a measured result?
Value ± uncertainty with units and the confidence basis, rounded so the uncertainty has one or two significant figures.
5. Given a finished budget, how do you improve the experiment?
Attack the dominant term: better instrument or method for that one input. Improving the others buys almost nothing.