Math for ME · Chapter 18 of 19 · Advanced
Probability, Statistics, and Engineering Uncertainty
Every measurement scatters and every part varies. Engineers who can quantify uncertainty make defensible decisions; the rest make guesses with confident faces.
The thread: Every method so far assumed exact numbers. Real measurements scatter and real parts vary, so the final tool is making sound decisions in the face of that doubt.
Readiness check
From Algebra, Derivatives, and Multivariable Calculus. Tick only what you can do closed-notes.
- Compute means and percentages quickly.
- Square, sum, and root a column of numbers without slips.
- Read a histogram and a scatter plot.
- Use the total differential for small-change estimates.
- Keep significant figures honest.
The core idea
Report every measured quantity as a value plus a stated doubt: x̄ ± u.
s = √(Σ(xᵢ − x̄)²/(n−1))The mean locates the result; the standard deviation measures the scatter; the confidence interval says how well the mean itself is known. For derived quantities, Series and Taylor's total differential becomes propagation of uncertainty: sensitivities times input doubts.
The skills, taught in order
18.1 A result is a value with a doubt
A measurement is not finished until it carries an uncertainty. The mean locates the result, the sample standard deviation measures the scatter, and dividing s by √n gives the standard uncertainty of the mean itself.
x̄ = Σxᵢ/ns = √(Σ(xᵢ − x̄)²/(n − 1))The divisor is n − 1, not n, because one degree of freedom was spent estimating the mean from the same data.
18.2 Random versus systematic error
Two errors behave completely differently, and confusing them is the costliest mistake in measurement.
| Error | Looks like | Cured by |
|---|---|---|
| Random | scatter between repeats | averaging more readings |
| Systematic | a constant offset (bias) | calibration, never averaging |
18.3 The normal distribution
Random errors pile into a bell curve, and most engineering uncertainty assumes it. The useful summary is the 68-95-99.7 rule: about 68% of values fall within ±1s of the mean, 95% within ±2s, and 99.7% within ±3s. A z-score counts how many standard deviations a value sits from the mean.
18.4 Propagating uncertainty
When a result is computed from several measured inputs, their doubts combine through Series and Taylor's total differential, added in quadrature:
u_f = √((∂f/∂x · u_x)² + (∂f/∂y · u_y)²)For a pure product or quotient this simplifies to adding relative uncertainties, with any exponent acting as a multiplier: a quantity raised to a power amplifies that input's relative uncertainty by the power.
18.5 Regression and decisions
Least-squares regression fits the line that minimises the squared residuals; small, patternless residuals signal a trustworthy fit, while a pattern in them signals the wrong model. Reliability then combines scatter with a limit: the probability that strength falls below load, read from the normal curve, turns uncertainty into a design decision rather than a hope.
Engineering connection: Experiments, Measurement, Reliability, Data Science, AI for engineers.
Worked example: how round is the shaft, really?
Five micrometer measurements of a turned shaft: 25.02, 24.98, 25.01, 24.99, 25.00 mm. Report the diameter with a 95% confidence interval, and judge it against a tolerance of 25.00 ± 0.05 mm.
- ProblemSummarize the five readings in Figure 1 and compare with the tolerance.
- Given / findData: 25.02, 24.98, 25.01, 24.99, 25.00 mm. Find x̄, s, the 95% CI, and the verdict.
- AssumptionsIndependent repeats, random errors, micrometer free of bias (calibrated).
- ModelSample statistics, then the t-distribution for a small sample (n = 5, so 4 degrees of freedom, t = 2.776).
- Equationsx̄ = Σxᵢ/n s = √(Σ(xᵢ − x̄)²/(n−1)) CI = x̄ ± t·s/√n
- Solvex̄ = 125.00/5 = 25.000 mm. Deviations: +0.02, −0.02, +0.01, −0.01, 0; squares sum to 0.0010, so s = √(0.0010/4) = 0.0158 mm. CI: 25.000 ± 2.776 × 0.0158/√5 = 25.000 ± 0.020 mm.
- CheckUnits consistent; the CI (±0.020) is wider than s/√n alone (±0.007) because five samples deserve humility (the t factor). All five raw points sit inside the tolerance with room to spare.
- ConclusionReport d = 25.000 ± 0.020 mm (95%). The tolerance band ±0.05 is 2.5 times wider than the confidence band: the process comfortably meets spec, and the data proves it rather than asserts it.
Worked example 2: the density and its uncertainty
A solid cylinder is measured as mass m = 250 ± 2 g, diameter d = 30.0 ± 0.1 mm, length L = 80.0 ± 0.2 mm. Find its density and the uncertainty in that density.
- Given / findm = 250 ± 2 g, d = 30.0 ± 0.1 mm, L = 80.0 ± 0.2 mm. Find ρ and u_ρ.
- ModelDensity is mass over volume, and a cylinder's volume is (π/4)d²L, so ρ = 4m/(πd²L), a pure product and quotient.
- Best valueV = (π/4)(3.00 cm)²(8.00 cm) = 56.55 cm³, so ρ = 250/56.55 = 4.42 g/cm³.
- Relative uncertaintiesu_m/m = 2/250 = 0.0080; u_d/d = 0.1/30 = 0.0033; u_L/L = 0.2/80 = 0.0025.
- Combine in quadrature, doubling the diameter termsince d enters squared: (u_ρ/ρ)² = (0.0080)² + (2 × 0.0033)² + (0.0025)² = 0.000111, so u_ρ/ρ = 0.0105.
- Uncertaintyu_ρ = 0.0105 × 4.42 = 0.046 g/cm³, so ρ = 4.42 ± 0.05 g/cm³ (about 4420 ± 50 kg/m³).
- CheckThe mass term (0.0080) dominates, with the diameter close behind despite d being measured tightly, because its uncertainty is doubled by the square. Tightening the mass measurement would shrink the total fastest.
- ConclusionPropagation tells the engineer not only the doubt in the answer but which input to measure better. The exponent on each variable is the lever: squared terms deserve the tightest tolerances.
Misconceptions and diagnostics
| Mistake | Symptom | Diagnostic question | Correction |
|---|---|---|---|
| Averaging away a biased instrument | Tight CI around a wrong value | "When was this instrument last checked against a standard?" | Repeats kill random error only. Bias needs calibration, not statistics. |
| Dividing by n instead of n − 1 | Scatter slightly understated, always | "Sample or whole population?" | Samples use n − 1: you spent one degree of freedom estimating the mean. |
| CI read as "95% of parts are inside" | Tolerance decisions made on the wrong band | "Is this band about the mean, or about individual parts?" | The CI bounds the mean. Individual parts scatter with s, a much wider band. |
| Correlation crowned as cause | "X drives Y" from a scatter plot alone | "What experiment would isolate the mechanism?" | Correlation nominates suspects; controlled experiments convict. |
Practice ladder
Compute mean and standard deviation of: 12, 15, 11, 14, 13 (N·m, bolt torques).
Show answer
x̄ = 13. Deviations −1, 2, −2, 1, 0; squares sum 10; s = √(10/4) = 1.58 N·m.
For a process with mean 50.0 mm and s = 0.2 mm, within what range do about 95% of parts fall?
Show answer
±2s about the mean: 50.0 ± 0.4 mm, that is 49.6 to 50.4 mm, by the 68-95-99.7 rule.
A length is 100.0 ± 0.4 mm and a width 50.0 ± 0.3 mm. Propagate the uncertainty into the area A = LW.
Show answer
∂A/∂L = W = 50, ∂A/∂W = L = 100. u_A = √((50×0.4)² + (100×0.3)²) = √(400 + 900) = 36 mm². A = 5000 ± 36 mm²: Series and Taylor's differential wearing its statistics uniform.
A power is P = I²R with I = 2.0 ± 0.1 A and R = 10 ± 0.2 Ω. Use relative uncertainties to find u_P.
Show answer
P = 40 W. u_P/P = √((2×0.1/2)² + (0.2/10)²) = √(0.01 + 0.0004) = 0.102, so u_P = 4.1 W and P = 40 ± 4 W. The current's 5% becomes 10% because it is squared.
A spring's load-deflection data: (2 mm, 39 N), (4, 81), (6, 118), (8, 162). Fit k by least squares through the origin (k = Σxy/Σx²) and judge the fit.
Show answer
Σxy = 78 + 324 + 708 + 1296 = 2406; Σx² = 4 + 16 + 36 + 64 = 120. k = 20.05 N/mm. Residuals: −1.1, +0.8, −2.3, +1.6 N: small and patternless, so the linear model holds over this range.
A rod's strength is normal with mean 400 MPa, s = 20 MPa; the applied stress is a steady 340 MPa. Estimate the failure probability and decide whether the design is acceptable for a noncritical bracket.
Show answer
z = (340 − 400)/20 = −3: failure when strength falls below load, probability ≈ 0.13%. For a noncritical bracket this 3-sigma margin is normally acceptable; for a lifting hook it would not be, and the margin would be raised or the scatter reduced.
Working with AI, and proving it yourself
Use AI as an examiner, not a solver
Portfolio task
Run a real measurement campaign: one dimension on ten nominally identical objects (coins, bolts, printed parts) with the best instrument you have. Report mean, s, the 95% CI, a histogram, and a tolerance you would publish for the batch.
Retrieval and spaced review
Closed notes. Answer out loud, then reveal.
1. Distinguish random and systematic error, with the cure for each.
Random: scatter between repeats, reduced by averaging. Systematic: a consistent offset, reduced only by calibration and method changes.
2. State the 68-95-99.7 rule.
For a normal distribution, about 68% of values lie within ±1s of the mean, 95% within ±2s, 99.7% within ±3s.
3. Why n − 1 in the sample standard deviation?
One degree of freedom is consumed estimating the mean from the same data; n − 1 corrects the systematic underestimate.
4. Write the propagation rule for f(x, y) with independent uncertainties.
u_f = √((∂f/∂x · u_x)² + (∂f/∂y · u_y)²): sensitivities times doubts, combined in quadrature.
5. What does a 95% confidence interval bound, exactly?
The mean of the population, not individual values: individual parts scatter with the full s.
Textbook mapping
| Item | Mapping |
|---|---|
| Main sources | Kreyszig, Advanced Engineering Mathematics, Ch 24 to 25 (data analysis, probability, mathematical statistics). Deeper companion: Montgomery and Runger, Applied Statistics and Probability for Engineers |
| Core topics | 18.1 Why uncertainty · 18.2 Mean, variance, s · 18.3 Normal distribution · 18.4 Measurement error · 18.5 Propagation · 18.6 Confidence intervals · 18.7 Regression · 18.8 Correlation versus causation · 18.9 Reliability basics · 18.10 Decisions under uncertainty |
| Engineering connection | Experimentation and Measurements, Reliability, Quality, Data Science and AI for engineers. |
| Skip on first pass | Hypothesis-testing zoo, Bayesian machinery, design-of-experiments theory: later courses. |
| Read next | Engineering Optimization, finding the best design within its constraints. |