MEGR 3171 — Week 3: Uncertainty Analysis & Calibration

Learning Objectives

Apply the Kline-McClintock method to propagate elemental uncertainties through a multi-variable model.
Distinguish single-sample from multiple-sample uncertainty and design-stage from end-of-test analysis.
Define calibration traceability and explain the ISO/ANSI calibration hierarchy.
Develop a static calibration curve and report calibration results in standard engineering format.
Identify sources of uncertainty in a calibration: interpolation error, hysteresis contribution, and environmental effects.

1. The Kline-McClintock Method

When a result R is calculated from multiple independently measured variables x₁, x₂, ..., x_n, each with uncertainty w_xi, the total uncertainty in R is computed by root-sum-of-squares (RSS) combination of the partial derivative sensitivities:

Kline-McClintock Uncertainty Propagation

w_R = √[ (∂R/∂x_1 · w_x1)² + (∂R/∂x_2 · w_x2)² + ... + (∂R/∂x_n · w_xn)² ]

where:
  w_R   = total uncertainty in the result
  ∂R/∂x_i = partial derivative (sensitivity coefficient)
  w_xi  = elemental uncertainty in variable x_i

Physical Meaning Each term under the radical represents the contribution of one variable's uncertainty to the overall result uncertainty. The RSS combination reflects the statistical independence of the error sources: random errors do not add linearly; their effects combine in quadrature.

Example: Power Dissipation

Power P = V · I. If V = 12.0 ± 0.1 V and I = 2.00 ± 0.05 A:

Example Calculation

∂P/∂V = I = 2.00 A     contribution = (2.00 × 0.1)² = 0.04
∂P/∂I = V = 12.0 V    contribution = (12.0 × 0.05)² = 0.36

w_P = √(0.04 + 0.36) = √0.40 ≈ 0.632 W

Reported: P = 24.0 ± 0.63 W

2. Single-Sample vs. Multiple-Sample Uncertainty

Single-sample uncertainty applies when only one measurement can be taken. It relies on instrument specifications (resolution, accuracy class) rather than experimental statistics. Multiple-sample uncertainty uses repeated measurements to estimate the random component statistically. When both are present, they combine by RSS:

Combined Uncertainty

w_total = √(w_systematic² + w_random²)

Design-stage analysis is performed before testing to verify that the proposed measurement system can achieve the required uncertainty with the available instruments. End-of-test analysis uses actual data to report the uncertainty of the final result.

3. Calibration Standards and Traceability

Traceability means every calibration instrument can be linked through an unbroken chain of comparisons to a national or international measurement standard (in the U.S., NIST). Traceability does not guarantee accuracy, but it ensures that the calibration is anchored to a known reference.

The calibration hierarchy: International standards → National standards (NIST) → Transfer standards → Working standards → Instrument under calibration. Each step introduces additional uncertainty, which must be accounted for in the instrument's stated accuracy.

Static Calibration Procedure

Apply known reference inputs spanning the full measurement range in both increasing and decreasing directions (to capture hysteresis).
Record corresponding instrument outputs at each level.
Plot the calibration curve (output vs. input).
Fit a curve (linear or polynomial) and compute residuals.
Report: sensitivity, zero offset, linearity error, hysteresis, and uncertainty of each.

Common Exam Trap Linearity error is the maximum deviation from the best-fit line, not from the ideal theoretical line. Always fit the best straight line to the data, then measure deviations from that line.

Practice Problems

Problem 1 — Kline-McClintock The density of a fluid is calculated as ρ = m / V. A mass m = 0.500 ± 0.002 kg is measured in a volume V = 0.000400 ± 0.000005 m³. Find the uncertainty in density.

ρ = m/V = 0.500/0.000400 = 1250 kg/m³

∂ρ/∂m = 1/V = 1/0.000400 = 2500 m³ → contribution = (2500 × 0.002)² = 25

∂ρ/∂V = −m/V² = −0.500/(0.000400)² = −3,125,000 → contribution = (3125000 × 0.000005)² = 243.14

w_ρ = √(25 + 243.14) = √268.14 ≈ 16.4 kg/m³

ρ = 1250 ± 16.4 kg/m³ (1.3% uncertainty)

Problem 2 — Dominant Error Identification Flow rate Q = A · v where A = 0.0100 ± 0.0002 m² and v = 3.50 ± 0.10 m/s. Which variable contributes more to the total uncertainty in Q?

∂Q/∂A = v = 3.50 → contribution = (3.50 × 0.0002)² = 4.9 × 10&sup-⁶

∂Q/∂v = A = 0.0100 → contribution = (0.0100 × 0.10)² = 1.0 × 10&sup-⁶

w_Q = √(4.9×10&sup-⁶ + 1.0×10&sup-⁶) ≈ 7.68×10&sup-⁶ → w_Q ≈ 0.000877 m³/s

Area measurement dominates (contributes ~83% of total uncertainty). Improving area measurement would yield the greatest reduction in overall uncertainty.