Measurement Errors, Uncertainty & Calibration

No measurement is exact. Learn gross, systematic and random errors, statistics of repeated readings, the GUM method for Type A and Type B uncertainty, combined and expanded uncertainty, propagation of uncertainty, traceability to national standards, calibration procedures (as-found / as-left, 5-point checks), calibration intervals, and test uncertainty ratio — with a real uncertainty budget in Python.

A measurement result is incomplete without a statement of how uncertain it is. '100.11 kPa' means little; '100.11 ± 0.06 kPa (k = 2)' tells an engineer exactly how far to trust it.

Calibration compares an instrument with a more accurate reference that is itself traceable, through an unbroken chain, to national standards (NPL, NIST, PTB). In regulated industries (pharma, food, oil & gas custody transfer) calibration records are legal evidence.

An analogy

Measurement uncertainty is like a weather forecast's 'chance of rain': it does not say you are wrong, it says how confident you can be — and the forecaster (instrument) must itself be checked against reality (calibration) from time to time.

Types of error

Errors fall into three families:

Uncertainty the GUM way

The Guide to the Expression of Uncertainty in Measurement (GUM, JCGM 100) treats every source as a standard uncertainty. Type A comes from statistics of repeated readings; Type B from everything else — calibration certificates, specifications, resolution, temperature effects — converted using an assumed distribution (rectangular: a/√3, triangular: a/√6, normal: a/k). Sources are combined by root-sum-square with sensitivity coefficients, then multiplied by k = 2 for ≈95% confidence.

Calibration procedure

A typical 5-point transmitter calibration:

Traceability, decision rules and MSA

Traceability requires every reference to have a valid certificate with its uncertainty, back to SI units. A TUR of at least 4:1 keeps false-accept risk low; when it is smaller, guard-banding tightens the acceptance limit (ISO/IEC 17025 decision rules). In manufacturing, measurement system analysis (Gauge R&R) separates instrument and operator variation from process variation.

Formulas

Worked example: Type B from a spec

Given: Calibrator accuracy ±0.05 kPa (rectangular)

Find: Standard uncertainty.

Answer: 0.029 kPa

Worked example: Propagation

Given: Power P = V·I, V = 10 V ± 0.05 V (u), I = 2 A ± 0.01 A (u)

Find: u(P).

Answer: u(P) = 0.14 W on 20 W

Worked example: TUR

Given: Transmitter tolerance ±0.25% of span, calibrator uncertainty ±0.05%

Find: TUR.

Answer: 5:1 ✓ (≥ 4:1)

GUM uncertainty budget

import math, statistics
# Type A (repeat readings) + Type B (spec sheet) uncertainty, GUM method
readings = [100.12, 100.08, 100.15, 100.10, 100.09, 100.13, 100.11, 100.07, 100.14, 100.10]
mean = statistics.mean(readings)
s = statistics.stdev(readings)
uA = s / math.sqrt(len(readings))                 # standard uncertainty of the mean
uB_acc = 0.05 / math.sqrt(3)                      # +/-0.05 accuracy, rectangular
uB_res = 0.01 / (2 * math.sqrt(3))                # 0.01 resolution
uc = math.sqrt(uA**2 + uB_acc**2 + uB_res**2)
U = 2 * uc                                        # k = 2, ~95 % coverage
print(f"mean {mean:.3f}, s {s:.4f}, uA {uA:.4f}")
print(f"uB(acc) {uB_acc:.4f}, uB(res) {uB_res:.4f}, uc {uc:.4f}")
print(f"result: {mean:.3f} +/- {U:.3f} (k=2)")
# mean 100.109, s 0.0260, uA 0.0082
# uB(acc) 0.0289, uB(res) 0.0029, uc 0.0302
# result: 100.109 +/- 0.060 (k=2)

The instrument's accuracy spec (Type B) dominates here; taking more readings only shrinks u_A. An uncertainty budget shows where to spend money: a better reference would help, more averaging would not.

Common mistakes

Applications

Key takeaways

Going further