Measurement Errors & Statistics of Readings

Every measurement is uncertain. Learn the difference between error and uncertainty, accuracy and precision, systematic and random errors, significant figures, how repeated readings reduce uncertainty (standard error of the mean), Type A and Type B evaluation, combining components, coverage factors and how to report a result like a professional lab.

A micrometer reading of 10.12 mm does not mean the part is exactly 10.12 mm. The instrument has resolution limits, the operator varies the clamping force, the part expands with temperature. Engineering decisions (accept/reject, safety margins) depend on knowing how wrong a number could be.

The internationally agreed approach (the GUM — Guide to the Expression of Uncertainty in Measurement) turns all these doubts into a standard uncertainty, combines them, and reports a result like 10.115 ± 0.018 mm (k = 2, ≈95%).

An analogy

Accuracy and precision are like darts: precise throws cluster tightly; accurate throws centre on the bullseye. A precise but inaccurate thrower has a systematic error — a bias you can calibrate out; random scatter you can only average down.

Types of error

Knowing the type tells you how to fix it.

Accuracy, precision, resolution

Accuracy = closeness to true value; precision (repeatability) = closeness of repeated readings to each other; resolution = smallest change the instrument displays. A 6-digit display does not mean 6-digit accuracy.

Significant figures: report uncertainty to 1–2 significant figures and round the value to the same decimal place: 10.115 ± 0.018, not 10.11500 ± 0.0183.

Type A and Type B

Type A uncertainties are evaluated statistically from repeated data (s/√n). Type B come from other knowledge — calibration certificates, resolution, manufacturer specs, temperature effects — converted to a standard deviation (rectangular: a/√3, triangular: a/√6).

Coverage and degrees of freedom

For a normal distribution, k = 2 gives ≈95% coverage. With few readings, use the Student t-value for the effective degrees of freedom (Welch–Satterthwaite): for n = 8 alone, t₀.₉₇₅,₇ = 2.365. Correlated inputs require covariance terms, and non-normal outputs call for Monte Carlo methods (GUM Supplement 1).

Formulas

Worked example: Standard error of the mean

Given: 8 readings, s = 0.0245 mm

Find: u_A.

Answer: u_A = 0.0087 mm

Worked example: Resolution uncertainty

Given: Digital caliper resolution 0.01 mm

Find: u_B.

Answer: u_B = 0.0029 mm

Worked example: Reporting

Given: x̄ = 10.115 mm, u_c = 0.0091 mm

Find: Final statement.

Answer: d = 10.115 ± 0.018 mm (k = 2, ≈95%)

Uncertainty budget for repeated readings

import math, statistics as st

d = [10.12, 10.08, 10.15, 10.11, 10.09, 10.14, 10.10, 10.13]   # mm
n = len(d)
mean, s = st.mean(d), st.stdev(d)
uA = s / math.sqrt(n)                  # Type A: scatter of the mean
uB = 0.005 / math.sqrt(3)              # Type B: 0.01 mm resolution, rectangular
uc = math.hypot(uA, uB)
print(f"mean = {mean:.3f} mm, s = {s:.4f} mm")
print(f"uA = {uA:.4f}, uB = {uB:.4f}, uc = {uc:.4f} mm")
print(f"result: {mean:.3f} +/- {2 * uc:.3f} mm (k = 2)")
# mean = 10.115 mm, s = 0.0245 mm
# uA = 0.0087, uB = 0.0029, uc = 0.0091 mm
# result: 10.115 +/- 0.018 mm (k = 2)

The scatter of the readings dominates the budget; improving resolution alone would barely help. Taking more readings would (uA ∝ 1/√n).

Common mistakes

Applications

Key takeaways

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