4. Common probability distributions in metrology

Every measurement result carries an uncertainty statement, and that statement is only as credible as the probability distribution assigned to each error source. In metrology, a distribution is not chosen for mathematical convenience; it is chosen because the physical mechanism that generates the error has a known shape. This chapter examines the four distributions that appear most often in uncertainty budgets — normal, uniform (rectangular), triangular, and Student's t — and shows how each is applied to model measurement errors in practice.

Why the Distribution Shape Matters

The standard uncertainty u of a quantity is the standard deviation of its assumed probability distribution. Two error sources can have identical half-widths but different standard uncertainties if their shapes differ. For a source bounded by ±a, the uniform distribution gives u = a/√3 ≈ 0.577a, while a triangular distribution over the same bounds gives u = a/√6 ≈ 0.408a. Choosing the wrong shape therefore changes the reported uncertainty by up to 40 % for that component. The shape also determines the coverage factor needed to reach a stated confidence level, so distribution selection propagates directly into the final expanded uncertainty.

The Normal Distribution

The normal (Gaussian) distribution is the default model for random errors that arise from the sum of many small, independent influences. Its probability density is

p(x) = 1/(σ√(2π)) · exp( −(x−μ)² / (2σ²) )

where μ is the mean and σ the standard deviation. In metrology the normal distribution is used in three situations: repeated observations whose scatter has been tested for normality, calibration certificates that state an expanded uncertainty U with a coverage factor k = 2 (implying u = U/2), and the combined standard uncertainty u_c of a measurand influenced by many independent components. The central limit theorem justifies the last case: as the number of independent contributions grows, their sum approaches normality regardless of the individual shapes.

A practical check is the normalized histogram of 20–50 repeated readings. If the histogram is roughly symmetric and bell-shaped, and the sample skewness and excess kurtosis are small, the normal assumption is reasonable. For small samples, formal tests such as Shapiro–Wilk are preferred over visual inspection.

The Uniform (Rectangular) Distribution

The uniform distribution models an error known only to lie within limits ±a, with no reason to expect values near the center more than near the edges. Its density is constant over [−a, +a], and its standard uncertainty is u = a/√3. Typical metrological sources include:

• Resolution of a digital display: if the last digit steps by d, the quantization error lies within ±d/2, so u = d/(2√3).
• Manufacturer's tolerance limits when no distribution is stated and the error is equally likely anywhere in the interval.
• Temperature drift specified as "±2 °C over the operating range" when the actual temperature is unknown within that band.
• Rounding of a reference value quoted to a fixed number of digits.

The uniform distribution is the most conservative common choice: for a given bound it yields the largest standard uncertainty among the symmetric shapes discussed here.

The Triangular Distribution

When two independent uniform contributions of similar magnitude combine, their sum is triangular. The triangular distribution also applies when an error is more likely near the center of its interval than at the extremes — for example, a tolerance that is rarely approached in practice, or the sum of two quantization errors. Its standard uncertainty is u = a/√6, smaller than the uniform case because probability mass is concentrated toward the center.

A common example is the mismatch error of a connector or the alignment error of a gauge block stack, where the extreme values are physically possible but statistically rare. Using a triangular model instead of uniform reduces the component's contribution by a factor of √2, which can be significant when several such components dominate the budget.

Student's t Distribution

Student's t distribution accounts for the additional uncertainty introduced when σ is estimated from a finite sample. If n repeated observations yield a sample standard deviation s, the standard uncertainty of the mean is u = s/√n, but the coverage factor required for a 95 % confidence interval is not 1.96 — it is the t-value with ν = n − 1 degrees of freedom. For n = 5, ν = 4 and t₉₅ ≈ 2.78; for n = 10, ν = 9 and t₉₅ ≈ 2.26. As ν → ∞, the t distribution converges to the normal distribution and t₉₅ → 1.96.

In the GUM framework, the t distribution is used at two points: converting the standard deviation of the mean into a confidence interval for a Type A evaluation, and computing the effective degrees of freedom of the combined uncertainty via the Welch–Satterthwaite formula before selecting the final coverage factor.

Comparing the Four Distributions

The table shows why distribution choice cannot be separated from uncertainty calculation. A component bounded by ±0.5 µm contributes 0.289 µm if uniform, 0.204 µm if triangular, and would contribute 0.250 µm if treated as normal with σ = a/2. These differences accumulate across the budget.

Worked Example: Distribution Selection in a Budget

Consider a micrometer calibration at 25 mm. Three error sources are identified:

1. Repeatability: 10 repeated readings give s = 0.8 µm. Type A, Student's t with ν = 9. u₁ = 0.8/√10 = 0.253 µm.
2. Resolution: digital display steps by 1 µm. Uniform, a = 0.5 µm. u₂ = 0.5/√3 = 0.289 µm.
3. Reference gauge block: certificate states U = 0.3 µm with k = 2. Normal. u₃ = 0.3/2 = 0.150 µm.

Combining in quadrature: u_c = √(0.253² + 0.289² + 0.150²) = √(0.064 + 0.084 + 0.023) = √0.171 ≈ 0.414 µm. The effective degrees of freedom from Welch–Satterthwaite is dominated by the Type A term, giving ν_eff ≈ 18, so t_95,18 ≈ 2.10 and U = 2.10 × 0.414 ≈ 0.87 µm. Reporting U = 0.9 µm (k = 2) would be slightly optimistic; the correct statement is U = 0.87 µm with k = 2.10.

Selecting the Right Distribution: A Decision Path

The decision path above reflects the logic used in most calibration laboratories. The first question separates Type A from Type B evaluations; the second distinguishes between bounded-only knowledge and center-weighted knowledge. Once every source has a distribution and a standard uncertainty, the combination step is identical regardless of shape — only the degrees of freedom and the final coverage factor depend on the mix of distributions.

Practical Guidance

Document the distribution choice and its justification for every component in the uncertainty budget. A reviewer must be able to see why a uniform rather than triangular model was used, or why a normal model was justified by a certificate. When in doubt between uniform and triangular, the uniform choice is more conservative and easier to defend; when in doubt between normal and t, use t unless the degrees of freedom exceed about 30. Consistency between the stated distribution and the reported coverage factor is the single most common audit finding in uncertainty statements, and it is entirely avoidable with the selection logic presented here.