WEBVTT

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Scatter, mean and common bias. Repeat readings: 2.8, 3.0, 3.2, 2.9, 3.1 m/km.

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MATHEMATICS · define the relationship and the meaning of each symbol before calculating.

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Five repeat readings in m/km; n−1 = 4 degrees of freedom. uA assumes independent repeat errors and excludes other uncertainty components.

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INPUT · individual repeats · read the axis units and the numerical values.

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Repeat readings: 2.8, 3.0, 3.2, 2.9, 3.1 m/km.

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Mean = 3.000; sample s = 0.158; uA(mean) = 0.071 m/km, assuming independent repeat errors.

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Add +0.300 m/km to every reading: mean = 3.300; s stays 0.158. Repetition does not remove common bias.

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OUTPUT · bias leaves scatter unchanged · read the axis units and the numerical values.

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Mean = 3.000; s = 0.1581; uA = 0.0707 m/km. Adding 0.300 to every reading changes the mean but not the scatter.

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CALCULATION · substitute the defined inputs; keep units through each intermediate step.

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Mean: (2.8 + 3.0 + 3.2 + 2.9 + 3.1) / 5 = 3.0 m/km

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Squared deviations: 0.04 + 0 + 0.04 + 0.01 + 0.01 = 0.10 (m/km)²

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Sample variance: 0.10 / (5 − 1) = 0.025 (m/km)²

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Standard quantities: s = √0.025 = 0.158114; uA = s/√5 = 0.070711 m/km

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SENSITIVITY · change one declared input · read the axis units and the numerical values.

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n = 5: 0.0707107 m/km Type A uncertainty of mean. Hypothetical independent repeats with the same sample standard deviation 0.158114.

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n = 10: 0.05 m/km Type A uncertainty of mean. Hypothetical independent repeats with the same sample standard deviation 0.158114.

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n = 20: 0.0353553 m/km Type A uncertainty of mean. Hypothetical independent repeats with the same sample standard deviation 0.158114.

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Check: Verify invariance of scatter under a constant offset and scaling of scatter when every reading is multiplied by a factor. Separate observed repeats from hypothetical sample-size projections and record common uncertainty sources.

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ONE ROW · Two hundred passes sound persuasive. Shared error can make that number misleading.

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PARAMETER MAP · 12200 evaluated cells. The colour limits stay fixed across every row.

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Expand Var(mean) as n variance terms plus n(n−1) equal covariance terms, then divide by n². With covariance ρσ² the plotted expression follows. At ρ = 0 it recovers σ/√n. Positive

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correlation creates a floor approaching σ√ρ. This is a covariance assumption, not a correlation estimated from the five observations above.

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VALUE LANDSCAPE · Height and colour encode Standard uncertainty of mean (m/km). This is a graph over the two input axes, not automatically a physical pavement shape.

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Equal variance and equal nonnegative pairwise correlation are imposed. Unknown fixed bias still requires separate investigation; repeatability does not establish accuracy.

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The spread is unchanged by common bias; uA is not the complete uncertainty budget.