Repeatability, bias and uncertainty in road measurements

A small change needs a measurement argument

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

A later survey reports a higher roughness value. The subtraction is easy; the interpretation is harder. Was the same wheelpath measured? Was the instrument operating under comparable conditions? How much short-term scatter does the measurement process produce? Could an offset affect every reading in the same direction? These questions determine whether a difference is evidence of change or a reason to examine the measurement process.

Repeatability concerns results obtained under specified repeat conditions. A sample standard deviation can describe their scatter, but a small standard deviation does not establish agreement with a reference value. NIST distinguishes repeatability, systematic effects and the wider uncertainty of a reported result. (NIST, n.d.)

Five passes, one mean and a useful warning

Governing relationship: s = √[Σ(xi − x̄)²/(n − 1)]; uA(x̄) = s/√n.

Five repeat readings in m/km; n−1 = 4 degrees of freedom.

uA assumes independent repeat errors and excludes other uncertainty components.

The film first plots reading (m/km), then follows the calculation and plots m/km. Mean = 3.000; s = 0.1581; uA = 0.0707 m/km. Adding 0.300 to every reading changes the mean but not the scatter.

Measurement experiment

Scatter, mean and common bias

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

Download example data (CSV)

Model, calculations and transcriptEquations, worked steps, source notes and a timed reading guide

Understand the model

Repeat scatter and uncertainty of the mean describe random variation. A shared offset can shift every reading without changing that scatter.

Work through the inputs and equations below, then compare the calculated output with the film. All numerical inputs in this study are synthetic.

Inputs and symbol key

xi; n

Individual repeat reading (m/km); number of readings.

x̄; s

Sample mean and sample standard deviation (m/km).

uA(x̄)

Type A standard uncertainty of the mean (m/km), assuming independent repeat errors.

Values used in this film

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

The mathematical relationships

1. Equation 1

   Source and scope
   $$\begin{gathered}\bar x=\frac{1}{n}\sum_i x_i\\s=\sqrt{\frac{\sum_i(x_i-\bar x)^2}{n-1}}\end{gathered}$$

   Estimate the mean and sample standard deviation. The n−1 divisor accounts for estimating the mean from the same sample.

2. Equation 2

   Source and scope
   $$u_A(\bar x)=s/\sqrt n$$

   Divide sample standard deviation by the square root of n for the Type A standard uncertainty of the mean under the independence assumption. It excludes shared bias and other uncertainty components.

Work through the example

Use the relationships above with consistent units. These substitutions are our worked example, not values quoted from the source.

1. Mean

   (2.8 + 3.0 + 3.2 + 2.9 + 3.1) / 5 = 3.0

   m/km

2. Squared deviations

   0.04 + 0 + 0.04 + 0.01 + 0.01 = 0.10

   (m/km)²

3. Sample variance

   0.10 / (5 − 1) = 0.025

   (m/km)²

4. Standard quantities

   s = √0.025 = 0.158114; uA = s/√5 = 0.070711

   m/km

Read the output

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

Check your understanding

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.

Assumptions and limits

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

Return to the full references and method notes to check the cited edition and its applicability.

Timestamped film transcript 26 short entries · select a time to revisit the film

1. 0:00

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

2. 0:01

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

3. 0:07

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

4. 0:14

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

5. 0:23

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

6. 0:32

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

7. 0:39

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

8. 0:43

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

9. 0:52

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

10. 0:59

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

11. 1:05

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

12. 1:08

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

13. 1:11

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

14. 1:14

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

15. 1:17

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

16. 1:26

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

17. 1:29

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

18. 1:33

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

19. 1:36

    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.

20. 1:42

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

21. 2:00

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

22. 2:10

    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

23. 2:21

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

24. 2:28

    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.

25. 2:40

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

26. 2:49

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

Download the timed captions (VTT)

The transcript records the on-screen wording. Formula provenance and interpretation are explained in the source note above.

Calculate the scatter before discussing the change

Consider five invented repeat readings for the same reporting interval: 2.8, 3.0, 3.2, 2.9 and 3.1 m/km. Assume for this calculation that the repeat errors are independent and the measurand is unchanged. Their arithmetic mean is 3.000 m/km. The deviations from that mean are −0.2, 0, +0.2, −0.1 and +0.1 m/km; their squared sum is 0.100 (m/km)².

The sample standard deviation uses four degrees of freedom: s = √(0.100 / 4) = 0.1581 m/km. The estimated Type A standard uncertainty of the mean from this repeat scatter is s / √5 = 0.0707 m/km. These two quantities answer different questions. The first describes the scatter of individual repeats. The second concerns their mean under the stated independence assumption.

Quantity	Calculation	Result
Mean	15.0 / 5	3.000 m/km
Squared deviations	0.04 + 0 + 0.04 + 0.01 + 0.01	0.100 (m/km)²
Sample variance	0.100 / (5 − 1)	0.025 (m/km)²
Sample standard deviation	√0.025	0.1581 m/km
Repeat-scatter component for the mean	0.1581 / √5	0.0707 m/km

Why a common bias survives repetition

Now add 0.300 m/km to every reading. The new mean is 3.300 m/km. Each deviation from the new mean is unchanged, so the sample standard deviation is still 0.1581 m/km. Repeating the measurement has not revealed or removed that common offset. The animation makes this visible by translating the whole group while retaining its spread.

The offset is deliberately prescribed here so the result can be checked. In practice, establishing a correction needs reference evidence, and the correction has its own uncertainty. A repeatability experiment alone cannot establish every systematic contribution. This is why uncertainty evaluation should describe the measurement procedure and relevant components, rather than report the standard error as if it were the complete budget. (NIST, n.d.)

Case review

A tidy cluster can still be in the wrong place

Do repeated readings test scatter, bias, or both?

Pass 1 · 2.8 m/km

The discriminating check

Separate repeatability from reference checks and account for shared effects before applying a 1/√n reduction.

Evidence supports the comparison

With independent repeat errors, calculate the uncertainty contribution of the mean and report other components separately.

Evidence is missing or incompatible

With correlated or common effects, use the covariance structure and investigate bias instead of relying on more repeats.

What you can conclude. Precision describes agreement among repeats. Accuracy requires evidence about agreement with an appropriate reference.

Original worked case. Read the source editions and method notes.

Correlation changes what averaging achieves

The familiar division by the square root of the number of readings assumes independent repeat errors. If several readings share an error source, the variance of their mean also includes covariance terms. Collecting more nearly identical observations may therefore add less information than the simple formula suggests. Preserve the sequence, timing and test conditions so that the independence assumption can be examined.

This distinction is useful when comparing repeat traverses. A short sequence may reveal local scatter while missing between-day effects, positioning differences or changes in processing. A longer experiment should be designed around the sources of variation relevant to the intended comparison. It should not merely maximise the number of rows.

Report a result another engineer can interrogate

A useful report states the quantity being measured, the reference interval, repeat conditions, sample size, mean, dispersion and the uncertainty components included. It also identifies which components remain outside the estimate. If an expanded uncertainty is reported, state the coverage factor and its basis; do not silently multiply by two and imply a guaranteed probability.

For the five readings above, a defensible conclusion is that the observed short-term sample standard deviation is approximately 0.158 m/km, with a repeat-scatter contribution of approximately 0.071 m/km to the mean under the stated assumptions. It is not yet a complete claim about instrument accuracy or the significance of a change between two surveys. That final interpretation requires the measurement context and the additional uncertainty components relevant to the comparison.

Calculation laboratory

Why do tight repeats not prove an unbiased measurement?

Calculation step	Substitution or result	Units and interpretation
Mean	(2.8 + 3.0 + 3.2 + 2.9 + 3.1) / 5 = 3.0	m/km
Squared deviations	0.04 + 0 + 0.04 + 0.01 + 0.01 = 0.10	(m/km)²
Sample variance	0.10 / (5 − 1) = 0.025	(m/km)²
Standard quantities	s = √0.025 = 0.158114; uA = s/√5 = 0.070711	m/km

The divisor n − 1 accounts for estimating the mean from the same sample when forming the sample variance. The resulting standard deviation describes observed repeat scatter. Dividing by √n describes the Type A standard uncertainty of the mean under the independence assumption; it does not turn the scatter into a bound on every individual reading.

Adding a shared offset of 0.300 m/km to all five readings changes the mean to 3.300 while every deviation from the mean remains unchanged. Neither s nor uA detects that offset. Repeating the measurement more times can reduce the independent random contribution, but it cannot eliminate a common calibration contribution. A calibration uncertainty therefore belongs in a broader uncertainty budget, with sensitivity coefficients and dependence considered.

The n = 10 and n = 20 cases hold s fixed to isolate the square-root relationship. They are hypothetical sample-size scenarios, not extra measurements. Correlation between repeats would invalidate the simple independence reduction: for equal pairwise correlation ρ, the mean variance is σ²[1 + (n − 1)ρ]/n. This explains why repeated passes sharing the same conditions may offer less independent information than their count suggests. NIST distinguishes these measurement and uncertainty concepts. (NIST, n.d.)

Controlled sensitivity experiment

Controlled case	Calculated output	What changes
n = 5	0.070711 m/km Type A uncertainty of mean	Hypothetical independent repeats with the same sample standard deviation 0.158114.
n = 10	0.050000 m/km Type A uncertainty of mean	Hypothetical independent repeats with the same sample standard deviation 0.158114.
n = 20	0.035355 m/km Type A uncertainty of mean	Hypothetical independent repeats with the same sample standard deviation 0.158114.

CALCULATION LABORATORY

Why do tight repeats not prove an unbiased measurement?

Compare the cases on a fixed scale. Select a case to inspect its input change and calculated output.

Sensitivity case1. n = 52. n = 103. n = 20

n = 5: 0.070711 m/km Type A uncertainty of mean. Hypothetical independent repeats with the same sample standard deviation 0.158114.

Reproduce and challenge the result

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.

These are original synthetic calculations. The sensitivity cases vary the stated input while retaining the other assumptions; they are not field observations or treatment thresholds. The film’s calculation and sensitivity chapters use the same numerical cases as this table.

Find the floor beneath the averaging curve

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

Reproduce the experiment. u(mean) = σ√[(1+(n−1)ρ)/n], σ²=0.025 (m/km)². The downloadable table contains 12,200 evaluated cells, with the two input coordinates and output value retained on every row. These are parameter combinations, not that many independent field observations. The equation extends the worked example above; it is original teaching arithmetic, not a new standard or calibrated prediction.

Computed parameter atlas · 12,200 cells

More passes cannot average away everything

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

Download every calculated cell (CSV) · Open full-size explorer · Equation source and tests (ZIP)

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

References and method notes

1. NIST (n.d.). NIST Technical Note 1297: Appendix D1, Terminology. Guidelines for evaluating and expressing measurement uncertainty. Original source (Accessed: 5 September 2026).Repeatability, standard uncertainty and reporting terminology
2. NIST (1994). NIST Technical Note 1297: Appendix A, Law of Propagation of Uncertainty. 1994 edition; online Appendix A. Original source (Accessed: 5 September 2026).Section 3; Appendix A, equation (A-5), for independent repeated observations

Use the source edition and project requirements that apply to your work. Educational examples are not site-specific design advice.