Random and systematic uncertainty

Separate error and uncertainty

Measurement error is the measured quantity value minus a reference quantity value. That difference can be signed. When the intended comparison is with an unknown value of the measurand, the corresponding error is also unknown; a calibrated reference is not automatically a known true value. A difference from the average of a few readings is not, by itself, that unknown error.

Measurement uncertainty describes the dispersion of quantity values attributed to the measurand using the available information. Standard uncertainty expresses that dispersion as a standard deviation and is nonnegative. It is not the signed error of the particular result, and it is not automatically a guaranteed maximum error. A result can happen to have a small unknown error while still carrying substantial uncertainty.

The title names two important sources of uncertainty: effects that vary unpredictably across specified repetitions and effects that remain common or vary predictably. We will describe components arising from those effects rather than treating random uncertainty and systematic uncertainty as two universal mathematical types. All numbers below are original stipulated teaching records, not measurements performed by this article.

Specify the repetitions

Random measurement error varies unpredictably in replicate measurements. Systematic measurement error remains constant or changes predictably in those repetitions; measurement bias is an estimate of systematic error. These descriptions concern behavior in a defined process. A positive error is not necessarily systematic, and alternating signs do not establish an independent random process.

Repeatability conditions specify such features as the procedure, operator, measuring system, operating conditions, location and short time interval. Our first case concerns the elapsed time between two designated trigger events in repeated operations of a hypothetical timing source. The duration q is assumed unchanged, and the same specified timing setup and conditions are used. Values are in milliseconds, written ms.

For the selected calculation, also stipulate that the reading variations are independent, follow one unchanged distribution and have finite variance. Independence means that one variation provides no information about another under this model. These assumptions do not follow merely from using the same instrument. A shared timing offset can coexist with the observed spread.

Keep two classifications distinct

A Type A evaluation obtains an uncertainty component through statistical analysis of measured values under defined conditions. Type B uses other relevant information, such as a suitable calibration result, specifications or justified knowledge of an input. Both methods can produce a standard uncertainty. Their letters describe how a component is evaluated, not whether it is an error or whether it must be random or systematic.

NIST explicitly warns against identifying Type A with random and Type B with systematic: either evaluation method can apply to a component arising from either kind of effect. For example, a common offset can be investigated by statistical comparisons with a reference. In another use, a suitable external calibration result can supply information about that offset. Its role in the current model and the information actually used must be stated.

A short series does not make Type A automatically more reliable than a well-founded Type B evaluation.

Evaluate one repeated series

Supply four hypothetical timing readings r₁, r₂, r₃, r₄: 50.98 ms, 50.98 ms, 50.98 ms and 51.06 ms. Their arithmetic mean is r̄ = (r₁ + r₂ + r₃ + r₄)/4 = 51.00 ms. The table subtracts that mean from each reading. These deviations describe this series; they are not known errors relative to q.

For n readings, the usual sample variance estimate is the sum of squared deviations from the sample mean divided by n − 1. Its positive square root s describes the spread of individual readings. For independent repetitions of one unchanged distribution, the estimated standard uncertainty of the mean due to this variation is u_A(r̄) = s/√n. Here √ means the nonnegative square root; s and u_A have time units, while a variance has squared time units.

The squared deviations total 0.0048 ms². With n = 4, divide by 3 to obtain s² = 0.0016 ms² and s = 0.040 ms; then u_A(r̄) = 0.040 ms/2 = 0.020 ms. The sample estimate has n − 1 = 3 degrees of freedom, associated here with the three freely specifiable deviations when their sum is constrained to zero. Four readings provide a limited estimate, not a known population spread or a complete uncertainty budget.

Ask what averaging changes

Use the simple model rᵢ = q + b + εᵢ. Here q is the unchanged duration being sought, b is an unknown common timing offset during the run, and εᵢ is the varying part of reading i. Stipulate that the εᵢ are independent, have mean zero, and share a finite variance σ². The symbol σ denotes their population standard deviation, distinguished from its sample estimate s.

Averaging gives r̄ = q + b + ε̄. Under these assumptions the standard deviation of ε̄ is σ/√n. This describes dispersion across repeated series under the model, not a promise that every larger series has a smaller actual error. With the same σ, increasing n from 4 to 16 halves this standard deviation. The common b remains in the mean and is not divided by 4 or 16.

The reduction requires the stated conditions. If all readings share the same uncertain varying contribution E, then averaging that contribution gives (E + … + E)/n = E. If readings drift, are correlated or concern a changing duration, the simple series formula may be inappropriate. Here correlated variations have nonzero covariance, a measure of paired variation. Similar readings alone do not establish absence of a common effect.

Evaluate information beyond the series

A Type B evaluation still needs evidence and a justified model; it is not an arbitrary allowance. In a separate hypothetical timing case, supply information that represents a residual additive offset B as uniform on −0.060 ms ≤ B ≤ +0.060 ms. Uniform, also called rectangular, means equal probabilities for subintervals of equal width within those bounds and zero probability outside them in the supplied model. This distribution describes the information about B, not observed frequencies from a new experiment.

The midpoint estimate is zero. For a uniform distribution of half-width a, its standard deviation is a/√3. Here a = 0.060 ms, so u(B) = 0.060 ms/√3 = 0.020√3 ms ≈ 0.03464 ms. The bound's half-width and the standard uncertainty are different quantities. The approximate decimal is a calculation display; it does not establish instrument precision.

Assume this B is shared by every reading in the current run. Its standard uncertainty is not divided by √n merely because n readings are taken. The given bounds and distribution are specific assumptions, not a claim that every specification supplies guaranteed bounds or that every Type B component is uniform or common to all repetitions. New information can require a different model.

Retain correction uncertainty

A correction compensates for an estimated systematic effect. In a third stipulated case, return to the four-reading series and supply an applicable estimate b̂ = +0.100 ms of its common timing offset, with standard uncertainty u(b̂) = 0.030 ms from suitable external calibration information. The hat indicates an estimate. The chosen corrected estimate is q̂ = r̄ − b̂ = 50.900 ms, within the stated setup and calibration conditions.

Subtracting the same estimate from every reading subtracts it from the mean too. The corrected values are 50.880 ms, 50.880 ms, 50.880 ms and 50.960 ms; their deviations from 50.900 ms are unchanged. Thus the sample s = 0.040 ms and the repeatability component u_A(r̄) = 0.020 ms are unchanged. These are exact shifts of the stipulated record, not evidence that the actual offset has been removed perfectly.

Uncertainty in the estimated correction remains because its required value is not exactly known. It is different from the signed correction itself. Under this common-offset model, u(b̂) = 0.030 ms is not reduced to 0.015 ms by averaging the four readings. The two identified components and any dependencies must enter a full budget; this article does not compute their combined uncertainty or claim q̂ is the known true duration.

Record evidence and limits

A useful uncertainty record names the measurand, repetition conditions, input model, corrections, evaluation method, information source and stated uncertainty measure. It distinguishes the spread of individual readings, the component for their mean and other components. Record n and the degrees of freedom for the sample-based estimate, as well as limitations of a short series.

Checks should address different possible effects. Repeating the same procedure tests behavior within those repetitions. Suitable reference comparisons, changed conditions or independent procedures can investigate effects that one repeated series does not expose. Such checks need their own scope and uncertainty; agreement alone does not certify a true value or prove that every source of uncertainty has been found.

Recognized significant systematic effects should be corrected where the model and information justify doing so, and their remaining uncertainty retained. More observations are useful only for components and conditions to which the statistical model applies. The next task is to combine justified components through an explicit measurement model, without double counting or losing dependencies. That is a separate uncertainty-budget problem.

Worked examples

Try it yourself

1. In the first series, r₁ − r̄ = −0.02 ms. Is that the known measurement error relative to the unknown duration q?

   Show the answer and reasoning

   No. It is a deviation from this series' sample mean.

   The mean is an estimate of the series' center, not a known true duration. A common offset can shift all readings and their mean without changing these deviations.

2. Add the same +0.100 ms to every reading in the first series. What are the new mean, s and u_A of the mean under the same variation assumptions?

   Show the answer and reasoning

   The mean becomes 51.100 ms; s remains 0.040 ms and u_A remains 0.020 ms.

   Every reading and its mean shift equally, so the deviations and squared-deviation sum are unchanged. Unchanged spread does not show that a shared effect is absent.

3. In a separate theoretical model, stipulate independent zero-mean variations with one population standard deviation σ = 0.040 ms. Compare the standard deviation of the mean for n = 4 and n = 16. Does this divide a common fixed offset by the same factors?

   Show the answer and reasoning

   The mean's varying component has standard deviation 0.020 ms for n = 4 and 0.010 ms for n = 16. A common fixed offset remains unchanged.

   The model gives σ/√n, so quadrupling n halves this standard deviation. The supplied population σ is a model value; it is not made known by the four-reading sample, and no particular series' error is guaranteed to halve.

4. For another supplied uniform information model, use bounds −0.030 ms and +0.030 ms for a common offset B. Find its standard uncertainty and distinguish it from the half-width.

   Show the answer and reasoning

   u(B) = 0.030 ms/√3 = 0.010√3 ms ≈ 0.01732 ms; the half-width is 0.030 ms.

   The conversion uses the explicitly supplied uniform distribution. The standard uncertainty is a standard-deviation measure, not the interval's half-width, and a common component is not reduced just by repeating readings.

5. An offset is common during a run. One evaluation obtains its uncertainty by statistical analysis of suitable reference comparisons; another uses suitable external calibration information without analysing a measured series. Must both be Type B because the effect is systematic?

   Show the answer and reasoning

   No. Under these specified methods the first is Type A and the second is Type B.

   The common effect describes its role in the repetitions. A/B describe the evaluation method and information. The original provider's method is not inferred merely from the fact that a later user receives a calibration result.

6. A timing sequence shows drift and correlated variations. Can you automatically use s/√n as the repeatability component of its mean just because there are n readings?

   Show the answer and reasoning

   No. The simple independent, unchanged-distribution model has not been established.

   The drift, dependence and measurand conditions need investigation or an appropriate statistical model. Taking more readings does not by itself establish the assumptions or remove a common effect.

What to carry forward

• Measurement error is a signed comparison with a reference; uncertainty describes information-based dispersion and is not that unknown error.
• Random and systematic effects are described relative to stated repetitions and conditions.
• Type A/B label evaluation methods; either can evaluate a component associated with either kind of effect.
• The worked sample gives s = 0.040 ms and u_A(r̄) = 0.020 ms, with n = 4 and 3 degrees of freedom.
• Independent averaging has a conditional √n reduction; common effects, drift and dependence require separate attention.
• Corrections retain uncertainty, and identified components need a justified full budget before a complete result is claimed.