Type A and Type B Measurement Uncertainty Explained

Section 1. The GUM Classification: Why Two Types?

The Guide to the Expression of Uncertainty in Measurement (GUM) was first published in 1993 and is jointly authored by seven international organisations: BIPM (Bureau International des Poids et Mesures), IEC, IFCC, ISO, IUPAC, IUPAP, and OIML. It is the authoritative international document governing how measurement uncertainty is evaluated and reported in calibration certificates, and it is referenced directly in ISO/IEC 17025:2017. The standard against which calibration laboratories are accredited.

Before the GUM, the metrology community used the terms "random errors" and "systematic errors" to describe measurement imperfections. These terms were widely understood but problematic: they implied something about the nature of the error (whether it fluctuated randomly between measurements or remained fixed), rather than simply describing how the error was quantified. This led to inconsistent treatment across different laboratories, sectors, and countries, making it difficult to compare results or combine uncertainties from different sources in a rigorous way.

The GUM resolved this by replacing the random/systematic distinction with a classification based purely on the method of evaluation:

• Type A evaluation: the uncertainty contribution is evaluated by statistical analysis of a series of observations. Specifically, by calculating a mean and standard deviation from repeated measurements taken under the same conditions.
• Type B evaluation: the uncertainty contribution is evaluated by all other means. Using information from calibration certificates, manufacturer datasheets, published scientific data, physical reasoning, or expert judgement.

This distinction is conceptually simple but practically important. Both types of evaluation produce a standard uncertainty (a value expressed as one standard deviation), and the two types are combined using the same mathematical framework (root-sum-of-squares). There is no mathematical difference in how the two types are treated once they have been converted to standard uncertainties. The classification simply describes where the number came from.

The key point that many practitioners misunderstand: the classification into Type A and Type B is not a ranking of reliability or importance. A Type B contribution derived from an NMI (National Metrology Institute) calibration certificate, where the uncertainty has been evaluated with full traceability through the international measurement system and is extremely small, may be far better characterised and more precisely known than a Type A contribution derived from only five repeated readings on a variable measurement system. The GUM is explicit on this point, and it is worth keeping in mind whenever you interpret an uncertainty budget.

Section 2. Type A Uncertainty Evaluation in Detail

When a calibration technician takes repeated readings at the same nominal measurement point (for example, measuring a thermometer against a calibration bath set to 100°C ten times in succession), the spread of those readings reflects real variability in the measurement process. This variability arises from many sources: electronic noise in the measurement system, thermal fluctuations in the reference medium, mechanical vibration, small changes in contact between the sensor and the reference, and the repeatability limits of the operator and the equipment. The Type A evaluation captures the net effect of all these sources of variability in a single statistical quantity.

The Step-by-Step Procedure

1. Take N repeated readings of the measurand under the same conditions (same operator, same equipment, same environment, same measurement point): x₁, x₂, ... x_N.
2. Calculate the arithmetic mean: x̄ = (x₁ + x₂ + · · · + x_N) / N.
3. Calculate the experimental standard deviation: s = √[Σ(x_i − x̄)² / (N − 1)]. Note the denominator is N − 1 (Bessel's correction), not N. This gives an unbiased estimate of the population standard deviation.
4. The Type A standard uncertainty is: u_A = s / √N. This is the standard deviation of the mean, also known as the standard error of the mean.

The reason for dividing by √N rather than simply using s is that the calibration result is the mean of the N readings, not any single reading. The standard deviation s describes the spread of individual readings around the mean. But the mean itself is more stable than any individual reading. If you were to repeat the entire set of N measurements many times, the means would cluster more tightly than the individual readings. The quantity s/√N correctly describes the uncertainty in the mean, which is the quantity being used as the calibration result.

Worked Example: Temperature Calibration

A calibration technician measures a reference platinum resistance thermometer (PRT) against a calibration bath stabilised at 100°C. Ten readings are recorded: 99.96, 99.98, 100.01, 99.97, 100.02, 99.99, 100.00, 99.98, 100.01, and 99.99°C.

The mean: x̄ = (99.96 + 99.98 + 100.01 + 99.97 + 100.02 + 99.99 + 100.00 + 99.98 + 100.01 + 99.99) / 10 = 999.91 / 10 = 99.991°C.

The experimental standard deviation: s = 0.019°C (calculated from the sum of squared deviations from the mean, divided by N − 1 = 9, then square-rooted).

The Type A standard uncertainty: u_A = 0.019 / √10 = 0.019 / 3.162 = 0.0060°C.

This 0.0060°C is the standard uncertainty contribution from repeatability. It represents the uncertainty in the mean value 99.991°C due to the variability observed across the ten readings.

The Effect of N on Type A Uncertainty

Because u_A = s / √N, increasing the number of readings reduces the Type A uncertainty in proportion to the square root of N. With 10 readings, u_A ≈ 0.316 × s. With 25 readings, u_A = 0.2 × s (a 37% reduction from 10 readings. With 100 readings, u_A = 0.1 × s), a 68% reduction from 10 readings, but requiring ten times the measurement effort.

This diminishing return means that for practical calibration work, N = 5 to 10 readings per calibration point is the typical standard. The gain from taking more than 20 readings is rarely worth the additional time, particularly when the Type A contribution is already small relative to the Type B contributions in the budget. Most calibration procedures specify a minimum N (often 3 or 5) and laboratories commonly use 10 as a default for temperature, pressure, and electrical calibrations.

Degrees of Freedom and the t-Distribution

The Type A evaluation based on N readings has ν = N − 1 degrees of freedom. When N is small (say, 3 to 6), the estimated standard deviation s is itself uncertain. There is a real possibility that s underestimates the true variability. To account for this, the correct coverage factor for a given confidence level comes from the t-distribution rather than the normal distribution.

For N = 5 (ν = 4 degrees of freedom), the t-factor for approximately 95% confidence is 2.78. Significantly larger than the normal-distribution value of 2.0. For N = 10 (ν = 9), the t-factor is 2.26. Only when ν ≥ 30 (N ≥ 31) does the t-distribution converge to the normal distribution value of 2.0. Most practical calibrations use N = 10, so the normal approximation introduces a small but often acceptable overconfidence. When N is small and the Type A term dominates, the Welch-Satterthwaite equation should be used to calculate effective degrees of freedom for the combined uncertainty, and the appropriate t-factor selected accordingly.

Section 3. Type B Uncertainty Evaluation in Detail

Type B uncertainty sources are almost always more numerous than Type A in a typical calibration uncertainty budget. Where a budget might have one Type A component (repeatability), it typically has three to six or more Type B components: the reference standard's calibration uncertainty, the instrument resolution, temperature effects, reference standard drift, and potentially other influences such as humidity, electromagnetic interference, or alignment effects.

The challenge in Type B evaluation is converting each information source into a standard uncertainty, which requires choosing an appropriate probability distribution to represent the state of knowledge about that contribution. The GUM provides guidance on the most commonly applicable distributions.

Normal Distribution

Used when the source of information is a calibration certificate that states an expanded uncertainty U at a known coverage factor k. The standard uncertainty is simply:

u_B = U / k

This is the most straightforward Type B evaluation. The calibration certificate for the reference standard states, for example: "U = 0.04°C at k = 2 (approximately 95% confidence, assuming a normal distribution)." The standard uncertainty is therefore u_B = 0.04 / 2 = 0.020°C. No distribution conversion is needed because the certificate already describes the uncertainty in terms appropriate to a normal distribution; dividing by k recovers the standard deviation.

Rectangular (Uniform) Distribution

Used when all values within a stated range are considered equally likely, that is, when there is no information to suggest that values near the centre of the range are more probable than values near the extremes. The standard uncertainty for a rectangular distribution with half-width a is:

u_B = a / √3

The rectangular distribution arises in several common situations:

• Digital instrument resolution: a digital display with a resolution of 0.1°C means the last digit could represent any value within ±0.05°C of the displayed value with equal probability. Standard uncertainty: u = 0.05 / √3 = 0.029°C.
• Manufacturer's accuracy specification: when a manufacturer states "accuracy ±0.5% of reading" without specifying a distribution, the GUM recommends the rectangular distribution as the default. Standard uncertainty: u = limit / √3.
• Environmental limits: when the laboratory's temperature is controlled within ±0.5°C, all temperatures within that range are equally likely, so the rectangular distribution is appropriate.
• Reference standard drift: when drift between calibrations is bounded by historical records (e.g. "drift does not exceed ±0.06 kPa between annual calibrations"), a rectangular distribution is a reasonable assumption in the absence of more specific information.

Triangular Distribution

Used less commonly, the triangular distribution applies when values near the centre of a range are more likely than values at the extremes, for example, when two independent rectangular distributions are combined, the result is triangular. The standard uncertainty is:

u_B = a / √6

where a is the half-width of the triangular distribution. The triangular distribution gives a smaller standard uncertainty than the rectangular distribution for the same half-width, reflecting the higher probability near the centre.

U-Shaped Distribution

The U-shaped (arcsine) distribution arises for sinusoidal signals and certain oscillating phenomena where values near the extremes of the range are actually more probable than values at the centre. The standard uncertainty is:

u_B = a / √2

This distribution is encountered less frequently in routine calibration work but appears in RF and AC electrical calibrations where sinusoidal waveforms are involved.

Section 4. Worked Examples: From Source to Standard Uncertainty

The following five examples trace the calculation from a raw uncertainty source through to a standard uncertainty value, using a pressure gauge calibration at 100 kPa as the running scenario. Each example identifies the source, the evaluation type, the appropriate distribution, and the resulting standard uncertainty.

Example 1. Type A: Repeatability of Gauge Readings

Ten readings of the pressure gauge under calibration at a nominal 100 kPa are taken using the reference standard as the comparison point. The ten readings yield an experimental standard deviation of s = 0.15 kPa.

Evaluation type: Type A. Distribution: normal (derived from data). Standard uncertainty: u_A = s / √N = 0.15 / √10 = 0.15 / 3.162 = 0.047 kPa.

Example 2. Type B: Reference Pressure Standard Calibration Certificate

The reference pressure standard used in the calibration was itself calibrated against a national standard. Its calibration certificate states: expanded uncertainty U = 0.08 kPa at coverage factor k = 2 (approximately 95% confidence, normal distribution).

Evaluation type: Type B. Distribution: normal (stated on certificate). Standard uncertainty: u_B = U / k = 0.08 / 2 = 0.040 kPa.

Example 3. Type B: Resolution of the Gauge Under Calibration

The pressure gauge under calibration has a digital display with a resolution of 0.1 kPa, that is, the smallest increment that can be read from the display. The displayed reading could differ from the true value by up to ±0.05 kPa, with all values in that range equally likely.

Evaluation type: Type B. Distribution: rectangular. Half-width: a = 0.05 kPa. Standard uncertainty: u_B = a / √3 = 0.05 / 1.732 = 0.029 kPa.

Example 4. Type B: Temperature Effect on the Gauge

The calibration is performed at a laboratory temperature of 20°C ± 2°C. The gauge's temperature coefficient (from its datasheet) is 0.01 kPa/°C. A ±2°C temperature variation could therefore introduce a pressure reading error of up to ±0.02 kPa. Since all temperatures within the ±2°C range are equally likely (rectangular distribution):

Evaluation type: Type B. Distribution: rectangular. Half-width: a = 0.02 kPa. Standard uncertainty: u_B = 0.02 / √3 = 0.02 / 1.732 = 0.012 kPa.

Example 5. Type B: Reference Standard Drift Between Calibrations

The reference pressure standard is calibrated annually. Historical records of the reference standard's calibration results over the past five years show that its reading has drifted by no more than ±0.06 kPa between any two consecutive annual calibrations. In the absence of information suggesting the drift follows a specific distribution, a rectangular distribution is assumed.

Evaluation type: Type B. Distribution: rectangular. Half-width: a = 0.06 kPa. Standard uncertainty: u_B = 0.06 / √3 = 0.06 / 1.732 = 0.035 kPa.

Section 5. Combining Type A and Type B: The RSS Method and a Full Budget Table

Once every uncertainty source has been evaluated as a standard uncertainty (regardless of whether it is Type A or Type B), all contributions are combined into a single combined standard uncertainty u_c using the root-sum-of-squares (RSS) method. The RSS method reflects the mathematical principle that if the uncertainty contributions are independent (uncorrelated), the total variance equals the sum of individual variances, and the combined standard uncertainty is the square root of the total variance.

For our pressure gauge example with five independent contributions:

u_c = √(u_A² + u_B1² + u_B2² + u_B3² + u_B4²)

Substituting the values from the five examples above:

u_c = √(0.047² + 0.040² + 0.029² + 0.012² + 0.035²)

u_c = √(0.002209 + 0.001600 + 0.000841 + 0.000144 + 0.001225)

u_c = √(0.006019)

u_c = 0.0776 kPa

The expanded uncertainty at coverage factor k = 2 (approximately 95% confidence for a normal distribution):

U = k × u_c = 2 × 0.0776 = 0.155 kPa ≈ 0.16 kPa

The value 0.16 kPa is the expanded uncertainty that would appear on the calibration certificate, rounded to two significant figures as recommended by the GUM. The certificate would state something like: "The reported expanded uncertainty is 0.16 kPa, estimated at approximately 95% confidence and determined in accordance with GUM methodology using a coverage factor k = 2."

Full Uncertainty Budget Table

The table illustrates a pattern typical of well-controlled calibrations: four of the five contributions are Type B, and the largest individual contribution is from the reference standard's calibration certificate (u = 0.040 kPa, contributing 0.001600 to u²). Not from the Type A repeatability term. The reference standard drift (Type B, u = 0.035 kPa) is also larger than the repeatability contribution. This is the normal situation: in a laboratory with well-controlled processes, the dominant uncertainty contributors are the reference standard and its associated Type B terms, not the day-to-day variability of the measurement.

Section 6. Common Misconception: Type A is Not Better Than Type B

Perhaps the most widespread misunderstanding about Type A and Type B uncertainty is the assumption that statistical evaluation (Type A) is somehow more rigorous, more objective, or more reliable than non-statistical evaluation (Type B). This is incorrect, and the GUM explicitly says so.

Consider two contributions to a temperature calibration budget. The first is the Type B uncertainty from the reference thermometer's calibration certificate, which traces back through an accredited laboratory to the National Metrology Centre (NMC) of A*STAR, singapore's national metrology institute. The certificate states U = 0.008°C at k = 2, giving u_B = 0.004°C. This value has been determined using primary measurement standards, with comprehensive uncertainty budgets evaluated by metrologists with decades of experience. The second contribution is a Type A uncertainty from five readings of a variable measurement system, giving s = 0.05°C and u_A = 0.05/√5 = 0.022°C.

The Type B contribution (0.004°C) is more than five times smaller than the Type A contribution (0.022°C), and it is far better characterised. Backed by a full technical record, traceability to the SI, and peer-reviewed methodology. The fact that it comes from a certificate (Type B) rather than from statistics (Type A) does not make it less reliable. If anything, the opposite is true in this case.

The practical lesson is this: when you read an uncertainty budget, do not focus on whether a contribution is labelled Type A or Type B. Focus instead on the magnitude of u² for each contribution (because it is u² that adds in RSS, not u directly), and ask which contributions dominate the combined uncertainty. The dominant contributions are the ones worth investigating if you want to reduce the total uncertainty.

When a Large Type A Contribution is a Warning Sign

While Type A is not inherently better than Type B, a large Type A contribution relative to the Type B contributions in the budget is a meaningful signal. It tells the laboratory that the measurement process has significant variability. More than the reference standard, the instrument resolution, and the environmental effects combined. This can indicate environmental instability (temperature fluctuations, vibration, air currents), poor contact between the sensor and the reference medium, electromagnetic interference, or operator effects such as inconsistent handling or connection technique.

A large Type A contribution is therefore not simply something to include in the budget and move on. It is an invitation to investigate the measurement setup. Can the environment be better controlled? Can the contact be improved? Is there an identifiable source of noise that can be eliminated? Reducing the process variability reduces u_A, and if u_A is currently the dominant term, this directly reduces the combined uncertainty and improves the laboratory's measurement capability (CMC). Conversely, if u_A is already the smallest term in the budget, taking more readings or reducing variability will have minimal impact on the combined uncertainty.

This is why understanding the structure of an uncertainty budget (knowing which contributions are large and which are small, and being able to identify Type A versus Type B), is practically valuable. It points calibration laboratories toward the right investments for improving measurement quality, and it helps customers understand what is driving the uncertainty on the certificates they receive.