Almost everyone’s first guess about Type A and Type B is wrong, and it’s a reasonable guess: that Type A means random and Type B means systematic.
It doesn’t. The classification has nothing to do with what causes an uncertainty. It’s about how you worked the number out.
- Type A — evaluated statistically, from repeated observations you made yourself.
- Type B — evaluated by any other means: certificates, specifications, published data, documented judgement.
That’s the entire distinction. A single physical effect can be Type A in one budget and Type B in another, depending purely on how you got a number for it. There’s a section below with examples of exactly that.
The Short Answer
| Type A | Type B | |
|---|---|---|
| How it’s evaluated | Statistical analysis of repeated readings | Everything else |
| Where the data comes from | Your own measurements, now | Certificates, specs, published data, experience |
| Typical formula | u = s ÷ √n | u = value ÷ divisor |
| Distribution | Assumed normal | Whatever you judge it to be |
| Degrees of freedom | n − 1 | Usually treated as ∞ |
| Needs repeated readings? | Yes, that’s the definition | No |
| Usually the larger share? | No | Yes, typically |
Both are standard uncertainties once evaluated. Once they’re in the budget, they’re treated identically — squared, summed, rooted. The classification affects how you get the number, not what you do with it afterwards.
What Is Type A Uncertainty?
Type A uncertainty is evaluated by the statistical analysis of a series of observations.
In practice: you measure the same thing several times, the readings scatter a bit, and you use that scatter to quantify how repeatable your measurement is.
Six readings of a voltage reference: 10.0012, 10.0015, 10.0011, 10.0014, 10.0013, 10.0012 V
The spread across those six readings is real information about your measurement process, and Type A evaluation turns it into a number.
What it captures: everything that varies between one reading and the next — electrical noise, small changes in how you position the instrument, thermal drift over the measurement period, your own judgement varying slightly each time.
What it misses entirely: anything that affects every reading identically. If your instrument reads 0.002 V high, all six readings are high by the same amount and the scatter tells you nothing about it. Type A is blind to consistent effects.
What Is Type B Uncertainty?
Type B uncertainty is evaluated by means other than statistical analysis of repeated observations.
That’s a definition by exclusion, which is unusual but accurate. It’s everything you look up, read off a document, or judge from experience.
Common sources:
- Calibration certificates — the stated uncertainty of your instrument or reference standard
- Manufacturer’s specifications — accuracy, linearity, temperature coefficient, one-year stability
- Instrument resolution — the smallest increment displayed
- Published data — thermal expansion coefficients, material properties, reference values
- Previous measurement data — results from earlier studies on the same setup
- Documented judgement — an honest, reasoned estimate where nothing else exists
That last one deserves a word, because people are uncomfortable with it. The GUM explicitly permits estimates based on experience, and a documented estimate is far better than pretending a source doesn’t exist. Omitting a contribution claims it equals zero, which is never true. Write down your reasoning and put a number in.
The Real Difference: How You Got the Number
Everything above reduces to one question: did this number come out of a statistical calculation on your own repeated readings, or from somewhere else?
Statistical calculation on your readings → Type A. Anything else → Type B.
That’s it. Not whether the effect is random. Not whether it’s correctable. Not whether it’s large or small. Just where the number came from.
This is deliberately mechanical, and the next section explains why.
Why Not “Random” and “Systematic”?
Older textbooks split uncertainty by cause: random errors scatter, systematic errors shift. Useful for understanding what’s going wrong, and the GUM replaced it as a classification for one specific reason.
The random/systematic split isn’t stable. It depends on where you draw the boundary around your measurement.
Take an instrument that reads 0.02 mm high.
For you, using that one instrument, this is unambiguously a systematic effect. Every reading is shifted. Repeating won’t reveal it.
Now look at a population of a hundred such instruments coming off the production line. Each has its own small offset, and across the population those offsets scatter randomly around zero. The same physical phenomenon is now a random effect.
Which is it? Depends entirely on the scope of the question which makes it useless as a filing system.
Type A and Type B don’t have this problem. They describe how you obtained the number, and that’s a fact about your process, not a judgement about causes. It’s unambiguous regardless of where you draw the boundary.
This is why modern calibration certificates state uncertainties with a coverage factor and never mention random or systematic error. The concepts remain useful for diagnosing problems knowing that averaging fixes one and not the other is practical knowledge. They’re just no longer how uncertainty is classified.
How to Calculate Type A Uncertainty
Mean:
x̄ = Σxᵢ ÷ n
Sample standard deviation:
s = √[ Σ(xᵢ − x̄)² ÷ (n − 1) ]
Standard uncertainty of the mean:
u_A = s ÷ √n
Two things decide whether this is right.
The divisor in the standard deviation is n − 1. That’s STDEV.S in Excel, not STDEV.P. The n−1 version is the sample standard deviation and it’s the correct one. With five readings, using n instead understates your uncertainty by about 12%.
The ÷√n is not optional. You’re reporting the mean of your readings, and a mean is more reliable than any individual reading. Stopping at s gives you the spread of single measurements a different quantity, roughly twice as large with five readings.
Degrees of freedom: ν = n − 1. Keep this; you may need it later.
How to Calculate Type B Uncertainty
Every Type B source arrives in a different format, so each gets converted:
u_B = a ÷ divisor
where a is the half-width of the interval.
| Source | Distribution | Divisor |
|---|---|---|
| Certificate stating k = 2 | Normal | 2 |
| Certificate stating k = 3 | Normal | 3 |
| Instrument resolution | Rectangular | √3 |
| Manufacturer’s ± limits | Rectangular | √3 |
| Temperature between two limits | Rectangular | √3 |
| Value most likely near the centre | Triangular | √6 |
| Value most likely near the extremes | U-shaped | √2 |
Resolution has an extra step people miss. The half-width is half the resolution:
u = (resolution ÷ 2) ÷ √3 ≈ resolution × 0.289
A 0.0001 V display gives 0.0000289 V, not 0.0000577 V.
When you don’t know the distribution, choose rectangular. It’s the conservative choice and what the GUM recommends when you have limits and nothing more.
Where the Divisors Come From
Most guides give you √3 and √6 as facts to memorise. They’re not arbitrary each is the standard deviation of its distribution, and knowing why makes them impossible to mix up.
Rectangular (uniform) distribution. The true value is equally likely to be anywhere between −a and +a. The variance of a uniform distribution over that interval is a²/3, so the standard deviation is:
σ = a ÷ √3 ≈ 0.577a
Use it when you have limits and no reason to think the middle is more likely than the edges — a tolerance band, a resolution window, a temperature range.
Triangular distribution. The true value is more likely near the centre and tapers to zero at ±a. Variance is a²/6:
σ = a ÷ √6 ≈ 0.408a
Use it when you have limits and good reason to believe the centre is favoured for instance when a value results from several combined effects that tend to cancel.
U-shaped (arcsine) distribution. The value spends most of its time near the extremes typical of a cyclically varying quantity, like temperature in a room with a thermostat that overshoots. Variance is a²/2:
σ = a ÷ √2 ≈ 0.707a
Normal distribution from a certificate. The certificate already tells you the coverage factor, so you divide by it. That’s not a distribution property it’s just undoing the expansion the calibration laboratory applied.
Notice the pattern: the more the probability clusters at the centre, the smaller the standard deviation, so the larger the divisor. Triangular (√6) gives a smaller uncertainty than rectangular (√3) from the same limits, because you’re claiming more knowledge about where the value sits.
Degrees of Freedom for Type A and Type B
Degrees of freedom describe how reliable your uncertainty estimate itself is. They matter when you calculate the coverage factor.
Type A: ν = n − 1. Six readings give five degrees of freedom.
Type B: usually treated as infinite. A well-characterised source a calibration certificate from an accredited laboratory is taken as ν = ∞, meaning you’re confident in the value.
But that’s a convenience, not a law. If you’re genuinely unsure about a Type B estimate, the GUM gives a formula:
νᵢ ≈ ½ [ Δu(xᵢ) ÷ u(xᵢ) ]⁻²
where Δu/u is how uncertain you are about your own uncertainty estimate, as a fraction.
| How sure are you of the estimate? | Δu/u | ν |
|---|---|---|
| Very confident (±10%) | 0.10 | 50 |
| Reasonably confident (±25%) | 0.25 | 8 |
| Rough estimate (±50%) | 0.50 | 2 |
So a source you’d describe as “a rough guess, could easily be half or double” carries about 2 degrees of freedom and if it dominates your budget, that will push your coverage factor well above 2.
Most people never do this calculation. It’s worth knowing it exists for the case where a large, poorly known Type B source is carrying your budget.
Sources That Could Be Either
This is the section that makes the classification click. The same physical effect can be Type A or Type B depending only on how you evaluated it.
Temperature effects. Log the temperature every minute for two hours and calculate the standard deviation of the variation → Type A. Take the environmental chamber’s specification of ±1 °C and divide by √3 → Type B. Same physical effect. Different route to the number.
Operator variation. Run a study with five operators measuring the same part, calculate the standard deviation of their means → Type A. Estimate from experience that operators differ by about ±0.005 mm and treat it as rectangular → Type B.
Instrument drift. Compare the last four calibration certificates and calculate the standard deviation of the changes → Type A. Use the manufacturer’s one-year stability specification → Type B.
Workpiece form error. Measure at eight angular positions and take the standard deviation → Type A. Use the roundness tolerance from the drawing, divided by √3 → Type B.
Some sources can only be Type B. Resolution is the clearest case it’s a fixed property of the display, and no amount of repeating will produce a statistical estimate of it.
The practical point: if a source dominates your budget and you evaluated it as a rough Type B estimate, consider measuring it properly and converting it to Type A. You’ll usually get a smaller, better-justified number — and more degrees of freedom.
Pooled Standard Deviation: Type A Without Repeating
A real situation the textbooks rarely cover. What if you can only take one reading?
Destructive tests. Production measurements where time is money. Anything where repeating isn’t practical.
You can still have a Type A term, using historical repeatability data. If you’ve previously characterised this measurement process across several sets of readings, pool those standard deviations:
s_p = √[ Σ(nᵢ − 1)sᵢ² ÷ Σ(nᵢ − 1) ]
That pooled figure describes your process, not any single session. Then for a measurement where you take just one reading:
u_A = s_p ÷ √1 = s_p
Two advantages. You get a Type A term without repeating anything today. And the degrees of freedom are the sum of all the historical ones often fifty or more, which comfortably justifies k = 2.
This is standard practice in production metrology and accredited laboratories. It requires the process to be genuinely stable, and the historical data to have been gathered under the same conditions but where those hold, it’s a much better answer than guessing.
Worked Example: Both Types in One Budget
A digital multimeter measuring a 10 V DC reference standard.
Type A — six readings:
10.0012, 10.0015, 10.0011, 10.0014, 10.0013, 10.0012 V
Mean = 10.00128 V Sample standard deviation, s = 0.000147 V u_A = 0.000147 ÷ √6 = 0.000060 V (ν = 5)
Type B — four sources:
| Source | Value | Distribution | Divisor | Standard uncertainty |
|---|---|---|---|---|
| Reference standard certificate | 0.00010 V (k=2) | Normal | 2 | 0.000050 V |
| DMM one-year specification | 0.00015 V | Rectangular | √3 | 0.000087 V |
| Resolution (0.0001 V) | 0.00005 V | Rectangular | √3 | 0.000029 V |
| Temperature coefficient | 0.00005 V | Rectangular | √3 | 0.000029 V |
Combine everything:
u꜀ = √(0.000060² + 0.000050² + 0.000087² + 0.000029² + 0.000029²) u꜀ = 0.000124 V
U = 2 × 0.000124 = 0.0002 V
Result: 10.0013 ± 0.0002 V (k = 2)
The breakdown by contribution:
| Source | Type | % of variance |
|---|---|---|
| DMM specification | B | 49.1% |
| Repeatability | A | 23.6% |
| Reference certificate | B | 16.4% |
| Resolution | B | 5.5% |
| Temperature coefficient | B | 5.5% |
Type A total: 23.6%. Type B total: 76.4%.
Which Type Usually Dominates?
Type B, in most careful measurements and by a wide margin.
That surprises people, because Type A is the visible one. You can see your readings scatter. You can’t see your instrument’s specification quietly contributing half the total.
The example above is typical: repeatability contributed under a quarter, while the multimeter’s own specification contributed nearly half.
The practical consequence: taking more readings is the improvement everyone reaches for first, and it’s frequently among the least effective. Doubling from six readings to twenty-four would have halved the Type A term and reduced the total uncertainty in that example by about 9%.
Getting a better DMM specification, or calibrating against a lower-uncertainty reference, would have done far more.
When Type A does dominate: unstable processes, manual techniques with high operator variability, low-resolution instruments used on noisy signals, and any measurement where you’ve taken very few readings.
How to Decide Which Type a Source Is
One question: did I calculate this number from a standard deviation of my own repeated readings?
Yes → Type A. No → Type B.
That’s genuinely all there is to it. The confusions come from asking the wrong questions:
- “Is the effect random or systematic?” — irrelevant to the classification.
- “Is it correctable?” — irrelevant. A correctable bias should be corrected, and the residual doubt goes into the budget as either type.
- “Is it big or small?” — irrelevant.
- “Is it from the instrument or the environment?” — irrelevant. Either can be evaluated either way.
Common Misconceptions
“Type A means random error and Type B means systematic error.” The most common misunderstanding. The classification is by evaluation method, not cause. A systematic effect measured statistically is Type A.
“Type B is less rigorous than Type A.” No. A calibration certificate from an accredited laboratory is a very strong piece of evidence — usually stronger than six readings you took this morning.
“You can’t use judgement in an uncertainty budget.” The GUM explicitly allows documented estimates as Type B. A reasoned estimate beats omitting a source, which claims it contributes zero.
“Type A and Type B are combined differently.” They aren’t. Once each is a standard uncertainty, all sources are squared, summed and rooted identically.
“Type B sources always have infinite degrees of freedom.” It’s a convenient default for well-characterised sources. A rough estimate can carry as few as 2 degrees of freedom, and if it dominates, your coverage factor will need to rise.
“A source is permanently one type or the other.” The same effect can be either, depending on how you evaluated it. Temperature is the clearest example.
“You need repeated readings to have a Type A component.” Not with a pooled standard deviation from historical data on a stable process.
Frequently Asked Questions
What is the difference between Type A and Type B uncertainty?
Type A is evaluated by statistical analysis of repeated observations you made. Type B is evaluated by any other means certificates, specifications, published data or documented judgement. The distinction is about method, not cause.
Is Type A random error and Type B systematic error?
No. That’s the most common misconception. The GUM deliberately replaced the random/systematic split because it depends on how you define the measurement, whereas Type A and Type B describe how the number was obtained.
How do you calculate Type A uncertainty?
Take repeated readings, calculate the sample standard deviation using n − 1, then divide by √n.
How do you calculate Type B uncertainty?
Divide the source value by a factor based on its assumed distribution 2 for a k = 2 certificate, √3 for rectangular sources, √6 for triangular.
Why do we divide by √3 for a rectangular distribution?
Because the standard deviation of a uniform distribution over ±a is a ÷ √3. The variance works out to a²/3, so the divisor follows from the mathematics of the distribution.
Why divide by √6 for triangular?
The variance of a triangular distribution over ±a is a²/6, giving a standard deviation of a ÷ √6. Because probability is concentrated near the centre, it gives a smaller uncertainty than rectangular from the same limits.
What are the degrees of freedom for Type B uncertainty?
Usually taken as infinite for well-characterised sources. If you’re genuinely unsure of an estimate, the GUM gives ν ≈ ½(Δu/u)⁻², so a rough estimate you’d trust to ±50% carries about 2 degrees of freedom.
Can one source be both Type A and Type B?
Not simultaneously, but the same physical effect can be evaluated either way. Temperature variation measured with a logger is Type A; taken from an environmental specification, it’s Type B.
Which type is usually larger?
Type B, in most careful measurements. Instrument specifications and calibration certificates typically contribute more than repeatability.
Do I need repeated readings for a Type A component?
Normally yes, but a pooled standard deviation from historical data lets you include a Type A term even when you take a single reading, provided the process is stable.
Conclusion
Type A and Type B is a filing system for how you got the number, and nothing more.
- Type A — from a statistical calculation on your own repeated readings. u = s ÷ √n, with ν = n − 1.
- Type B — from everything else. Value divided by 2, √3 or √6 depending on the distribution, usually with ν = ∞.
- It’s not random versus systematic. That older split was abandoned because it changes depending on where you draw the boundary around the measurement.
- The divisors aren’t arbitrary. √3 and √6 are the standard deviations of the uniform and triangular distributions over ±a.
- The same effect can be either type. How you evaluated it decides, not what causes it.
- Once evaluated, both are treated identically — squared, summed, rooted.
- Type B usually dominates. Which means taking more readings is often not the improvement you need.
