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Type A / Type B Combined Uncertainty

Combine a statistical (Type A) uncertainty with an instrument/spec (Type B) contribution per the GUM — get u_c, expanded U, and effective degrees of freedom.

Inputu_A = s/√n u_B = a/√3 (rect), a/√6 (tri), a/√2 (U), a (normal 1σ) u_c = √(u_A² + u_B²) U = k·u_c ν_eff = u_c⁴ / (u_A⁴/(n−1))

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The engineering

The GUM splits uncertainty by how you got the number, not what causes it. Type A comes from statistics on your own repeated readings: the standard deviation of the mean, s/√n. Type B comes from everything else — a manufacturer's accuracy spec, a cal certificate, digital resolution — converted to a standard uncertainty by assuming a distribution over the stated limits. A ±a spec with no other information is a rectangular distribution, so u_B = a/√3. The two combine in quadrature, and multiplying by k = 2 gives roughly 95% coverage.

The dominant-contribution row is the one to act on. If Type A dominates, more averaging helps (u_A drops as 1/√n); if Type B dominates, no amount of extra readings will fix it — you need a better instrument or a tighter cal. Also check ν_eff: if it's below about 10, k = 2 is optimistic and you should pull k from the Student-t table instead.

Common gotcha: a spec quoted as '±0.1% of reading, 95% confidence' is already expanded — divide by 2 (normal), don't divide by √3. The rectangular divisor is only for limits with no stated confidence.

Where this math comes from

Before the 1980s every national lab combined 'random' and 'systematic' errors its own way — some added linearly, some in quadrature — and cal certificates from different countries couldn't be compared. In 1977 the CIPM asked the BIPM to sort it out; Recommendation INC-1 (1980) proposed the Type A / Type B split by method of evaluation rather than error cause, deliberately sidestepping the random-vs-systematic argument that had deadlocked metrologists for decades.

The full Guide to the Expression of Uncertainty in Measurement — the GUM — landed in 1993 under seven international bodies (ISO, IEC, BIPM, and others) and was reissued essentially unchanged as JCGM 100:2008, the free PDF every cal lab works from today. The effective-degrees-of-freedom row rides on the Welch–Satterthwaite approximation, a 1940s answer to combining variances with unequal sample sizes that the GUM pressed into metrology service.

  1. 1908William Gosset ('Student')Publishes the t-distribution — the reason small-n uncertainties need k > 2.
  2. 1946B. L. Welch & F. E. SatterthwaiteIndependently derive the effective-degrees-of-freedom approximation for combined variances.
  3. 1980BIPM Working Group / CIPMRecommendation INC-1 defines Type A and Type B evaluation, ending the random-vs-systematic stalemate.
  4. 1993ISO / BIPM / IEC et al.First edition of the GUM published; quadrature combination becomes the international rule.
  5. 2008JCGMJCGM 100:2008 reissues the GUM as the freely available standard this card implements.

See the full timeline of the math behind every calculator →

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