How to use the combined and expanded uncertainty calculator
An uncertainty budget starts by converting every component into a standard uncertainty in the same unit. A Type A component from repeat readings becomes standard deviation ÷ √(number of readings). A Type B component taken from a specification or calibration certificate is divided by a factor that depends on its distribution: √3 for a rectangular interval, √6 for a triangular one, and 2 for a value already stated at k = 2.
Square those standard uncertainties, add them, and take the square root to get the combined standard uncertainty u(c); multiply by the coverage factor to get the expanded uncertainty U. Worked check: a 0.0005 in standard deviation over 10 readings gives 0.000158 in, and adding rectangular components of 0.0002 in and 0.0001 in yields u(c) = 0.000204 in with U ≈ 0.00041 in at k = 2, where repeatability accounts for 60% of the variance.
This follows the basic procedure of the GUM and NIST Technical Note 1297, and it assumes the components are independent with sensitivity coefficients of one. Components measured in a different quantity must be multiplied by their sensitivity coefficient before entry, and correlated inputs need covariance terms. When the effective degrees of freedom are low the coverage factor has to be recomputed from the t distribution, so confirm any value destined for a calibration certificate against your laboratory procedure.
Frequently asked questions
A specification written only as plus or minus a is treated as a rectangular distribution, meaning any value inside the interval is equally likely. That distribution has a standard deviation of a divided by the square root of 3, so dividing the half-width by 1.732 converts it to a standard uncertainty. Use the square root of 6 for a triangular component and divide by 2 for a value already reported at k = 2.
Multiplying the combined standard uncertainty by 2 gives an interval with roughly 95 percent confidence when the result is approximately normal. With few components or few repeat readings the effective degrees of freedom are low, so a t-based coverage factor larger than 2 may be required.