How this calculation works
Convert each input magnitude to a standard uncertainty using its stated distribution. Combine signed sensitivity-weighted terms through a validated correlation matrix rather than automatically using independent root-sum-squares.
Combined variance = sumᵢ sumⱼ (cᵢuᵢ) correlationᵢⱼ (cⱼuⱼ); expanded uncertainty = entered coverage factor × combined standard uncertaintyA worked example
Independent standard components three and four with unit sensitivities combine to five. With coverage factor two, expanded uncertainty is ten; positive or negative correlation changes that result.
Good to know
First-order propagation only. Components must share output units after sensitivities. Coverage factor does not automatically mean 95%; no effective degrees of freedom, nonlinear Monte Carlo or completeness certification.
Reference: NIST: combining standard uncertainties (opens in a new tab) ↗
Distinguish a standard deviation from a half-width
A standard input is already a standard uncertainty. A rectangular or triangular input is a symmetric half-width, converted by dividing by square root three or square root six respectively.
Applying the half-width divisor to an input that is already a standard deviation would understate its contribution. The source table preserves both original magnitude and converted standard value.
Correlation and sensitivity signs interact
A sensitivity can be negative even though its source uncertainty is nonnegative. Correlation then determines whether the cross term increases or decreases combined variance.
Pairwise correlations between minus one and one are not enough by themselves: the complete matrix must be positive semidefinite. Inconsistent sets are rejected instead of producing an invalid variance.
Coverage is a separate assumption
The entered coverage factor multiplies combined standard uncertainty. It is not an automatic probability statement for every distribution, data size or nonlinear measurement model.
A relative expanded uncertainty is undefined when the reported estimate is zero. The page labels that condition instead of dividing by zero or pretending the result is zero percent.
Use the result with the next measurement
The Welch two sample t test calculator estimates sampling uncertainty for a difference of independent means. A general measurement budget can include other sources and correlations that its standard error does not capture.
First-order propagation only. Components must share output units after sensitivities. Coverage factor does not automatically mean 95%; no effective degrees of freedom, nonlinear Monte Carlo or completeness certification.
A few questions, answered
Are rectangular inputs standard deviations?
No, enter their symmetric half-widths.
Are sensitivity signs retained?
Yes, they matter in covariance terms.
Does factor two always guarantee 95%?
No, coverage interpretation needs additional assumptions.