How do I propagate uncertainty through calibration?

You propagate uncertainty through calibration corrections by using the calibration equation y = m x + b, then combining the uncertainties from the slope m, intercept b, and the measured x with standard error propagation formulas to find the uncertainty in y.

Think of the calibration like a straight line that turns the number you measure into the number you want. If the line has uncertainty in its slope and intercept, and your measurement also wiggles a bit, each wiggle can change the final answer. Mathematically you compute the partial derivative of y with respect to each variable, square those derivatives times the variance of each variable, and add them up, and then take the square root to get the combined standard uncertainty. For the linear calibration y equals m x plus b, the uncertainty in y is the square root of x squared times the variance of m plus the variance of b plus m squared times the variance of x plus twice x times the covariance of m and b if those two are correlated. If the measurement uncertainty in x is independent of the fit parameters, you can omit covariances with x.

A common mistake is forgetting the uncertainty in the calibration fit and treating the slope and intercept as exact, which underestimates the true uncertainty, so always include fit parameter variances; another mistake is ignoring covariance between slope and intercept when the calibration fit provides it, and that covariance can increase or decrease the combined uncertainty so include it when available; a third mistake is using absolute instead of relative units inconsistently, so check units and propagate variances in the same unit system to avoid arithmetic errors.

For more videos and downloadable worksheets that walk through worked examples of propagating uncertainty through calibration corrections, please visit w w w dot gumball degree dot com.

Propagate Uncertainty Through Calibration Corrections

1. Use calibration line y = m x + b
2. Include slope and intercept uncertainties
3. Include measurement uncertainty in x
4. Use partial derivatives for propagation
5. Sum squared terms then take root
6. Include covariance of m and b if correlated

Common Calibration Uncertainty Mistakes

1. Ignoring fit parameter uncertainty
2. Omitting slope-intercept covariance
3. Mixing inconsistent units

calibration, uncertainty, errorpropagation, measurement, metrology, statistics, instrumentation, dataanalysis, linearfit, uncertaintyanalysis