The distribution of residuals divided by their uncertainties; a check that the errors themselves are right.A pull distribution is a diagnostic tool used in physics and other sciences to check whether the uncertainties assigned to measurements are realistic. It is constructed by taking each data point, subtracting the expected value (the model prediction), and dividing by the uncertainty of that point. The resulting 'pull' values should follow a standard normal distribution (mean 0, standard deviation 1) if the model is correct and the uncertainties are properly estimated.
The term 'pull' has been used in high-energy physics since at least the 1980s, though the underlying idea of standardizing residuals is older and appears in statistics as 'studentized residuals'.
If your uncertainties are too small, the pulls will be larger than 1 on average, producing a distribution that is too wide. If your uncertainties are too large, the pulls will cluster near zero, making the distribution too narrow. By fitting a Gaussian to the pull distribution and checking its width, you can detect miscalibrated errors. A width significantly different from 1 signals that your error estimates need adjustment — a common issue in complex analyses where systematic effects are underestimated.
It is routinely used in particle physics experiments (e.g., at CERN) to validate the uncertainty estimates in fits to particle masses or cross-sections, and in astronomy for checking the calibration of telescope measurements.
A pull distribution that looks perfect — a clean Gaussian with width exactly 1 — is so rare in real analyses that physicists often become suspicious it was 'too good' and check for hidden correlations.
