Probability plot

A probability plot compares order statistics with plotting positions or model quantiles to diagnose whether a proposed probability distribution describes data. If the model can be transformed into a linear identity $T(F(x))=a+bS(x)$, plot $T(i/(n+1))$ against $S(X_{(i)})$. Approximate linearity supports that model's shape; systematic curvature suggests a mismatch.

The plotting position $i/(n+1)$ comes from the exact mean of the $i$th uniform order statistic, by the probability integral transform. It does not assert equality between the expectation of a nonlinear transformation and that transformation of the expectation. Fitted slope and intercept can estimate model parameters, but the visual plot alone is not a calibrated hypothesis test.