Documentation › The fit report
The fit report
The Results step shows the plot with the fitted curve, and the report: the convergence status, each parameter with its uncertainty, the goodness-of-fit statistics, and optional diagnostics. Each part answers a different question, and none of them answers all of them.
Convergence status
The status is Minuit's own verdict on the minimisation, reported verbatim. Fit converged (status 0) means the minimiser met its stopping conditions and the covariance matrix is accurate. The others return numbers that should not be used as they stand: covariance matrix was made positive-definite (1) means the uncertainties are approximate; Hesse matrix is invalid (2) means they are not reliable at all; estimated distance to minimum is above the limit (3) and maximum number of function calls (4) mean the fit stopped before reaching a minimum. The usual causes are starting values far from the answer, a parameter the data cannot determine (see correlations below), or a model that cannot describe the data. Convergence says the procedure finished; it says nothing about whether the model is right. That is what the statistics and diagnostics are for.
Parameters and uncertainties
Each parameter is reported with its standard uncertainty, the square root of the corresponding diagonal element of the covariance matrix. Minuit obtains the covariance from the second derivatives of χ² at the minimum (the Hesse matrix), inverted: this is the uncertainty for which χ² rises by one when that parameter is moved away from its best value and all the others are re-optimised. It is a 68% interval under the usual assumptions (Gaussian errors, a model that is approximately linear near the minimum). With no Y uncertainties supplied, the fit assumes σ = 1 for every point and the reported parameter uncertainties inherit that assumption; they are then only meaningful if your data really do scatter by about one unit.
Quote a result to a precision consistent with its uncertainty: one or two significant figures on the uncertainty, and the value rounded to the same decimal place. 2.4 ± 0.08 s, not 2.3871 ± 0.0813 s.
χ², NDF and the p-value
χ² at the minimum is the sum over fitted points of (residual / uncertainty)². NDF, the number of degrees of freedom, is the number of fitted points minus the number of free parameters. If the model is correct and the uncertainties are right, each point contributes about 1 on average, so χ² ≈ NDF and the reduced chi-square χ²/NDF is near 1; more precisely, χ² is then distributed as a chi-square variable with NDF degrees of freedom, which has mean NDF and standard deviation √(2·NDF). The p-value is the probability that a correct model would give a χ² at least this large.
- χ²/NDF ≫ 1 (p ≈ 0): the points scatter about the curve more than their error bars allow. Either the model is missing something, or the uncertainties are underestimated. The residuals tell which: a trend means the model, random excess means the uncertainties.
- χ²/NDF ≪ 1 (p ≈ 1): the points sit closer to the curve than their error bars say they should. The uncertainties are overestimated, or the data have been smoothed, or the model has enough free parameters to follow noise.
- χ²/NDF ≈ 1: consistent with the model and the stated uncertainties. This does not prove the model; it means the data cannot distinguish it from the truth at this precision.
How far from 1 is "far" depends on NDF: with 5 degrees of freedom a χ²/NDF of 1.6 is unremarkable (p ≈ 0.15); with 500 it is decisive. Read the p-value rather than the ratio alone. For a histogram fitted by Poisson likelihood, the report gives the deviance in place of χ², which behaves the same way at high counts, and no p-value.
Residuals and pulls
Optional panels under the plot show, for each point, the residual (data − fit), the pull (residual ÷ its effective uncertainty), the data/fit ratio, the percentage difference, or a histogram of the residuals. These are the most informative part of the report. For a correct model with correct uncertainties the pulls are a sample from a standard normal distribution: scattered about zero with no trend in x, about two-thirds within ±1, and hardly any beyond ±3. A bow, a wave or a step in the residuals is systematic disagreement, a sign that the model is wrong or incomplete, even when χ²/NDF looks acceptable. A single pull of ±4 is an outlier worth investigating. The diagnostics are computed with the fitted curve held fixed; they are not uncertainty bands.
Correlations
The correlation matrix (the covariance normalised to ±1) shows how the fitted parameters trade off. A correlation near ±1 between two parameters means the data constrain only a combination of them: moving one and compensating with the other leaves the curve almost unchanged. This happens when a feature is outside the measured range (the centre of a peak you only caught the tail of), or when two parameters play the same role (an amplitude and a normalisation). The individual uncertainties are then large and should be read together with the correlation; the uncertainty of a derived quantity that combines both parameters can be much smaller, which Analyze Data accounts for by carrying the full covariance.
Confidence band
An optional band at 68%, 95% or 99% shows the uncertainty of the fitted curve: at each x, the curve's value is propagated from the parameter covariance to first order (ROOT's GetConfidenceIntervals). It is drawn only when the fit converged with an accurate covariance matrix. It is the region in which the true curve plausibly lies given your data; it is not a prediction interval for where a new measurement would fall, which is wider by the measurement uncertainty. No χ² rescaling is applied.
Point exclusions
Point exclusions… removes individual measurements from an XY fit without deleting them. Select rows (Shift-click for a range, or type row numbers), mark them excluded, and record a reason. Excluded points stay on the plot as open grey markers, are left out of χ², the diagnostics and the point count, and their reasons are saved with the session and printed in exports. Legitimate reasons are a known fault (a saturated detector, a mis-set instrument), a point outside the regime the model describes, or a transcription error you cannot correct. Excluding a point because it spoils the fit is not one; the reason field exists so that the decision is recorded and can be judged. Exclusions are checked against the data each time you fit, so if the excluded point's values change, you are asked to review them. For a contiguous region, a fit range is simpler.
When a fit fails or looks wrong
Check the obvious first: parameter order and signs, units consistent between the data and the model, and the function itself. Then try other physically reasonable starting values. If several different starts reach the same parameters, the minimum is well defined; if they reach different ones with similar χ², the data do not distinguish them, and the measured range or precision, not the fitting, is the limitation. Large uncertainties and strong correlations say the same thing. Do not adjust the uncertainties to make χ²/NDF come out near 1: they are measurements of your apparatus, not tuning parameters.