Calculator
Example data table
| # | x | Observed y | σ | Expected y (linear example) |
|---|---|---|---|---|
| 1 | 0 | 1.02 | 0.05 | 1.00 |
| 2 | 1 | 2.05 | 0.05 | 2.00 |
| 3 | 2 | 2.98 | 0.05 | 3.00 |
| 4 | 3 | 4.02 | 0.05 | 4.00 |
| 5 | 4 | 5.00 | 0.05 | 5.00 |
Formula used
The chi-square statistic measures weighted disagreement between data and a model:
The reduced chi-square divides by degrees of freedom:
Here, N is the number of points, p is the number of fitted parameters, σ are uncertainties, and f are expected values from your model.
How to use this calculator
- Choose an input mode: provide expected values directly, or compute them from a model.
- Select weighting. If you know uncertainties, use provided σ for meaningful goodness-of-fit.
- Paste data rows. Use commas or spaces; one row per line. Comments can start with #.
- If using a model, choose a model type and either fit parameters or enter them manually.
- Press Submit. Results appear below the header and above the form.
- Use Download CSV or Download PDF to save your report.
Notes for interpretation
- χ²red ≈ 1 often indicates agreement within stated uncertainties.
- χ²red > 1 can indicate model mismatch or underestimated σ.
- χ²red < 1 can indicate overestimated σ or an overly flexible model.
- The p-value is the probability of seeing χ² this large or larger, assuming the model and σ are correct.
1) Why reduced chi-squared matters
Reduced chi-squared, χ2red, is a goodness-of-fit metric in physics. It compares observed data with expected values while accounting for uncertainty. Because it is normalized by degrees of freedom, it supports comparisons across datasets and models with different parameter counts.
2) Inputs: observed, expected, and uncertainty
The core statistic is χ2 = Σ ((yi - fi)2 / σi2). Each term measures how many standard deviations a point misses the model. When uncertainties represent one-sigma errors, the weighting matches least-squares assumptions.
3) Degrees of freedom and parameter count
Degrees of freedom are ν = N - p, where N is the number of points and p is the number of fitted parameters. For example, N = 25 with p = 3 gives ν = 22. The reduced value is χ2red = χ2 / ν, which prevents extra parameters from making a fit look better.
4) How to interpret typical values
Values near 1 often indicate consistency between the model, data scatter, and uncertainties. Values much larger than 1 can signal model mismatch, underestimated σ, drift, or unmodeled systematics. Values well below 1 can suggest overestimated σ or correlated points that reduce scatter.
5) Screening ranges and experimental context
Many teams use screening bands such as 0.7 to 1.3, but the right tolerance depends on ν and the experiment. With small ν, χ2red can fluctuate widely. With large ν, modest departures from 1 can be meaningful.
6) Using the p-value as a companion metric
The p-value is the probability of obtaining a χ2 at least as large, assuming the model and uncertainties are correct. A very small p-value suggests the mismatch is unlikely from random noise alone. A very large p-value can warn that uncertainties are inflated.
7) Workflow with direct and model modes
Use direct mode when you know expected values and only need the statistic and degrees of freedom. Use model mode to generate expected values from standard forms and optionally fit parameters by least-squares (weighted when σ is provided). Ensure ν stays positive; if N is not greater than p, reduced chi-squared is undefined.
8) Reporting and exporting results
For reporting, record the model form, parameter values, N, p, ν, χ2, and χ2red. Describe how uncertainties were estimated. Export the CSV to preserve per-point residuals and contributions, and attach the PDF summary for traceability and complete documentation.
1) What does χ2red ≈ 1 indicate?
It usually means residuals are consistent with stated uncertainties under the chosen model. It does not prove the model is correct; it suggests the mismatch is compatible with random measurement noise.
2) Can I use this calculator without uncertainties?
Yes. Select no σ and the calculator assumes σ = 1 for each point. This supports relative comparisons on the same dataset, but it is not a strict statistical goodness-of-fit test.
3) Why is ν = N - p so important?
ν sets the normalization and the expected spread of χ2. If p is large compared with N, ν becomes small and χ2red becomes unstable. If ν is not positive, it is undefined.
4) My χ2red is much larger than 1. What should I check?
Confirm data entry, units, and that σ values reflect realistic one-sigma errors. Then inspect residual patterns for nonlinearity, offsets, drift, or missing physics. Large values often mean model mismatch or underestimated uncertainty.
5) My χ2red is far below 1. Is that a problem?
It can indicate uncertainties are overstated or points are correlated, reducing effective scatter. It may also occur when a model is overly flexible for the dataset. Revisit how σ was estimated and any averaging steps.
6) When should I fit parameters instead of entering them?
Fit parameters when you want the best agreement under the chosen model. Enter parameters when they come from theory, calibration, or independent measurements and you want to test consistency of new data against those values.
7) What should I include when reporting results?
Report the model form, parameter values, N, p, ν, χ2, and χ2red, plus how σ was obtained. Attach the CSV or PDF export to preserve the calculation trail and residual details.