Chi-Squared Calculator for Physics Data

Example data table

A small sample showing observed values, model expectations, and one-sigma uncertainties.

Formula used

Uncertainty-weighted chi-square:

χ² = Σᵢ (Oᵢ − Eᵢ)² / σᵢ²

Goodness-of-fit chi-square:

χ² = Σᵢ (Oᵢ − Eᵢ)² / Eᵢ

Degrees of freedom:

ν = N − p (or another chosen rule)

Reduced chi-square:

χ²ν = χ² / ν

How to use this calculator

1. Choose a calculation mode that matches your dataset.
2. Enter observed and expected values for each measurement point.
3. In uncertainty mode, enter σ for every row.
4. Select a degrees-of-freedom rule and parameter count, if needed.
5. Press Compute to view χ², ν, reduced χ², and p-value.
6. Use the export buttons to download CSV or PDF summaries.

Chi-square in physics data analysis

1) Purpose in experimental comparison

Chi-square summarizes how strongly measurements deviate from a model after you account for expected scatter. It is used for response curves, background-subtracted spectra, and validation of simulations against laboratory data. Compute it after calibration and consistent normalization.

2) Weighted form for measurement uncertainty

For most instrument readings, use the uncertainty-weighted form Σ(O−E)²/σ². Smaller σ values naturally carry more influence, matching standard least-squares fitting. If σ represents one-sigma errors and points are independent, chi-square connects to likelihood-based confidence intervals. Many fitting tools minimize χ² to find best parameters reliably.

3) Count data and the E denominator

For count-like bins where variance is near the expectation, a goodness-of-fit form Σ(O−E)²/E is common. It appears in histogram checks and detector event counts. Keep expected values positive, and avoid extremely small expected bins that distort the approximation.

4) Degrees of freedom and fitted parameters

Degrees of freedom ν describe how many independent constraints remain after fitting. A typical rule is ν = N − p, where p is fitted parameters. If you impose extra constraints or smoothing, effective ν can be smaller than the simple count suggests.

5) Reduced chi-square as a scale diagnostic

Reduced chi-square χ²ν = χ²/ν helps compare fit quality across datasets. Values near one often align with well-estimated uncertainties. Much larger values can indicate model mismatch or underestimated σ, while much smaller values can reflect inflated σ or correlated points. In stable setups, χ²ν between 0.8 and 1.2 is common. Treat it as guidance, not a strict acceptance threshold.

6) p-value and decision context

The p-value gives the upper-tail probability of obtaining a chi-square at least as large, assuming the model is correct. Small p-values flag tension, but always check residual plots and known systematics. Very large p-values may hint at overestimated uncertainties or dependence.

7) Practical workflow checks

Before interpreting results, confirm consistent units and uncertainty definitions, and verify σ is positive for every row. Inspect outliers and repeat calculations with and without questionable points. When combining runs, account for shared calibration systematics that violate independence assumptions.

8) Reporting for reproducibility

Report χ², ν, reduced χ², and p-value together, plus the model equation and how σ was obtained. State the data range, exclusions, and the fitted parameter count. Include a residual plot and export the summary so others can reproduce the calculation exactly. Document the degrees-of-freedom rule used, especially when p is debated.

FAQs

1) Which mode should I use for typical measurement data?

Use uncertainty mode when each point has a one-sigma uncertainty. It matches common lab reporting and supports reduced chi-square as a quick scale check.

2) When is the denominator E appropriate?

It suits count-like data where variance is close to the expected value. Keep expected values positive and avoid very small bins, where approximations can break down.

3) What does reduced chi-square near one indicate?

It suggests residual scatter is consistent with stated uncertainties, assuming errors are independent. Always confirm with residual plots and checks for systematics.

4) Why can reduced chi-square be far below one?

Uncertainties may be overestimated, data points may be correlated, or the model was tuned on the same data without accounting for constraints.

5) How should I choose degrees of freedom?

A common choice is ν = N − p, where p is fitted parameters. Some contexts use ν = N − 1; select the rule consistent with your model and constraints.

6) What p-value is considered significant?

There is no universal cutoff. Many analyses use 0.05, but physics often demands stronger evidence and independent cross-checks before rejecting a model.

7) Can I export and share my results?

Yes. After computing, use the CSV and PDF buttons. The export includes summary metrics and per-row contributions for clear reporting and reproducibility.