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Chi-Square (χ²) Fit Calculator

Evaluate goodness-of-fit between measured data and theoretical model predictions

Parameters

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Controls

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Calculated Values

χ²:
2.76;2.76;
Degrees of freedom ν:
3.00;3.00;
Reduced χ²:
0.92;0.92;
Fit quality:
1.00;Acceptable1.00;Acceptable

Examples

Linear model check

Four points with σᵢ ≈ 1; Eᵢ from best-fit line.

  • χ²: 2.762.76
  • χ²_red: 0.920.92

Ideal gas at four temperatures

Compare measured P to PV=nRT prediction.

  • Accept if χ²_red ≈ 1: 1.001.00

Visualization

Chi-Square Goodness of Fit

The chi-square statistic measures how well experimental data agree with a theoretical model, weighted by measurement uncertainty. Large χ² means residuals (O − E) are large compared to your stated errors.

For n data points and p fitted parameters, χ² = Σᵢ (Oᵢ − Eᵢ)² / σᵢ². Each term is dimensionless because the residual is divided by the expected standard deviation σᵢ.

Degrees of freedom ν = n − p. For a simple consistency check with no fitted parameters (p = 0), use ν = n − 1. Reduced chi-square χ²_red = χ² / ν should be near 1 if the model and uncertainties are correct.

Rule of thumb: 0.5 ≲ χ²_red ≲ 1.5 suggests acceptable agreement. χ²_red ≫ 2 often means underestimated σᵢ, wrong model Eᵢ, or systematic errors. χ²_red ≪ 1 may mean overestimated uncertainties.

Always use σᵢ from Error Propagation or instrument specs — not the spread of repeated trials unless those trials are independent measurements of the same quantity.

χ² is widely used in particle physics, astronomy, and any lab comparing a curve (Hooke’s law, ideal gas, radioactive decay) to measured points.

Key Concepts

  • χ² = Σ (O − E)² / σ²
  • ν = n − p (degrees of freedom)
  • χ²_red = χ² / ν ≈ 1 for good fit
  • Each point contributes (O−E)²/σ² to the total
  • Underestimated σ inflates χ²_red artificially

Real-World Applications

  • Verifying linear fit (F = kx) through spring data
  • Checking ideal gas law at multiple temperatures
  • Comparing measured decay rates to exponential model
  • Assessing whether outliers dominate a dataset

Explore Further

More measurement uncertainty tools

  • Error Propagation

    Combine measurement uncertainties for sums, products, quotients, and powers in experimental formulas.

  • Significant Figures

    Round measurements and calculated results to the correct precision for lab reports.

Physics Equations

Chi-square:
χ2=∑i(Oi−Ei)2σi2\chi^2 = \sum_i \frac{(O_i - E_i)^2}{\sigma_i^2}
Reduced:
χred2=χ2ν,ν=n−p\chi^2_{\text{red}} = \frac{\chi^2}{\nu}, \quad \nu = n - p
Per-point term:
χi2=(Oi−Ei)2σi2\chi^2_i = \frac{(O_i - E_i)^2}{\sigma_i^2}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Organize data table

For each point: observed Oᵢ, model prediction Eᵢ, uncertainty σᵢ.

Result:

4datapoints,3degreesoffreedom(ν=n−1)4 data points, 3 degrees of freedom (ν = n − 1)

Explanation:

σᵢ should come from measurement error or propagated uncertainty, not arbitrary guesses.

2

Step 2: Point 1 contribution

Compute squared residual weighted by uncertainty.

Equation:

χi2=(Oi−Ei)2σi2\chi^2_i = \frac{(O_i - E_i)^2}{\sigma_i^2}

Calculation:

(10−9.5)212=0.25001.0000=0.2500\frac{(10 - 9.5)^2}{1^2} = \frac{0.2500}{1.0000} = 0.2500

Result:

χ12=0.2500\chi^2_1 = 0.2500

Explanation:

Residual = 0.500; a 1 error bar means this point agrees well with the model.

3

Step 3: Point 2 contribution

Compute squared residual weighted by uncertainty.

Equation:

χi2=(Oi−Ei)2σi2\chi^2_i = \frac{(O_i - E_i)^2}{\sigma_i^2}

Calculation:

(15−14)21.22=1.00001.4400=0.6944\frac{(15 - 14)^2}{1.2^2} = \frac{1.0000}{1.4400} = 0.6944

Result:

χ22=0.6944\chi^2_2 = 0.6944

Explanation:

Residual = 1.000; a 1.2 error bar means this point agrees well with the model.

4

Step 4: Point 3 contribution

Compute squared residual weighted by uncertainty.

Equation:

χi2=(Oi−Ei)2σi2\chi^2_i = \frac{(O_i - E_i)^2}{\sigma_i^2}

Calculation:

(8−9)20.82=1.00000.6400=1.5625\frac{(8 - 9)^2}{0.8^2} = \frac{1.0000}{0.6400} = 1.5625

Result:

χ32=1.5625\chi^2_3 = 1.5625

Explanation:

Residual = -1.000; a 0.8 error bar means this point is acceptable with the model.

5

Step 5: Point 4 contribution

Compute squared residual weighted by uncertainty.

Equation:

χi2=(Oi−Ei)2σi2\chi^2_i = \frac{(O_i - E_i)^2}{\sigma_i^2}

Calculation:

(12−11.5)212=0.25001.0000=0.2500\frac{(12 - 11.5)^2}{1^2} = \frac{0.2500}{1.0000} = 0.2500

Result:

χ42=0.2500\chi^2_4 = 0.2500

Explanation:

Residual = 0.500; a 1 error bar means this point agrees well with the model.

6

Step 6: Sum all contributions

Add individual χ²ᵢ terms for total χ².

Equation:

χ2=∑iχi2\chi^2 = \sum_i \chi^2_i

Calculation:

0.2500+0.6944+1.5625+0.2500=2.75690.2500 + 0.6944 + 1.5625 + 0.2500 = 2.7569

Result:

χ2=2.7569\chi^2 = 2.7569
7

Step 7: Compute reduced chi-square

Normalize by degrees of freedom to assess overall fit quality.

Equation:

χred2=χ2ν\chi^2_{\text{red}} = \frac{\chi^2}{\nu}

Calculation:

2.75693=0.9190\frac{2.7569}{3} = 0.9190

Result:

χred2=0.9190\chi^2_{\text{red}} = 0.9190

Explanation:

χ²_red below 0.5 may mean overestimated σ; 0.5–1.5 is good; above 2 suggests model or σ issues.

8

Step 8: Interpret the fit

Compare χ²_red to unity.

Result:

Acceptable agreement between data and model within stated errors

Frequently Asked Questions (FAQ)

What if reduced χ² is very large?

Likely causes: underestimated σᵢ, wrong model Eᵢ, or systematic errors not included in σ.

Can χ²_red be less than 1?

Yes — often indicates conservative (too large) uncertainty estimates or too few data points.

Should I use σ = instrument resolution or standard deviation of trials?

Use the standard error of the mean √(s²/n) for n independent trials of the same point; use instrument resolution for single readings.

Practice MCQs

  1. Reduced χ² near 1 generally means:
  2. If all residuals are zero, χ² equals:
  3. Halving every σᵢ while keeping O and E fixed will:
  4. With n = 5 points and p = 2 fitted parameters, ν =
  5. A point with |O−E| = 2σ contributes approximately:

Related Calculators

These tools connect to the same physics concepts used in this calculator.

Error PropagationCombine uncertainties when multiplying measured quantities.σᵢ in the denominator should match propagated uncertainties.Ideal Gas LawFit P–V–T lab data and assess model quality with χ².Example: compare PV=nRT predictions to lab pressure readings.Projectile MotionKinematic trajectories under gravity — compare simulated vs measured positions.Numerical IntegrationTrapezoidal and Simpson rules for areas under physics curves.Monte Carlo IntroStochastic estimation of integrals and constants.Validate stochastic estimates (π, integrals) against theory with χ².Significant FiguresRound measurements and calculated results to the correct precision for lab reports.Gradient DescentIteratively minimize f(x) by following the negative gradient with animated path visualization.Related Computational Physics topic.Ohm's LawCalculate voltage, current, and resistance in electrical circuits.Related Electricity topic.