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Suppose we want to fit a model function <math>\hat{y}(x;\alpha)=\hat{y}(x;a_1,...,a_M)</math> that has M adjustable parameters <math>a_j</math>, <math>j=1,...,M</math> to N data points <math>(x_i,y_i)</math> <math>i=1,...,N</math>, where <math>N \ge M</math>. The model predicts a functional relationship between the measured and independent variables. The "best" values for the parameters <math>a_j</math> are those for which the model represents an estimator for the underlying real function <math>y(x)</math> giving rise to the data.

Suppose we want to fit a model function <math>\hat{y}(x;\alpha)=\hat{y}(x;a_1,...,a_M)</math> that has M adjustable parameters <math>a_j</math>, <math>j=1,...,M</math> to N data points <math>(x_i,y_i)</math> <math>i=1,...,N</math>, where <math>N \ge M</math>. The model predicts a functional relationship between the measured and independent variables. The "best" values for the parameters <math>a_j</math> are those for which the model represents an estimator for the underlying real function <math>y(x)</math> giving rise to the data.

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If the data points <math>y_i</math> are Gaussian distributed around the "true" model <math>\hat{y}(x;\bar{\alpha})</math> with variance <math>\sigma_i^2</math>, the best fit parameters <math>\bar{\alpha}</math> can be found by minimizing the quantity:

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If the data points <math>y_i</math> are Gaussian distributed around the "true" model  

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<math>\hat{y}(x;\bar{\alpha})</math>  

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with variance <math>\sigma_i^2</math>, the best fit parameters <math>\bar{\alpha}</math> can be found by minimizing the quantity:

<math>\chi^2=\sum_{i=1}^N\frac{[y_i-\hat{y}(x_i;a_1,...,a_M)]^2}{\sigma^2_i}</math> ,  

<math>\chi^2=\sum_{i=1}^N\frac{[y_i-\hat{y}(x_i;a_1,...,a_M)]^2}{\sigma^2_i}</math> ,  

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 | '''2025'''

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