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In summary, fitting data to a linear function with least squares gives coefficients a0 and a1. To calculate the uncertainty of a0, the diagonal elements of the covariance matrix C can be used, but the off-diagonal elements also come into play when evaluating expressions where both coefficients appear. A good reference for this is Eq 22 of Kirchner's note.

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Found your answers and a good reference in this thread

[edit] on second thought: the errors

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Thank you for welcoming and the reference. Seems like Eq 22 of Kirchner's note is what I was looking for.

Uncertainty of coefficients after a least square fit is a measure of the amount of error or variability in the estimated coefficients of a linear regression model. It is a reflection of how well the model fits the data and how reliable the estimated coefficients are.

Uncertainty of coefficients is typically calculated using the standard error of the estimated coefficients. This is a measure of the variation in the estimated coefficients from different samples of the same size. It takes into account both the variability in the data and the quality of the model fit.

Uncertainty of coefficients is important because it provides information about the reliability and accuracy of the estimated coefficients in a linear regression model. It allows us to assess the significance of the coefficients and make informed decisions about the model's predictive power.

The amount of data can have an impact on uncertainty of coefficients in a least square fit. Generally, as the amount of data increases, the uncertainty of coefficients decreases. This is because larger sample sizes provide more accurate estimates and reduce the variability in the coefficients.

Uncertainty of coefficients cannot be completely eliminated in a least square fit, but it can be reduced by increasing the amount of data, improving the quality of the model, or using more advanced regression techniques. It is important to keep in mind that some uncertainty is inevitable and should be considered when interpreting the results of a linear regression model.

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