# Understanding Confidence Intervals for Fit Parameters

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• BillKet
In summary, the conversation discusses the use of confidence intervals and error matrices in data analysis. While the error matrix can be used to estimate the uncertainty in a calculated result, it does not take into account the correlation between parameters. This can lead to incorrect confidence intervals. Bevington and Robinson's book suggests using a chi squared distribution to calculate confidence intervals, while other resources use an F distribution. It is important to carefully consider the method used for calculating confidence intervals to ensure accurate results.
BillKet
Hello! Can someone help me understand how are confidence intervals for some parameters of a fit different from the errors on the parameters obtained, for example, from the error matrix. I read Bevington and the whole book he mentions that we can use the error from the error matrix to define the confidence interval (e.g. ##68.3\%## confidence interval for 1 ##\sigma## of a parameter), then in the last chapter he says that, this is not generally correct and we should use confidence intervals which automatically take into account the correlation between parameters. I understand his argument and it makes sense to do that, but now I am not sure I understand what is the error matrix useful for anymore, if the estimates from the error matrix don't take into account the correlations among the parameters? I guess they are useful when the correlations are zero, but does that happen often? Thank you!

I assume that you are referring to Data Reduction and Error Analysis for the Physical Sciences, by Bevington and Robinson. It seems to me that equation 3.13 and the discussion around it make clear that covariance terms are important. Also, on page 125 the authors state, "The error matrix can be used to estimate the uncertainty in a calculated result, including the effects of the correlations of the errors." Could you give page numbers and brief quotes of the later contradiction? I am looking at the third edition of the book.

jim mcnamara and Stephen Tashi
tnich said:
I assume that you are referring to Data Reduction and Error Analysis for the Physical Sciences, by Bevington and Robinson. It seems to me that equation 3.13 and the discussion around it make clear that covariance terms are important. Also, on page 125 the authors state, "The error matrix can be used to estimate the uncertainty in a calculated result, including the effects of the correlations of the errors." Could you give page numbers and brief quotes of the later contradiction? I am looking at the third edition of the book.
Thank you for your reply! Actually I finally understand why using the error matrix is bad. However I am still a bit confused about how Bevington calculates the confidence intervals. He uses a chi squared distribution and for example he uses the fact that for one parameter an increase on 1 for the chi squared is equivalent to a 68% confidence level. However in other online resources (for example: https://lmfit.github.io/lmfit-py/confidence.html) they use an F distribution to define the confidence interval. But the 2 methods don't seem equivalent, as in Bevington the value of number of data points (N in the link I provided) doesn't come in the formula. So which one is the right formula?

Why is it bad to use the error matrix?
Can you tell me what page of Bevington you are looking at?

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## 1. What is a confidence interval for fit parameters?

A confidence interval for fit parameters is a range of values that is likely to include the true value of a parameter based on the data collected from a sample. It is a statistical measure used to estimate the precision of the estimated value and is expressed as a range of values with a specified level of confidence.

## 2. How is a confidence interval for fit parameters calculated?

A confidence interval for fit parameters is typically calculated using a formula that takes into account the sample size, standard deviation, and the desired level of confidence. The most commonly used formula is the t-distribution formula, which takes into account the sample size and the standard error of the estimate.

## 3. What is the significance of the confidence level in a confidence interval for fit parameters?

The confidence level in a confidence interval for fit parameters represents the probability that the true value of the parameter falls within the calculated interval. For example, a 95% confidence level means that if the same study was conducted 100 times, 95 of those times the calculated interval would contain the true value of the parameter.

## 4. How does the sample size affect the width of a confidence interval for fit parameters?

The sample size has a direct impact on the width of a confidence interval for fit parameters. A larger sample size results in a smaller standard error, which in turn results in a narrower confidence interval. This means that as the sample size increases, the precision of the estimated value also increases.

## 5. What is the difference between a confidence interval and a prediction interval?

A confidence interval for fit parameters is used to estimate the precision of the estimated value of a parameter, while a prediction interval is used to estimate the range of values that a future observation is likely to fall within. In other words, a confidence interval is used for estimating the precision of the estimate, while a prediction interval is used for making predictions about future observations.

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