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I have data that has error. The error bars in the data represent one standard deviation. I would like to fit a line to these data, while accounting for the fact that the data could vary between my error bars. Is this kind of reasoning statistically sound? Does it make any sense to do what I describe? Or should I just fit a regular line and use confidence intervals to describe the error in the data?

Below is an example data set that I would like to fit a line to.

enter image description here

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One method to account for variable measurement error between observations is weighted ordinary least squares regression. In general, weighted regression specifies a weights matrix. In this case, you can specify a weights matrix that is actually an (estimated) precision matrix. Many applications assume -- rightly or wrongly -- independence of observations, so the covariance matrix of observations is diagonal, so its precision matrix is also diagonal with positive values equal to inverse variances.

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