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.