How to calculate uncertainty in gradient and intercept of a fitted line

[Identity]

If you have several data points, each with a small uncertainty in the y-direction, and you want to find the uncertainty in the gradient and the uncertainty in the intercepts of the line of best fit, how would you go about doing that?


*I know with many points you would have to do something with regression, but could the simple, 2-data point case also be explained?

Here's what I'm thinking so far for the 2-data point case, can someone please tell me if I'm right:

Equation of the line, including uncertainties:

$$y -(y_0 \pm U(y_0)) = \frac{y_1 \pm U(y_1) - (y_0 \pm U(y_0))}{x_1 - x_0}(x - x_0)$$

So you would eventually get two separate "uncertainty" bits, one in the gradient and the other in the constant term.

$$y = (m \pm U(m))x + C \pm U(C)$$

Now do you just let 'y' or 'x' equal 0 and solve?


Thanks so much

 

[jackiefrost]

Wouldn't the best fit line for just two data points points, regardless of the uncertainties in the two points' y values, be a line through the two points themselves since their y values would necessarily be centered in the y error range? Therefore, wouldn't the gradient just be the slope of the line through the two points?

 

[Bob S]

Here is the general least squares fitting algorithm for an arbitrary number of data points:

http://mathworld.wolfram.com/LeastSquaresFitting.html

Also see regression analysis and weighted least squares in

http://en.wikipedia.org/wiki/Least_squares

Bob S