Deriving Uncertainties for m and b in Least Squares Method"

[ghery]

Hi there:

First of all, I thank all the people who have answer my questions until now, they were really helpful for me...

Now I have another Doubt, in the least squares method, in order to fit experimental data to find an straight line y = m*x + b. We need to obtain the values of m and b with their uncertainties.

How do you do in order to derive the uncertanties of m and b?

Thanks

 

[ghery]

And one more thing:

I know the equations to find the uncertanties, with this question I would like to know how to derive or figure out the equations ?

 

[RTW69]

Uncertainty

Each data point should be bracketed with error bars that correspond to the resolution uncertainity of the measuring instrument. If the variable you are measuring varies slowly with time it is reasonable to assume the uncertainty is one half the smallest division on a scale. For example the smallest division on a meter stick is 1 mm so the unceratinity could be 25 mm +/- 0.5 mm. Check the operating manual for the resolution uncertainty for digital measuring equipment.

You have determined the best fit line however two other lines must be drawn. Quoting from "Experimental methods: An introduction to the analysis and presentation of data" by Les Kirkup(John Wiley and sons, 1994) The other two lines are drawn so that they give the maximum and minimum gradient consistent with the error bars. ..The line with the maximum gradient (slope) is drawn so that it passes through all the error bars, but for the data or the extreme right the data passes through the top of the error bars, and for the points at the extreme left the line passes through the bottom of the error bars. For example the gradient of the steepest line may be 2.2 mm/hr

The minimum gradient is found from the line which passes through the bottom of the error bars for the data points on the extreme right, and the top of the error bars for the data on the extreme left. For example the gradient for this line may be 1.6 mm/hr

We can now write the gradient and the associated uncertainty as 1.9 +/- 0.3 mm/hr

In order to obtain the uncertainty in the intercept we locate where the three lines cross the y-axis, Identify the best fit y-intercept. Subtract the upper value y-intercept from the lower value y-intercept and divide by 2. For example if the best fit y-intercept is 0.5 mm, the upper y-intercept is 1.1 mm, the lower value y-intercept is .1 mm the uncertainty of the intercept would be 0.5 mm +/- 0.5 mm"