Deriving Uncertainties for m and b in Least Squares Method"

In summary, the author is asking how to find the uncertainty in the y-intercept of a best fit line. The author determines the gradient and associated uncertainty by locating the line with the maximum and minimum gradient and then subtracting the upper value y-intercept from the lower value y-intercept. The resulting uncertainty is then divided by 2.
  • #1
ghery
34
0
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
 
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  • #2
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 ?
 
  • #3
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"
 

What is the Least Squares Method?

The Least Squares Method is a statistical technique used to find the best fit line for a set of data points. It minimizes the sum of the squared differences between the actual data points and the predicted values on the line.

How do you derive uncertainties for m and b in the Least Squares Method?

The uncertainties for m and b in the Least Squares Method can be derived using the following equations:

Uncertainty for m = (sqrt(sum of squared residuals / (n-2)) / sqrt(sum of squared x deviations))

Uncertainty for b = (sqrt(sum of squared residuals / (n-2)) / sqrt(n))

What is the significance of uncertainties in the Least Squares Method?

Uncertainties in the Least Squares Method provide a measure of the reliability of the values for m and b. They indicate the range within which the true values of m and b are likely to fall.

What factors can affect the uncertainties in the Least Squares Method?

The uncertainties in the Least Squares Method can be affected by the number of data points, the distribution of the data, and the magnitude of the residuals.

Is the Least Squares Method the only way to find a best fit line for a set of data points?

No, there are other statistical techniques that can be used to find a best fit line, such as the Maximum Likelihood Method and the Method of Moments. However, the Least Squares Method is one of the most commonly used methods due to its simplicity and ease of calculation.

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