# Root Mean Square Error, a straight line fit, and gradient issue

1. Apr 11, 2010

### K29

I have some measurements from a physics lab experiment and I am coding in Matlab a fit for the data. [Note this is not a problem with Matlab, my problem here is theory]

In normal regression of statistics the RMSE is given by:

$$s=\frac{\sigma}{\sqrt{n}} =\sqrt{\frac{\Sigma (\epsilon _i)^2}{n(n-1)}}$$
where $$\sigma$$ is the standard deviation or Root Mean Square Deviation.

Now, according to my physics lab manual:

"For large n the standard error of the mean implies 68% confidence interval. For small n this is not reliable and it is necessary to multiply $$\sigma$$ by a certain factor t, to obtain the appropriate confidence interval."

They then give a table with t= 12.7 for n = 2; t = 4.3 for n =3 (t is reduced by a factor of 1/3.6 for each n)

Onwards...

The root mean square error for the straight line fit is given by:
$$S_{y}=\sqrt{\frac{\Sigma(\delta y_{i}^{2})}{n-2}}$$

The error in the gradient of the straight line fit is:
$$S_{m}=S_{y}\sqrt{\frac{\Sigma x_{i}^{2}}{n \Sigma (x_{i}^{2})-(\Sigma x_{i})^2 }}$$

Now for my plot I have only 3 data points. They are however, very accurate. The root square is about 0.98. (the fit explains 98% of the total variation in the data about the average.)

But the RMSE is quite large due to there being only 3 data points. My error for gradient is therefore ridiculously large. I can not find anywhere how the RMSE equation for the graph is actually derived, therefore I am having difficulty working out how/if/where I am to multiply the factor t into the RMSE equation for a straight line.