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

In summary: So, for example, for n=3, the multiplier is 4.3. So just multiply your standard error of the mean, which is ##S_y## by 4.3.In summary, the conversation discusses the standard error of the mean and the root mean square error for a straight line fit in statistics. It is noted that for small sample sizes, a multiplier must be applied to the standard deviation to obtain an appropriate confidence interval. The individual's plot has only 3 data points, which are considered to be very accurate, but the RMSE is still large due to the small sample size. The individual is unsure of how to incorporate the multiplier into the RMSE equation for a straight line.
  • #1
K29
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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:

[tex]s=\frac{\sigma}{\sqrt{n}} =\sqrt{\frac{\Sigma (\epsilon _i)^2}{n(n-1)}}[/tex]
where [tex] \sigma [/tex] 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 [tex]\sigma[/tex] 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:
[tex]S_{y}=\sqrt{\frac{\Sigma(\delta y_{i}^{2})}{n-2}}[/tex]

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

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.

Can anyone please help? Thanks in advance
 
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  • #2
The statistics of your experiment come from the fact that you took 3 readings. Think about it like this: out of the "universe of possible readings" you picked 3 of them. If you take enough readings, you expect some randomness to show up.

Your statement that they are "very accurate" may be true. It sounds like it may be based on your domain knowledge. Perhaps, you can only take 3 readings due to time or cost constraints. You should document your procedure, so that anybody else, who wants to repeat the procedure, can get comparable results.
 
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  • #3
A technical point. Standard deviation, at least to my knowledge, often denoted as ##\sigma##, is used for the population parameter, while S.E is used as the standard deviation of a random variable, obtained from sample data.
 
  • #4
K29 said:
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.)
By "very accurate" do you mean that you measured them very accurately or that you have some subject-matter reason to think that there is very little random variation in the results, or that the R-squared of the linear regression is large? Those three ways to interpret "very accurate" have very different implications.
K29 said:
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.
I am not familiar with those multipliers for such small samples, but it sounds like you should just multiply ##S_y## by them.
 
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1. What is Root Mean Square Error (RMSE)?

Root Mean Square Error (RMSE) is a statistical measure of how well a regression line fits the data points. It is calculated by taking the square root of the average of the squared differences between the predicted values and the actual values.

2. How is RMSE used in evaluating a straight line fit?

RMSE is used to determine the accuracy of a straight line fit by measuring the distance between the predicted values and the actual values. A lower RMSE value indicates a better fit, while a higher RMSE value indicates a poorer fit.

3. Can RMSE be negative?

No, RMSE cannot be negative. It is always a positive value as it is the square root of the sum of squared errors. If the predicted values are perfectly aligned with the actual values, the RMSE will be 0.

4. What is the significance of the gradient in a straight line fit?

The gradient, also known as the slope, is a crucial factor in determining the steepness of a straight line fit and the relationship between the independent and dependent variables. It represents the rate of change of the dependent variable with respect to the independent variable.

5. How is the gradient calculated in a straight line fit?

The gradient is calculated by dividing the change in the dependent variable by the change in the independent variable. It is represented by the symbol "m" in the equation of a straight line, y = mx + c, where "m" is the gradient and "c" is the y-intercept.

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