Statistics Linearity Question

[roam]

Homework Statement

I have two different experimental curves, and I would like to measure how closely a straight line fits each data, and which curve is more crooked. In statistics how can I measure this "linearity"?

By the way this is about stepper motor step linearity (ideally it has to be a straight line i.e. homogeneous step sizes). I am comparing the two plots made for two different speeds:

Homework Equations

The Attempt at a Solution

I'm new to stats and I'm not sure what method to use. I'm very confused because some websites say I have to calculate the $R^2$ value, while others say I need a some kind of regression line.

So, if the linearity could somehow be determined from the equation of regression line, what kind of regression do I need to use (linear or quadratic, cubic, etc)? And how exactly do I determine linearity from that equation?

Any explanation is greatly appreciated.

P.S. I am using Matlab.

 

[Borek]

I'm very confused because some websites say I have to calculate the R2R^2 value, while others say I need a some kind of regression line.

I am not sure of the nomenclature, but I assume R² is just a correlation coefficient, which in this case is a measure of how good the linear regression is. Two sides of the same coin.

 

[roam]

I am not sure of the nomenclature, but I assume R² is just a correlation coefficient, which in this case is a measure of how good the linear regression is. Two sides of the same coin.

Thank you for the clarification. A high $R^2$ is what I think I will need to show good linearity.

 

[Yanick]

R² may be one way to do it but remember that is just a measure of how "far" away your linear fit is from the data (in the R² it is squared to get rid of negative numbers and somehow normalized such that a perfect fit gets you a value of 1). You can have a lower R² from noisy data which are still linear or from data which are not described well by a linear equation. What I would do is to fit a line, calculate the residuals, then either show the residuals are just noise with respect to the independant variable (this would just be a plot showing that there is no pattern to the residuals) or you can make a histogram/frequency plot of the residuals and show that they follow a gaussian/normal type of distribution.

It all depends on how far you want to go to show the linearity of your data (sometimes just plotting your line and data on the same graph is enough).

 

[DeeNos]

In order to measure the linearity of your experimental curves, you can use the coefficient of determination (R^2). This value measures the proportion of the variation in the data that is explained by the regression line. In this case, a higher R^2 value would indicate a better fit to a straight line and therefore, a higher linearity.

You can use linear regression to determine the equation of the regression line for each of your curves. The equation will have the form y = mx + b, where m is the slope of the line and b is the y-intercept. The slope (m) represents the change in y for every unit change in x and can serve as a measure of linearity. If m is close to 0, then the line is almost horizontal and indicates low linearity. If m is large, then the line is steep and indicates high linearity.

You can also visually compare the two regression lines to see which one fits the data more closely. If one line is closer to the majority of the data points, then it would indicate a higher linearity compared to the other curve.

In Matlab, you can use the "fitlm" function to perform linear regression and obtain the R^2 value and the equation of the regression line. You can also plot the regression line on top of your data points to visually assess the linearity.

Ultimately, the choice of which method to use (R^2 value or regression line) depends on your specific research question and what information you are trying to obtain from your data.