I recently reported this on Quora but I'd like to share it here too to try to get more people to give their opinions about this. Basically I designed a simple equation for skewed Gaussian curves.
For context: lately I’ve been doing enzyme activity experiments, and when testing the activity of my enzyme at different pH values I came across this data trend. I tried to fit the points to a Gaussian curve, but the fit felt off. (Image)
It looks more like a skewed distribution, so I tried to find an equation to improve my fit, but the equations for a skewed normal distribution (Wikipedia) are complicated and impractical to apply when using a software like Origin or GraphPad (to my knowledge at least). So I did a little more digging and I came across a paper showcasing a simple equation. I have no idea where this equation came from, but it kinda worked, so I played a little bit with the parameters of that equation, optimized it, and ended up with this other equation. And it works! The fit is great! (Image)
$$y = y_0 + A \cdot e^{-\frac{1}{2}\left(\frac{x - x_0}{w + a \cdot (x - x_0)}\right)^2}$$
The idea is that it’s like a modified version of a typical Gaussian equation, but including parameter “a” for skewness. When a=0 you get your usual symmetric Gaussian graph, a<0 makes the graph negatively skewed (to the left), and a>0 positively skewed (to the right). The rest of the parameters (“x0”, “y0”, “A”, “w”) are the same as a typical Gaussian curve. I also made a simple diagram showcasing the role each parameter plays in the equation: (Image)
I also made a Desmos sheet so you can play with the parameters and figure it out yourselves: Simple skewed Gaussian.
It’s not really perfect like the more complicated equations, the graph isn’t really flat besides the peak, it’s got an inflexion point and another curve going on after it but I don’t think it’s relevant since the skewness is generally not large enough to affect the fit. The point is that it’s good enough for fits, it’s a simple equation appropiate for these cases, and I wanted to report it here, in case someone has the same problem as me and is looking for a simple solution. I hope this equation proves useful to anyone dealing with skewed Gaussian curves in their experiments!
Hope I explained myself well enough, and I welcome any suggestions and comments. I'd love to hear the opinions of people who have more knowledge than I do on this topic. Thank you for taking the time to read about my findings!