# What is the Graph Function of a Skewed Normal Distribution Curve?

I am looking for functions that, when plugged into a graphing calculator, draw the line of a normal distribution curve that is skewed to the right. I already have a function for a standard normal distribution curve, and I think what I need is either a new function and/or one or more functions manipulating variables within the main graph function in order to graph a skewed normal distribution curve.

Here is a function that draws a standard normal distribution curve: $$f\left(x\right)=\frac{e^{\left(-\frac{1}{2}\cdot\left(\frac{\left(x-μ\right)}{σ}\right)^{2}\right)}}{σ\cdot\sqrt{2\pi}}$$

Where

• "σ" is the standard deviation of your data
• "μ" is the average of your data
• "e" is Eulers' constant

I use this function to draw the normal distribution curve in this Desmos graph.

I need a function like this (and/or functions manipulating variables within the main function) that can graph a skewed normal distribution curve.

UPDATE: Thanks to Gerry Mason, I was able to get a working skewed normal distribution formula! The FULL formula for a skewed normal distribution curve is this massive equation:(you may need to zoom-in to see some of the variables!):

$$f\left(x\right)=\frac{2e^{\left(\frac{-\left(\left(\frac{x-ξ}{\omega}\right)-μ\right)^{2}}{2σ^{2}}\right)}\cdot\left(1+\frac{2}{\sqrt{\pi}}\int_{0}^{\left(\frac{α\left(x-ξ\right)}{\omega\sqrt{2}}\right)}e^{-t^{2}}dt\right)}{2\omegaσ\sqrt{2\pi}}$$

Where

• "σ" is the standard deviation of your data
• "μ" is the average of your data
• "e" is Eulers' constant
• "ξ" is the "location parameter (real)
• "ω" is the "scale" parameter (positive, real)
• "α" is the "shape" parameter

Here is a link to the updated Desmos graph with all the necessary functions: https://www.desmos.com/calculator/k5y9glwjee

• Replace $x$ by $\log x$? Apr 2, 2020 at 3:06
• Any thoughts on the answer I posted, Maxwell? Apr 3, 2020 at 11:42
• Are you still here, Maxwell? Apr 4, 2020 at 12:22
• I replied to your answer, @GerryMyerson Apr 4, 2020 at 22:51

Let $$\phi(x)$$ denote the standard normal probability density function $$\phi(x)={1\over\sqrt{2\pi}}e^{-x^2/2}$$ with cumulative distribution function $$\Phi(x)={1\over2}\left[1+{\rm erf}\left({x\over\sqrt2}\right)\right],$$ where "erf" is the error function. Then the probability density function of the skew-normal distribution with parameter $$\alpha$$ is given by $$f(x)=2\phi(x)\Phi(\alpha x).$$
• Yes, it's just multiplying by $1/2$. The graph at en.wikipedia.org/wiki/Skew_normal_distribution#/media/… seems to show the skew changing with $\alpha$. Apr 4, 2020 at 22:54