If its of real importance, I am trying to plot the data on gnuplot.
I have the following of some experimental data, obtained by octave:
- mean: $\overline{\mu} = 0.6058$
- median: $\tilde{\nu} = 0.6364$
- std: $\sigma_x = 0.1674$
- variance: $\sigma^2 = 0.028$
- skewness $= 0.3131$
I am trying to plot them, but AFAIK, the normal distribution does not account for skewness (left or right). I am using the typical Gaussian curve function:
$f(x) = \frac{1}{\sigma_x \cdot \sqrt{2 \pi}} e^{- \frac{(x-\mu)^2}{2\sigma^2}}$
How can I account the change in the curve shape, for my given skewness? For this data-set it is obvious that there exists a difference from median to mean, and I would like to plot that.
You can use the skew normal distribution with parameters $(ξ,ω,α)$ which can be estimated from the given data. If we set $δ=\dfrac{α}{\sqrt{1+α^2}}$, then the mean, variance and skewness of the skew normal distribution are given by (see the link)
Substitute your known values for the mean, variance and skewness to find proper values for the parameters $(ξ,ω,α)$ of the distribution. Approximate values will do (you do not need to solve exactly), because you are based on a sample, and the distribution that you will find does not need to fit exactly to the sample. So, trial and error (with a computer), may help, since this is not an easy to solve system.
Start from the formula of the skewness which depends only on $δ$. That is solve
$$0.3131=\dfrac{4-\pi}{2} \dfrac{\left(\delta\sqrt{\dfrac2\pi}\right)^3}{ \left(1-\dfrac{2\delta^2}{\pi}\right)^{3/2}}$$ to find (approximately $δ$). From $δ$ you can find directly $α$. Now, go to the variance and solve $$0.028=ω^2\left(1-\dfrac{2δ^2}{π}\right)$$ to find $ω$. Use the value for the $δ$ that you already have. Finally, use the formula for the mean (with $ω, δ$ known) to find $ξ$ in a similar way.