What's the meaning of $\sigma^3$ and $\sigma^4$ and how to find them

Question

I know you can use $\sigma^3$ and $\sigma^4$ with $\mu$₃ and $\mu$₄ respectively to find the skewness and the kurtosis, but by themselves what do they mean + what's the formula to find them?

Thank you in advance!

Answer 1

skewness=$\frac{\mu_3}{\sigma^3}$, kurtosis=$\frac{\mu_4}{\sigma^4}$, where $\mu_k=E((X-\mu)^k)$ and $\sigma^2=E((X-\mu)^2)=E(X^2)-\mu^2$

Using Google:

https://brownmath.com/stat/shape.htm

Answer 2

You presumably know how to find the variance $\sigma^2$. The others are powers of this. Put simply:

• $\sigma$ is the standard deviation, the square-root of the variance and measured in the same units as the original data
• $\sigma^2$ is the variance, the second central moment and the square of the standard deviation
• $\sigma^3$ is the cube of of the standard deviation, i.e. the variance raised to the power of $\frac32$
• $\sigma^4$ is the $4$th power of the standard deviation, i.e. the square of the variance

You divide the third central moment by $\sigma^3$ to get the skewness, and the fourth central moment by $\sigma^4$ to get the kurtosis, in order to remove scale effects from these statistics and make them unitless, so they can give information related to the shape of the distribution. Similarly, taking central moments rather than raw moments to remove location effects.