# Central moments of a normal distribution with mean and standard deviation known?

I want to evaluate central moments of a normal random variable till 8th order. From some search I came to know that for even order ($v$) central moment, it is $Sigma^v*(v-1)!!$ (double factorial) and zero for odd order.

Can some one tell me the expanded final version formula for 4th, 6th and 8th order central moment of normal distribution.

I am confused because from matlab experiment of generating samples from a normal distribution and then finding their central moments is giving me a solution (all 4,6,8 order central moments) which is independent of mean and standard deviation.

Thank You.

• There are something called standardized moment in which the central moment is divided by the standard deviation, powered to that order. So the scale parameter $\sigma^v$ is eliminated in the process and thus the standardized moment is independent of it. Make sure your mathlab experiment is generating the moment you want. – BGM Apr 18 '18 at 15:34
• Reply: Yes, I actually need standardized moments itself. But the formula having a double factorial is confusing me. It is giving different result from matlab's central moments. – Krishna Apr 19 '18 at 4:07
• Ya got it!! Actually I was taking that double factorial for whole formula, its only for (v-1) I guess. Then everything is matching. – Krishna Apr 19 '18 at 4:12