# what is the formula for calculation of fourth raw moment (or central moment) from variance and 3rd central moment (or raw moment)?

Basically the title. I can't seem to find any solution for this.

I have the mean, variance or the second central moment and third central moment and third raw moment. I need to find the fourth raw moment.

• There are universal inequalities among these moments but that's all one can say.
– whuber
Commented Apr 3 at 21:25
• Microsoft's beta version of its "Copilot" chatbot includes in its response (using your full title as the prompt) this passage: "Unfortunately, there isn’t a direct formula to calculate the fourth raw moment from the variance and the third central moment. These are different statistical measures that capture different aspects of the data distribution. "
– whuber
Commented Apr 3 at 21:34

Unfortunately this is not possible. For example, consider $$X_1$$ a unit normal distribution (zero mean, and variance=1) and $$X_2$$ a distribution which is $$\pm 1$$ with probability 50 percent. Both of these have the same mean, second moment, and third moment, but the 4th moment of $$X_1$$ is equal to 3, while the 4th moment of $$X_2$$ is equal to 1.