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The first cumulant is called the mean. The second is the variance.

Does the third cumulant have a name? The fourth?

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The third and fourth cumulants do not have specific sames, but their standardised versions do. The third and fourth standardised cumulants are given respectively by the skewness and the excess kurtosis:

$$\gamma = \frac{\mu_3}{\mu_2^{3/2}} \quad \quad \quad \quad \quad \kappa_* = \kappa - 3 = \frac{\mu_4}{\mu_2^2} - 3.$$

The skewness is a measure of the left or right skew in the data. The kurtosis is a measure of the fatness of the tails of the distribution, and the "excess kurtosis" is the amount in excess of mesokurtosis (i.e., the amount in excess of a normal distribution).

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The third cumulant is the third central moment, i.e. $\kappa_3=\mu_3=E[(X-E[X])^3]$. But, the fourth cumulant is not the fourth central moment. Specifically, it is the following: $$\kappa_4=\mu_4-3\mu_2^2$$ where $\mu_i$ is the i-th central moment. I haven't seen it being referred as other than cumulant (on its own). Otherwise, it's worth noting @whuber's comment:

When standardized as $\kappa_4/\mu_2^2 = \mu_4/\mu_2^2-3$, it is called "excess kurtosis." A common name for the standardized third cumulant $\kappa_3/\mu_2^{3/2}$ is "skewness." – whuber

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    $\begingroup$ When standardized as $\kappa_4/\mu_2^2 = \mu_4/\mu_2^2-3,$ it is called "excess kurtosis." A common name for the standardized third cumulant $\kappa_3/\mu_2^{3/2}$ is "skewness." $\endgroup$
    – whuber
    Nov 2, 2019 at 18:36

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