Order r Moment ${M}_{0}\left(r\right)$ Central Moment: ${M}_{\mu }\left(r\right)$ 1 ${\mathrm{e}}^{m+\frac{1}{2}{s}^{2}}$ 0 2 ${\mathrm{e}}^{2m+2{s}^{2}}$ ${\text{e}}^{2m}\left({\text{e}}^{2{s}^{2}}-{\text{e}}^{{s}^{2}}\right)$ 3 ${\mathrm{e}}^{3m+\frac{9}{2}{s}^{2}}$ ${\text{e}}^{3m}\left({\text{e}}^{\frac{9}{2}{s}^{2}}-3{\text{e}}^{\frac{5}{2}{s}^{2}}+2{\text{e}}^{\frac{3}{2}{s}^{2}}\right)$ 4 ${\mathrm{e}}^{4m+8{s}^{2}}$ ${\text{e}}^{4m}\left({\text{e}}^{8{s}^{2}}-4{\text{e}}^{5{s}^{2}}+6{\text{e}}^{3{s}^{2}}-3{\text{e}}^{2{s}^{2}}\right)$ 5 ${\mathrm{e}}^{5m+\frac{25}{2}{s}^{2}}$ ${\text{e}}^{5m}\left({\text{e}}^{\frac{25}{2}{s}^{2}}-5{\text{e}}^{\frac{17}{2}{s}^{2}}+10{\text{e}}^{\frac{11}{2}{s}^{2}}-10{\text{e}}^{\frac{7}{2}{s}^{2}}+4{\text{e}}^{\frac{5}{2}{s}^{2}}\right)$ 6 ${\mathrm{e}}^{6m+18{s}^{2}}$ ${\text{e}}^{6m}\left({\text{e}}^{18{s}^{2}}-6{\text{e}}^{13{s}^{2}}+15{\text{e}}^{9{s}^{2}}-20{\text{e}}^{6{s}^{2}}+15{\text{e}}^{4{s}^{2}}-5{\text{e}}^{3{s}^{2}}\right)$ 7 ${\mathrm{e}}^{7m+\frac{49}{2}{s}^{2}}$ ${\text{e}}^{7m}\left({\text{e}}^{\frac{49}{2}{s}^{2}}-7{\text{e}}^{\frac{37}{2}{s}^{2}}+21{\text{e}}^{\frac{27}{2}{s}^{2}}-35{\text{e}}^{\frac{19}{2}{s}^{2}}+35{\text{e}}^{\frac{13}{2}{s}^{2}}-21{\text{e}}^{\frac{9}{2}{s}^{2}}+6{\text{e}}^{\frac{7}{2}{s}^{2}}\right)$ 8 ${\mathrm{e}}^{8m+32{s}^{2}}$ ${\text{e}}^{8m}\left({\text{e}}^{32{s}^{2}}-8{\text{e}}^{25{s}^{2}}+28{\text{e}}^{19{s}^{2}}-56{\text{e}}^{14{s}^{2}}+70{\text{e}}^{10{s}^{2}}-56{\text{e}}^{7{s}^{2}}+28{\text{e}}^{5{s}^{2}}-7{\text{e}}^{4{s}^{2}}\right)$ Statistic Symbol Expression Standard Deviation $\sigma$ ${\text{e}}^{m}\sqrt{\left({\text{e}}^{2{s}^{2}}-{\text{e}}^{{s}^{2}}\right)}$ Skewness Sk $\left({\text{e}}^{3{s}^{2}}-3{\text{e}}^{{s}^{2}}+2\right)/{\left({\text{e}}^{{s}^{2}}-1\right)}^{\frac{3}{2}}$ Kurtosis K ${\text{e}}^{4{s}^{2}}+2{\text{e}}^{3{s}^{2}}+3{\text{e}}^{2{s}^{2}}-3$ Variance of Skewness Var (Sk) $\frac{Var\left({M}_{\mu }\left(3\right)\right)}{{S}^{6}}+\frac{3{M}_{\mu }{\left(3\right)}^{2}Var\left({M}_{\mu }\left(2\right)\right)}{4{S}^{13}}$ Variance of Kurtosis Var (K) $\frac{Var\left({M}_{\mu }\left(4\right)\right)}{{S}^{8}}+\frac{2{M}_{\mu }{\left(4\right)}^{2}Var\left({M}_{\mu }\left(2\right)\right)}{{S}^{16}}$