I'm using the following abbreviations:
- $n=$ number of observations
- $m = \bar{x}=$ sample mean of observations
- $s^2=$ (unbiased) sample variance of observations
- $y_2 = \frac{1}{n}\sum (x_i - m)^2=$ biased second central sample moment. Same as $\frac{n-1}{n}s^2$.
- $y_3 = \frac{1}{n}\sum (x_i - m)^3=$ biased third central sample moment, aka sample skewness
- $y_4 = \frac{1}{n}\sum (x_i - m)^4=$ biased fourth central sample moment, aka sample kurtosis
| mean | $m = \bar{x} = \frac{1}{n}\sum x_i$ |
| variance | $s^2 = \frac{1}{n-1}\sum\bigl(x_i - \bar{x}\bigr)^2$ |
| square of mean | $\frac{1}{n}\sum x_i^2 - s^2 = m^2 - \frac{s^2}{n}$ |
| fourth power of mean | $m^4 - \frac{6 }{n}m^2s^2 + \bigl(\frac{-1}{2n} - \frac{3}{n-2} + \frac{2}{n-3}\bigr)y_2^2 + \bigl(\frac{-8}{n-1} + \frac{8}{n-2}\bigr) m y_3 + \bigl(\frac{-3}{n-1} + \frac{6}{n-2} -\frac{3}{n-3}\bigr) (y_4 - 3) - \frac{9}{n-1} + \frac{18}{n-2} - \frac{9}{n-3}$ |
| square of population variance | $\bigl(1 + \frac{1}{2(n-1)} + \frac{5}{2(n-2)} + \frac{9}{2(n-2)(n-3)}\bigr) y_2^2 - \big(\frac{1}{n-2} + \frac{3}{(n-2)(n-3)}\bigr) y_4$ |
| variance of $s^2$ | $(s^2)^2 - \text{preceding line}$ |
| population kurtosis $\mathbb{E}[(X-\mathbb{E}(X))^4]$ | $ \bigl(\frac{3}{2(n-1)} + \frac{6}{n-2} - \frac{27}{2(n-3)}\bigr) y_2^2 + \bigl(1+\frac{1}{n-1}-\frac{6}{n-2} + \frac{9}{n-3}\bigr) y_4$ |
| variance of $s^2$ - alternative | $\frac{\text{preceding line}}{n} - \frac{\text{(second-to-last preceding line)} (n-3)}{n(n-1)}$ |
| variance of $s^2$ - another alternative, shortest version | $\bigl(\frac{1}{2(n - 2)} + \frac{1}{2n} - \frac{2}{n - 3} \bigr) (s^2)^2 + (\frac{3}{n - 3} - \frac{2}{n - 2}) y_4$ |
| error of a supervised learning algorithm | U-statistic associated with the kernel of degree $g+1$ where $g$ is the learning set size |