Mean, Variance, Skewness, Kurtosis

Statistical Moments

The moments of a distribution describe its shape. The first four moments are the foundation of quantitative analysis:

Mean — the average (first moment): $\mu = \frac{1}{n} \sum_{i=1}^{n} x_i$

Variance — spread around the mean (second moment, sample): $s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \mu)^2$

Skewness — asymmetry (third standardized moment): $\text{skew} = \frac{\frac{1}{n} \sum (x_i - \mu)^3}{\sigma^3}$

where $\sigma$ is the population standard deviation.

Excess Kurtosis — tail heaviness (fourth standardized moment minus 3): $\text{kurt} = \frac{\frac{1}{n} \sum (x_i - \mu)^4}{\sigma^4} - 3$

A normal distribution has skewness = 0 and excess kurtosis = 0.

Your Task

Implement all four moment functions. Use sample variance (divide by n-1) but population std (divide by n) for skewness and kurtosis normalization.