**Moment Statistics**

Moments are a set of statistical parameters to measure a distribution. Four moments are commonly used:

- 1st, Mean: the average
- 2d, Variance:
- Standard deviation is the square root of the variance: an indication of how closely the values are spread about the mean. A small standard deviation means the values are all similar. If the distribution is normal, 63% of the values will be within 1 standard deviation.

- 3d, Skewness: measure the asymmetry of a distribution
about its peak; it is a number that describes the shape
of the distribution.
- It is often approximated by Skew = (Mean - Median) / (Std dev).
- If skewness is positive, the mean is bigger than the median and the distribution has a large tail of high values.
- If skewness is negative, the mean is smaller than the median and the distribution has a large tail of small values.

- 4th: Kurtosis: measures the peakedness or flatness of a
distribution.
- Positive kurtosis indicates a thin pointed distribution.
- Negative kurtosis indicates a broad flat distribution.

Higher moments tend to be less robust. Press and others (1986, p.457) recommend that skewness and kurtosis be used "with caution or, better yet, not at all".

Related statistics for distributions:

- Average Deviation or mean absolute deviation: 1 / N * (sum(abs(value-mean))) This is a more robust estimate of the width of the distribution (Press and others, Numerical Recipes, 1986, Cambridge University Press)
- MAE, or mean absolute error: sum(abs(value))/N
- RMSE, or root mean square error: sqrt(sum(sqr(value))/N)

*Last revision 2/25/2022*