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)
Computing Moments in MICRODEM
Last revision 2/25/2022