Comparing all four moments of the frequency-distributions

Annual rainfall for Manilla, NSW, has varied widely from decade to decade, but it is not only the mean amounts that have varied. Three others measures have varied, all in different ways.

I based the graph on 21-year sub-populations of the 134-year record, centred on consecutive years. I analysed each sub-population as a frequency-distribution, to give values of the four moments: mean (drawn in black), variance (drawn in red), skewness (drawn in blue) and kurtosis (drawn in magenta).

[For more about the moments of frequency-distributions, see the recent post: “Kurtosis, Fat Tails, and Extremes”. See also the Note below: “Instability in the third and fourth moments.”]

Each trace of a moment measure seems to have a pattern: they are not like random “noise”. Yet each trace is quite unlike the others.

The latest values are on the right. They show that the annual rainfall is now remarkable in all four respects. First, the mean rainfall (black) closely matches the long-term mean, which has seldom happened before. By contrast, the other three moments are now near historical extremes: variance (red) is very low, skewness (blue) very negative, and kurtosis (magenta) very positive.

To my knowledge, such a result has not been observed or predicted, or even suspected, anywhere.

[SEE A REVISED VERSION OF THIS WORK
A revised version of this post uses twelve times as much data. It is “Moments of Manilla’s 12-monthly Rainfall” posted on 15 May 2018.]

The mean yearly rainfall (the first moment)

As I have shown before, the mean annual rainfall was low in the first half of the 20th century, and high in the 1890’s, 1960’s and 1970’s. Rainfall crashed in 1900 and again in 1980.

Yearly rainfall variance (the second moment)

The second moment of the frequency-distribution, the variance, measures the scatter or spread of the annual rainfall values. The pattern has some features in common with that of the mean rainfall: values of the variance fell rapidly at 1900, were very low at 1906 and 1936, and rose rapidly at 1940. Otherwise, the pattern was different: variance values were extremely high around 1950, and very low from 1980 onward.

One example of strong disagreement between the first and second moments is that, while the rainfall was very high around both 1958 and 1978, the scatter was extremely high around 1958 but very low around 1978.

Yearly rainfall skewness (the third moment)

The third moment of the frequency-distribution, the skewness, shows how the curve is lop-sided one way or the other. With positive skewness, the tail on the right side (high values) is heavier, being longer or fatter or both.

Skewness in annual rainfall declined throughout the period of record. A linear trend fits with an R-squared value of 0.67. The major deviations from the trend were a near-minimal value around 1968 and a positive value around 1982.

Such a rapid decline in skewness cannot be sustained, yet there is no sign of a reversal.

Yearly rainfall kurtosis (the fourth moment)

The fourth moment of the frequency-distribution is the kurtosis. As explained in the post “Kurtosis, Fat Tails, and Extremes”, kurtosis measures how frequent (or how rare) extreme values are in the data. With positive values of “Excess kurtosis”, extremes are more frequent than in a normal distribution that has the same variance.

The pattern of kurtosis in Manilla’s annual rainfalls is stark: extremely high values near the beginning and end of the record, a much lower, broader peak near 1940, and strongly negative values in the 1910’s, 1920’s, 1950’s, 1960’s and 1970’s.

I will use this graph to further explore extremes in Manilla’s frequency-distribution of annual rainfall.

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Note:
Instability in the third and fourth moments.

The book “Numerical Recipes in C: The Art of Scientific Computing” by William H. Press cautions strongly against using skewness, and especially kurtosis, when samples are small, as in the 25-point sub-populations used here. The results are highly unstable.

For kurtosis, minimal sample numbers in the thousands are suggested.
Nevertheless, I am encouraged to proceed by the clear, broad patterns that appear in this data set.

[SEE A REVISED VERSION OF THIS WORK

I have increased the sample size by a factor of 12 in the revised post: “Moments of Manilla’s 12-monthly Rainfall”.]