Twelve-monthly values of rainfall since 1883 at Manilla NSW yield the four moments of their frequency distributions: mean, variance, skewness, and kurtosis. I plotted the history of each moment (when smoothed) in an earlier post.
Here, I compare the moments in pairs. Connected scatterplots reveal the trajectory of each relationship with time.
Some linear and cyclic trends persist through decades, but none persists through the whole record.
The first image is an index to the suite of six graphs of pair-wise relationships that I present below.
Tag Archives: variance
Moments of Manilla’s 12-monthly Rainfall
REVISED, WITH MORE PRECISE DATA
Supersedes the post “Moments of Manilla’s Annual Rainfall Frequency” (15 November 2017). This post includes twelve times as much data.[See Note below: “Data handling”]
Comparing all four moments of the frequency-distributions
Yearly rainfall for Manilla, NSW, has varied widely from decade to decade, but it is not only the mean amounts that have varied. Three other measures have varied, all in different ways.
I based the graph on 125-month (decadal) sub-populations of the 134-year record. I plotted data for every month, at the middle month of each sub-population.
I analysed each sub-population as a frequency-distribution, to give values of the four moments: mean (drawn in indigo), variance (drawn in orange), skewness (drawn in green) and kurtosis (drawn in blue).
[For more about the moments of frequency-distributions, see the post: “Kurtosis, Fat Tails, and Extremes”.]
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.
Twenty-first century values are on the right. They are remarkable in three of the four moments. First, the mean rainfall (indigo) stays near the long-term mean, which has seldom happened before. By contrast, two moments are now near historical extremes: variance (orange) is very low and kurtosis (blue) very positive. Skewness (green) is rather negative.
To my knowledge, such a result has not been observed or predicted, or even suspected, anywhere.
[Note. The main difference from the earlier 4-moment graph based on more sparse data is that skewness does not trend downward.]
The mean 12-monthly rainfall (the first moment)
The first moment of the frequency-distribution of 12-monthly rainfall is the mean, or average. It measures of the amount of rain.
As I have shown before, the rainfall was low in the first half of the 20th century, and high in the 1890’s, 1950’s and 1970’s. Rainfall crashed in 1900 and again in 1980.
12-monthly rainfall variance (the second moment)
Moments of Manilla’s Yearly Rainfall History
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.