# SPI (Standardized Precipitation Index) for 40 years seasonal mean

I am working with SPI (Standardized Precipitation Index) and in particular I am interested in obtaining the seasonal mean observation over a period of 40 years.

The meteorological season I am considering is Winter and therefore the months of December-January-February.

The point is that for each location and for time scale I am using, the overall seasonal SPI winter mean should = ~0.

Is that possible?

I am expecting that the annual SPI averages would be = 0, because of the definition of SPI, but not for the overall seasonal mean.

The SPI statistic can be considered a standard normally distributed random variable $SPI \sim N(0,1)$. Thus the expected value, or mean, of the SPI statistic is $0$, and its standard deviation and variance are $1$.

If the mean of the distribution is zero, then by the central limit theorem, you would expect that the limit of your sample mean (the mean of ~40 years of data) would approach zero as your sample got larger. So yes, it is both possible and expected.

What may be causing your confusion is that SPI can be calculated for different time sets. If you are analyzing month by month, then I would assume that your dataset is divided by month. In that case, the SPI is calculated individually for each month, so that the mean of any month or set of months over many years would tend to zero, as per the last paragraph.

It would not make sense to calculate a monthly SPI from a 12-month mean divided by 12 (to give you a mean per month on an annual basis) In seasonal wet-dry climates you would have massively negative SPI value in the dry season, and highly positive numbers in the wet season. The monthly means are individually given an SPI statistic to allow apples-to-apples comparisons with the same month in other years.

• Thank you. So in your opinion would make sense to compare the (~0) 40 years seasonal mean with only 1 year average season and also 1 specific month average? – test Nov 17 '16 at 22:24
• Thank you. So in your opinion would make sense to compare the (~0) 40 years seasonal mean with only 1 year average season and also 1 specific month average? The point is that I want to show that the 1 season and 1 month SPI averages are well higher than the 40y mean. But maybe if they all tend to zero as much the data become more large this does not really make sense..or not? – test Nov 17 '16 at 22:29
• @test By definition, mean SPI needs to tend to zero for annual or any monthly SPI statistic. If a single season or month has an SPI of 1, that means it recieved one standard deviation more rain than expected; and SPI of -2 means two standards deviations less than expected for that single season or month. – kingledion Nov 17 '16 at 22:34
• Wouldn't a normally distributed random variable from 0 to 1 have a mean of 0.5?? – JeopardyTempest Apr 16 '18 at 17:31
• @JeopardyTempest That would be a uniform random variable on the range [0,1]. A standard normal random variable by definition has a mean of 0 and a variance (and standard deviation) of 1. The notation $N(0, 1)$ doesn't indicate the range, but the two parameters (mean and variance). – kingledion Apr 16 '18 at 17:37