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# Seasonal fluctuations. Seasonal indices. Constant mean method.

The study of periodic (seasonal) fluctuations. Calculation of average seasonal index by constant mean method.

Continuing the theme which was started in this article Analytical performance indicators.

Here we will talk about average seasonal index - analytical indicators of time series characterizing the seasonal fluctuations

The seasonal fluctuations are annual, constantly repeating change of the studied phenomena. During the analysis of the annual dynamics, you obtain the quantitative characteristics, reflecting the nature of the changes of indicators by months of the annual cycle.

Seasonal fluctuations are described by seasonal indices which are calculated as a ratio of the actual value of the indicator to some theoretical (predicted) level.

$I_{ij}=\frac{Y_{ij}}{Y_t_{ij}}$

Where i - the ordinal number of the seasonal cycle (years), j - the ordinal number of the intraseasonal period (months).

The obtained values are subject to random deviations that's why these values are averaged out by years so you get the average seasonal indices for each period of the annual cycle (months)

$I_{sj}=\frac{\sum_{i=1}^n I_{ij}}{n}$

Depending on the nature of the changes of the time series the formula can be calculated with different methods.

I'll review the easiest method - the constant mean method. The method can be used for the time series where are no downward/upward tendencies or they are negligible. In other words, the observed value fluctuates around a certain constant value.

$I_{sj}=\frac{Y_{sj}}{Y_{s0}}$,

where

$Y_{sj}=\frac{\sum_{i=1}^n Y_{ij}}{n}$, average for each intraseasonal period j (months) for all n seasons

$Y_{s0}=\frac{\sum_{i=1, j=1}^{i=n, j=m} Y_{ij}}{nm}$, an average for all seasons (n) and interseasonal periods (m)

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### Calculation of average seasonal index by constant mean method

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Seasonal index