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What does smoothed monthly mean sunspot number actually mean?

The smoothed monthly number results from an averaging of monthly mean values over the 13 months, from 6 months before to 6 months after a base month. All months are weighted equal except for the extreme ones, which are weighted by 1/2. This is expressed by the formula:
Rs= (0.5 Rm-6 + Rm-5 + Rm-4 + Rm-3 + Rm-2 + Rm-1 + Rm + Rm+1 + Rm+2 + Rm+3 + Rm+4 + Rm+5 + 0.5 Rm+6 ) / 12
In signal processing jargon, this would be called a "tapered box-car" smoothing function.

This smoothing formula was introduced in the early 20th century by the Zürich observatory, then in charge of the sunspot number production. It was probably chosen for its simplicity for manual calculations. Today, we know many other smoothing functions, often with better low-pass filtering charactristics. However, as it was used as a standard for so many decades, it remains the base reference allowing an easy comparison of various scientific analyses based on the sunspot number. Indeed, the smoothed series is meant for two main purposes:
- generating a series that reflects only the overal evolution of each solar cycle, by filtering out the fast variations (random surges and 27-day rotational modulation)
- defining the times of maximum and minimum for each cycle, thus providing the consistent timebase on which other series can be linked to the solar semi-regular periodicity.

NB: as the symmetrical smoothing process requires 6 monthly means around each base month, the smoothed values cannot be calculated for the first and last 6 months of the series. The last smoothed number thus always lags by 6 month behind the latest monthly mean sunspot number.