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After I made corrections (in accordance with Pro Tutorial) to some daily precipitation totals, both the monthly total and the annual total have changed, but no longer correspond to the total precipitation as shown in the History / Edit Hour Data tab. The monthly total indicates double the amount and the annual total 300 mm too much.
What could be the cause of this? And how can I remedy this?

So what corrections did you actually do to the daily precipitation totals?

Corrections tab History/Edit Hour Data. According to Tutorial Cleanup of History Data