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In this paper, in order to extend the application range and enrich the research area of variance, the concepts of variance bounds and semi-variances are defined and discussed from a theoretical point of view. With respect to some frequently-used fuzzy intervals, four relatively simple calculation formulas for upper and lower bounds of variance, and upside and downside semi-variances are put forward respectively, with the aid of which several correlation inequalities are subsequently presented and proved. 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