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Statistics are sometimes misinterpreted (or even intentionally misleading). One common way is when relationships that exist at a group level are assumed to hold at the individual level.
This is called the "ecological fallacy" A famous study by Holmes, et al and published in JAMA in 1999 followed over 88,000 nurses for 14 years. The study found that at an individual level, higher fat intake was not associated with a higher risk of cancer- dispelling previous beliefs based on population data that showed societies where people eat high fat foods have higher levels of breast cancer.
There was some controversy over the Holmes study, described in a letter to the editor of JAMA, with arguments about the statistics and methods in the study.