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Probably the worst error is thinking there is a correlation when that correlation is purely artificial. Take a data set with 100,000 variables, say with 10 observations. Compute all the (99,999 * 100,000) / 2 cross-correlations. You are almost guaranteed to find one above 0.999. This is best illustrated in may article How to Lie with P-values (also discussing how to handle and fix it.)
This is being done on such a large scale, I think it is probably the main cause of fake news, and the impact is disastrous on people who take for granted what they read in the news or what they hear from the government. Some people are sent to jail based on evidence tainted with major statistical flaws. Government money is spent, propaganda is generated, wars are started, and laws are created based on false evidence. Sometimes the data scientist has no choice but to knowingly cook the numbers to keep her job. Usually, these “bad stats” end up being featured in beautiful but faulty visualizations: axes are truncated, charts are distorted, observations and variables are carefully chosen just to make a (wrong) point.
Source for picture: here