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A negative correlation is when you compare 2 sets of data on a line graph (e.g. scores in a French test and scores in an English test), the higher one thing is, the lower the other is (e.g. someone might score 98% on the French test but only 12% on the English test (or visa versa)). A positive correlation is the other way around. A weak correlation is when there is a lot of deviation from the line of best fit (there will always be one with correlations as a line of best fit shows correlations after all) whereas with a strong correlation, there is little deviation.
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If the two distributions can be assumed to follow Gaussian (Normal) distributions then Fisher's F-test is the most powerful test. If the data are at least ordinal, then you can use the Kolmogorov-Smirnov two-sample test.
An example of weak positive correlation would be the relationship between the amount of time spent studying for a test and the grade achieved. While there may be a slight increase in grades as study time increases, the correlation is not very strong. This means that studying more does not guarantee a significantly higher grade, but there is still a positive trend between the two variables.
I was given this formula in college: IND Posttest score - IND pretest score ______________________________ = Improvement Score Highest score for all - IND pretest score