Summary
Various equations are used to calculate the correlation coefficient, these equations are presumed equally. However we find the extraordinary results when using\(r = \sqrt {\frac{{\Sigma (\hat y_i - \bar y)^2 }}{{\Sigma (y_i - \bar y)^2 }}} \) and\(r^2 = \frac{{\Sigma (y_i - \bar y)^2 - \Sigma (y_i - \hat y_i )^2 }}{{\Sigma (y_i - \bar y)^2 }}\) to calculate the correlation coefficient, for example, a line within 95% confidence band of a regressed line. The results are so extraordinary that we do not know whether or not we can still call the results as correlation coefficient, however we are sure that these results need to be presented.
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Wu, G. An extremely strange observation on the equations for calculation of correlation coefficient. Eur. J. Drug Metab. Pharmacokinet. 28, 85–92 (2003). https://doi.org/10.1007/BF03190494
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DOI: https://doi.org/10.1007/BF03190494