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In a recent article, it was noted that the similarities between A and B imply that A is more likely to be true than not. The statistics in this article were daunting, but extremely fascinating. In our study of the Pearson product-moment correlation coefficient, we found that out of 83 provided pairs, 41 relationships were significant enough to conclude that one was true and one false. Using all 83 pairs rather than just the 41 with significant statistics gave us an accurate range of probabilities for each relationship: from .005 all the way up to 100%. This analysis shows how curious correlations can produce deeper insights about human nature and society. According to Wikipedia, the Pearson product-moment correlation coefficient is a specific example of the Pearson partial correlation coefficient, where the unknown parameter is an X and the quantities of interest are Y and Z. According to , this statistic can be calculated by: Where: The r is a function of sample size and standard error formula_7. The "true number" of pairs that will give a significant value for this statistic is given by: The probability that such pairwise comparisons will yield a significant (positive or negative) result depends on what values x, y and z might take on. In this example, the model is a random process that has mean "μ" and variance "σ", as described by: Here, formula_10 is the standard deviation of the variable "X" without regard to its distribution. In words, if "Y" is a normal random variable with mean μ and variance σ, then the value of "Y" for any observation from model formula_11 will be related to the original variables X and Z through: The correlation coefficient can be calculated as: where:Five possible forms of Pearson's product-moment correlation coefficient are listed below. A correlation coefficient of 0 implies that there is no linear relationship, while a coefficient of 1 implies that the variables are perfectly correlated. The size of the coefficient also represents how strong or weak the linear association is. For example, if X and Y are perfectly correlated (the value of r = 1), then 2 units of X will always yield an increase in Y that is equal to one unit. If the correlation coefficient is 0, then this does not necessarily mean that there is no relationship at all; it means that no information about Y can be obtained by knowing X. The sign of a correlation shows whether there is a direct or inverse relationship between variables. In the case of a positive correlation, X is positively related to Y. Inversely, a negative correlation shows that when one increases, the other decreases. In general, the Pearson's product-moment correlation coefficient between two variables is always non-negative. A negative correlation implies that there exists a difference in order or function of values, and thus is "inverse". For example: If X and Y are negatively correlated (values of r < 0), then when we take action with respect to this relationship we will receive an effect regarding Y obtained by subtraction of the reaction value of X corresponding to subtraction from one unit value (or -1 for numerical purposes). cfa1e77820
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