The Fisher Z-Transformation is a way to transform the sampling distribution of Pearson’s r (i.e. the correlation coefficient) so that it becomes normally distributed. The “z” in Fisher Z stands for a z-score.
The formula to transform r to a z-score is: z’ = .5[ln(1+r) – ln(1-r)], where ln is the natural log.
Fisher’s z’ is used to find confidence intervals for both r and differences between correlations. But it’s probably most commonly be used to test the significance of the difference between two correlation coefficients, r1 and r2 from independent samples. If r1 is larger than r2, the z-value will be positive; If r1 is smaller than r2, the z-value will be negative.
While the Fisher transformation is mainly associated with Pearson’s r for bivariate normal data, it can also be used for Spearman’s rank correlation coefficients in some cases.
求出z1 和z2的差（Δz），再除以z1 和z2的联合标准误差，其结果也是一个z-值（即服从正态分布，因此可以根据其与正态分布的临界值来判断是否显著）
Fisher’s r-to-z ： Z_i=0.5 ln[(1+r_i )/(1-r_i) ],
and then Z statistics test was used to check z-values as
z=(Z_1-Z_2 ) √(1/(n_1-3)+1/(n_2-3)).