There is a great discussion on weighted averages of correlation coefficients at https://www.researchgate.net/post/average_of_Pearson_correlation_coefficient_values
My most recent comments there are given below.
“The main reason for not averaging correlation coefficients in the arithmetic sense follows.”
“Correlations coefficients cannot be averaged in the arithmetic sense as they are not additive in the arithmetic sense. This is due to the fact that a correlation coefficient is a cosine, and cosines are not additive. This can be understood by mean-centering a paired data set and computing the cosine similarity between the vectors representing the variables involved.”
“If a paired data set violates the bivariative normality assumption (often overlooked, as Seifert correctly asserted), that worsens the picture. However, even if it doesn’t violates bivariative normality the computed average is a mathematically invalid exercise. If a meta analysis study is based on these averages the results can be easily challenged on these grounds.”
“Sample-size weighting is a good start, as Seifert asserted. We can certainly do better. We may compute self-weighted averages from one, more than one, or all of the constituent terms of a correlation coefficient, to account for different types of variability information present in the paired data, which otherwise might be ignored by simply sample-size weighting or applying Fisher Transformations. Which self-weighting scheme to use depends on the source of variability information to be considered (https://www.tandfonline.com/doi/abs/10.1080/03610926.2011.654037).”