Can Pooled Correlation Exceed Every Group Correlation?

A dataset has 100,000 observations with variables x1 and x2. It is divided into three groups of approximately equal size. Within each group, the Pearson correlation between x1 and x2 is 0.05. Could the correlation after pooling all three groups be 0.10? ### Constraints and Clarifications The original equal-three-way split cannot be exact for 100,000 observations; group sizes 33,333, 33,333, and 33,334 preserve the intended comparison. Treat correlations as empirical population-moment correlations, using consistent weights. No equality of group means or variances is stated. ### What a Strong Answer Covers Give a constructive explanation using within-group covariance and variation in group means. Explain why averaging the three correlations does not generally produce the pooled correlation. ### Follow-up Questions - What additional assumptions would make pooled correlation equal the within-group value? - Can pooled correlation have the opposite sign from every within-group correlation?