Does duplicating 50 observations raise correlation

Question

Studying temperature and cacao tree growth, you have $50$ observations with correlation $0.75$ . You append an exact copy of all $50$ points and recompute the correlation. By how much does it rise?

Short answer

It does not change (increase of $0$ ). Duplicating every point leaves the means unchanged and doubles every centred sum in the correlation formula, numerator and denominator alike, so the ratio stays $0.75$ .

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Lessons behind this question

Covariance and Correlation
Populations, Samples, and Sampling Distributions

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