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When calculating Cronbach's $\alpha$, would I include the reverse scores? I have 20 items in total, including 11 reverse-scored items. Would I just calculate the original scores?

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You should reverse the scores of the reversed items before computing Cronbach's $\alpha$.

If you wish to reverse items into the original scale you can compute: $$ r_{jk} = (max_j-min_j) - x_{jk} $$

Where $max_j$ and $min_j$ are the theoretical maximum and minimum values of item $j$, $x_{jk}$ is the score of individual $k$ in item $j$ and $r_{jk}$ is the reversed score of individual $k$ in item $k$.

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