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I have two count variables. I want to calculate a Pearson correlation between them. However, one of the variables only has a range of 1-7. Is it inadvisable to use Pearson correlation to study a relationship between two variables when one of them has such a small range? If so what alternative method should I use?

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In theory, the range does not matter. You can scale your narrow-ranged variable without altering the correlation.

$$ \operatorname{cor}\left(X, Y\right) = \dfrac{ \operatorname{cov}\left(X, Y\right) }{ \operatorname{sd}\left(X\right) \operatorname{sd}\left(Y\right) }$$

Now, let $a\ne 0$.

$$ \operatorname{cor}\left(\left(aX\right), Y\right) = \dfrac{ \operatorname{cov}\left(\left(aX\right), Y\right) }{ \operatorname{sd}\left(\left(aX\right)\right) \operatorname{sd}\left(Y\right) } = \dfrac{ a\times\operatorname{cov}\left(X, Y\right) }{ a\times\operatorname{sd}\left(X\right) \operatorname{sd}\left(Y\right) }= \dfrac{ \operatorname{cov}\left(X, Y\right) }{ \operatorname{sd}\left(X\right) \operatorname{sd}\left(Y\right) } = \operatorname{cor}\left(X, Y\right) $$

Thus, you could multiply all of your values $1-7$ by a trillion $($so $a = 10^{12}$$)$ to give yourself a range of six-trillion instead of six, and the correlation would not change. This is good, because it means that unit conversions do not affect the correlation. That is, you get the same correlation if you measure in meters as you do if you measure in miles.

What would be concerning for real empirical work is if you only have a narrow range of the total possible range. In such a case, the conclusions you would draw from your range of $1-7$ might not apply for values above $8$ or below $1$, related to extrapolation. If you only care about that $1-7$ range, however, I see no issue.

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Range is not important for correlation. I would suggest first to plot the data and visually inspect the relationship. If it appears to be linear then a Pearson correlation coefficient is probably the best choice, if the relationship is not exactly linear then Spearman may be a better choice.

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