We can test the symmetry of a distribution around $0$ by Wilcoxon sign rank test, based on its sample.

But if we want to test if a distribution is symmetric around its mean, based on its sample $X_1, \dots, X_n$, is it valid to first normalize $X_i$ by the sample mean as $Y_i := X_i - \bar{X}$, and then apply Wilcoxon sign rank test to $Y_i$'s?

If not, what are some ways?

Thanks and regards!

We can test the symmetry of a distribution around $0$ by Wilcoxon sign rank test, based on its sample.

But if we want to test if a distribution is symmetric around its mean, based on its sample $X_1, \dots, X_n$, is it valid to first normalize $X_i$ by the sample mean as $Y_i := X_i - \bar{X}$, and then apply Wilcoxon sign rank test to $Y_i$'s?

If not, what are some ways?