I have data that ranges continuously from $-1$ to $+1$, with lots of zeros in the middle. I want to transform the data to a normal distribution. How would I do this? My normal approach with data containing zeros is to $+1$ then transform ($\log_{10}(\bullet)$, $\sqrt{\bullet}$, etc). However, if I add $1$ to my data, the values that were $-1$ become zero and the transformation produce infinite values.
Is it valid to add a number greater than $1$?