# Transform data distributed around zero

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 $1$?

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You can certainly add any number you want to the data. That will shift it horizontally, but it will not make the distribution any different except for changing the center point. In particular, it will be no closer to a normal distribution. Recall the normal distribution is defined with a pdf of $\frac 1{2\pi \sigma}\exp(\frac {(x-\mu)^2}{2\sigma^2})$. Adding a constant to all your data just adds the same constant to $\mu$. Your later transform can also be adapted to deal with zeros-instead of taking the log(data), you can take log(data+10), which has the same effect as adding 10 and then taking the log.