I'm working with some data (in excel) which is skewed right in order to get some Probability & Statistics practice in. The skew of the data prior to any transformation is 1.16. I then converted the data into Z-Scores using the standard formula:
$$ Z = \frac{x - \mu} \sigma $$
This results, as I would expect, in the mean being equal to zero and the standard deviation equal to 1. However, the new skew of the data is 1.15.
This leads me to believe I am misunderstanding the nature of the Z-transformation process. Should this process not result in a probability distribution that is symmetrical around the mean? If not, what the point of the Z-transformation? How can I find the probability of any given outcome? Am I missing something important regarding these concepts?
If it helps, I am working with stock trade volume data.
Standardizing simply applies a linear transformation to your data, so it cannot remove skew or any kind of asymmetry. Subtracting a constant $\mu$ from your data will only shift its location. Dividing by a constant $\sigma$ will only scale the data ("stretching" or "squeezing" it uniformly).
In order to remove skew, a nonlinear one-to-one transformation must be applied. The choice of transformation depends on the data. Sometimes, a logarithm is applied, such as when dealing with data that is always positive and has a very heavy right tail as we might encounter in the medical sciences (e.g., pharmacokinetic data). Sometimes, a square root function is applied.