Quote by Mark J.
Hi
Is there any normalization process I can make to change them from integers to real numbers as long as normal distribution is continuous?

If the meaning of your question is "Is there a way to fit a normal distribution to discrete data that can be proven correct or optimal by mathematics" , the answer is No, not without more information about what data is.
Have you tried simply converting each integer k to the corresponding "zscore" [itex] z = \frac{ (k  \mu)}{\sigma} [/itex] where [itex] \mu [/itex] is the mean of the data and [itex] \sigma [/itex] is the standard deviation of the data (or the unbiased estimator of the population standard deviation) ?
If you know that the discrete data comes from measurments of a continuous quantity that are rounded off, you might be able to do a process that essentially "smears out" each discrete data point to a possible distribution of continuous data points. Then you can try to fit a normal distribution to the superposition of these distributions. This is very sophisticated technique and I don't have the details of how to do it fresh in my mind. I think the method is called "using convolution kernels".
Explain more about the data.