If you must normalize the data, consider a transformation from X (from any distribution) to Z (standard normal distribution). The text by William Conover on nonparametric statistics suggests the use of a normal scores test similar to the original test by Van der Waerden. Many software package offer such tests. It is also easy to program yourself.

1. arrange the X values from smallest to largest.

2. obtain the inverse cdf for a standard normal variable for each X value

3. use the resulting z scores in any parametric test.

The data follow your method of collection and analysis. The data distributes along a scale you chose for analysis. The scale should have a meaning for the reason for collecting the data. The data are the data and no scale transformations to shape the data distribution alters the original distribution along the original scale. The transformation allows simpler analysis based on the scale of transform. The normal distribution is a common statistical distribution with well-known properties and is often imposed on data because testing the normal distribution is well-established.

To say the data follow a normal distribution or can be fit to a normal allows one to use established methods. Nevertheless, no data ever fits a normal distribution or any other distribution. Once we claim data representation by a distribution we are no longer talking about the data, but a representation of the data. Data departures from the distribution may be important and regions of the true data distribution may be more important than others. One must take care to stay grounded in the data, not the distribution.

Yes, you might be able to find groupings or rankings that force the data to appear to fit a normal distribution. Is it necessary? Is there an interpretive advantage? Do the current data answer the question asked? Have tried nonparametric analysis?

If you must normalize the data, consider a transformation from X (from any distribution) to Z (standard normal distribution). The text by William Conover on nonparametric statistics suggests the use of a normal scores test similar to the original test by Van der Waerden. Many software package offer such tests. It is also easy to program yourself.

1. arrange the X values from smallest to largest.

2. obtain the inverse cdf for a standard normal variable for each X value

3. use the resulting z scores in any parametric test.

thanks all for your valuable answers.