As initially the data set was positively skew, the sqrt transformation is applied to get data normalized. Result shows that data is not fully normalized and still there are effect of skewness. How to proceed for analysis?
Many transformations could be used. The log transformation is often not only useful, but also appropriate. If your data are compositions, three log-ratio approaches could be used: alr, clr and ilr. You can also use Box-Cox and Yeo-Johnson transformations. Finally, you can perform non parametric statistics.
I certainly agree with the use of non-parametric techniques. However, I would add that if you are looking at a reasonably large amount of data, it would be useful to know a little more about the distribution.
If the data are simply skewed you may be able to find a fit with the negative binomial distribution; then negative binomial regression would be more powerful then many of the alternatives. There are many well developed flavours of regression - so determining what distribution you have is a good step to invest in.
More generally I am a fan of randomization tests. For many situations they are the choice that most closely fits the hypotheses you may have. A randomization test would provide an exact p value for a given hypothesis and will perform well for _almost_ any distribution. Simple skew is not a problem. Understanding your hypothesis is.
So I would say it depends on how much data you have and what you can learn about it. Best of luck.
I have come across a web material that explain how to determine normality, by using standard error value (addition , substration) with skewness and kurtosis resulting range of confidence interval, which determine the normality. If the range contains "0" with in it then the data is normality distributed. Is this concept is correct to follow?
There are many better tests of normality than this. Depending on your sample size Shapiro-Wilks is often the best test. I can also recommend Kolmogorov-Smirnov for many situations (there is plenty of debate about this).
Although tests of normality are useful they do not directly answer your question. If you draw a sample from a normal population of scores then it is more or less irrelevant that your sample is not normal. For example, IQ is constructed to have a normal population distribution. You can count on this for your other analyses. Many variables have normal or nearly related distributions. Many others do not.
One often tests normality when one does not know what the underlying distribution is. Then a sample statistic is your best "guess". However, tests of normality, like all NHST, can correctly or incorrectly lead to inferences depending on your particular design of observation and "the luck of the draw."