Do I have to refer to normal Q-Q plot in order to find out the highest fitness of the scores?
not necessary, it is one of visual way, may be histograms of fitted CDF is instead and also goodness-of-fit statistics
I did the normality test in order to find the distribution of scores for each session that I have conducted. Since the p value is less than 0.05, I have used the normal q-q plot to verify the data. The normal q-q plot shows the highest r square fitness which is above 0.997. so i have assumed that my scores are not normally distributed.
I guess you try to make an (repeated measures?) ANOVA or alike and test the normality assumption. Is that correct?
If so, the normality assumption is not about the values but the residuals. So you can perform a linear model to non-normal data and it is very valid. Normality described in most sources is confusing and contradictory and not necessary (as argued by them) for such models. The important thing is the normality and constant variance of residuals.
Coming back to normality, if the sample size is very large, null hypothesis of your normality test might be easily rejected. This is not because the data is not normal, but the sample size is very large.
So you can draw histograms, Q-Q plots or make different tests, consider skewness & kurtosis, empirical rule etc to draw a conclusion about normality. Do not stick with one single test, do make different normality checks by above explained ways, if you still think that checking for normality is crucial.
About R squared, I guess you calculated a correlation coefficient between theoretical quantiles and quantiles from data and squared it. 0.997 is very large and must show a very large goodness of fit. So you must say, by just looking at this R squared value, that your data is normal, if that is the way what you have found R squared.
See my paper ``Confidence Bands for Distribution Functions When Parameters are Estimated from the Data: A Non Monte-Carlo Approach", published in the Biometrical Journal (2013), vol. 55, No. 3, pp. 370-385, Two important special cases are treated in some detail: the normal and exponential distributions. Graphical representations and comparisons with results obtained by Lillifors and Stephens via Monte-Carlo methods are discussed. You can email me to request a high resolution pdf reprint of the article.
Q-Q plot is in general used for visualizing whether normality assumption is acceptable for your model, but not a formal exact test for normality. It is often disturbed by outliers. So you may try to delete one or more outliers to see whether the fitness is improved, i.e., you may need to perform a detailed statistical diagnosis for your model-data set. Moreover, data transformation is often helpful for improving model settings.
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