Normality test for large samples So I working on a programming assignment that uses multiple algorithms to solve the floodit game. I have taken some of my data that I have collected thus far. I did a shapiro test:
   shapiro.test(x[1:5000])

   Shapiro-Wilk normality test

   data:  x[1:5000]
   W = 0.9806, p-value < 2.2e-16 

To my understanding, I must reject the null hypothesis, which means my data is not normal. I then used the hist function in R:

Does this Hist show a normal distributed data? if so does that mean i reject my results from the shapiro test? and If so, why do reject the shapiro test?
 A: Since the sample size is large, statistical hypotheses tests have a large power (1 - probability of II type error), and hence any small difference between your distribution and the null distribution (Normal distribution) is meaningful and leads to the rejection of the null hypothesis.
Your data looks (approximately*) Normally distributed, but considering the large sample size you can trust Shapiro-Wilk test: your data are not Normally distributed.
*your histogram has only 7 bins and thus your data looks approximately Normally distributed, but maybe if you increase the number of bins you can see a larger departure from the Normal distribution. Moreover, you could show the QQ-plot (your data VS theoretical Normal) to highlight the departures of your data from the Normal distribution.
A: When your sample-size is big enough (i.e. > 30) you can assume normality according to the Central Limit Theorem CLT.  Andy Field, author of Discovering Statistics using R, has an easy video on this question: https://www.youtube.com/watch?v=ermii2fQWOo. I know this answer may be 2 years too late, but hopefully it helps.  
A helpful document on the Central Limit Theorem is found here: http://www2.psychology.uiowa.edu/faculty/mordkoff/GradStats/part%201/I.07%20normal.pdf
Bottom line: if your sample size is greater than or equal to 30, you can assume normality.
