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Central Limit theory application

This is a probability distribution I have got. (Attachment, left is the value, right the frequency) The mean is 7648.875 and the standard deviation is 21542.6814

Now, I'd like to know for n trials, where X is the above probability distribution, carried out n times. (Or the chance that the average of n trials is smaller than 7420.236) Now with the central limit theorem this approaches the normal distribution as Another thing is that if , due to the law of large numbers. Now, here is where I get stuck. From which number onwards can I apply the central limit theorem? And due to the law of large numbers, shouldn't the deviation of the distribution become smaller and smaller? I think it has to do with me dividing by n, but I don't see why it goes wrong, nor how I can help it.