What does it mean for a stock to be normally distributed?

According to investopedia, a normal distribution is “a probability distribution that plots all of its values in a symmetrical fashion and most of the results are situated around the probability’s mean”.  If a firm’s returns are normally distributed, it means that if you create a histogram of a company’s returns, over a larger period of time, the histogram would take a bell shape, centered on the mean return.

If a stock’s return is normally distributed, it means that 68.26% chance that a return will be within one standard deviation ([pmath size=10]sigma[/pmath]) from the mean.  There is a 95.44% chance that the return will be within two standard deviations from the mean and a 99.74% chance that it will be within three standard deviations from the mean.