Currently I am very confused about the concept of daily volatility. First a quick background to the question. For a university paper I need to analyze several stock return volatilities (daily) using the GARCH Models. I understand that GARCH allows me to calculate a daily volatility based on the parameters and historical values. Nevertheless I would also like to compare the GARCH volatility estimations to the empirical volatilities.
The problem here is that I understand volatility as a measure of average squared deviation from the mean over a certain period. This would mean that in order to calculate volatility I would need a number of data points and not just one (daily closing prices). I have seen other papers compare GARCH volatility to the empirical volatility so I know that it is possible.
I just fail to understand how the empirical daily volatility based on one observation is calculated. If someone could explain this I would be very grateful!