I am interested in calculating high frequency 5-minute intraday volatility. I am going to use the standard Realized volatility which is the square root of the sum of squared log returns.
Given X is the log price of a stock the log return Y is calculated as $$ log \ returns = Y = \frac{x_i+1}{x_i} $$
Then the realized variance is the sum of the squared log returns:
$$ Realized\ Variance = \sum\limits_{i=1}^{n} (y_{t_{i}})^2 $$
$$ Realized \ Volatility = \sqrt{\sum\limits_{i=1}^{n} (y_{t_{i}})^2 } $$
For 5-minute realized volatility n = 78 (there are 6.5 hours in the NYSE trading day)
Now if Y is the log returns and the mean of Y is assumed to be zero you can also calculate a standard deviation
$$ standard \ deviation = \sqrt{\frac{1}{N}\sum\limits_{i=1}^{N} (y_i)^2}$$
So you can see the only difference between the Realized Volatility of Y and the standard deviation of Y is the $ \frac{1}{N} $ term in the standard deviation calculation.
Can you explain the significance of this? Why does realized variance not have 1/N and how can the 2 be interpreted?