Even monthly inflation numbers are imprecise and are based not on prices at a point of time at the end of the month and rather on surveys of prices over time within a month. Over many short periods, the geometric mean is approximately the arithmetic mean minus half the variance. The variance is the average of the squared deviation from the mean, and the standard deviation is the square root of the variance. If T is large, it obviously doesn't make much difference whether we divide by T or T-1 . For daily returns, the mean is much smaller than a typical Rt , and it doesn't make much difference if we don't subtract the mean.
Practitioners commonly refer to volatility as vol for short.