I'm very confused about this difference, and I wanted to know the reason behind it (If this is very rudimentary, I'm sorry but I can't seem to wrap my head around it).
If you were to transform a data set by multiplying it, the standard deviation would grow that same amount. For example, multiplying a data set by 3 would have the average multiplied by 3 as well as the standard deviation.
However, I learned that when you annualize monthly stock returns, you multiply the average monthly stock return by 12 to get the yearly stock return, and to get from the volatility (standard deviation) of the monthly stock return to a yearly stock return volatility you would have to multiply by the square root of 12. I'm not sure where you get the difference from. Aren't you just applying a transformation? I've checked every resource on annualizing stock returns and it gives me the same formula. But looking up linear transformation the standard deviation goes up by the multiplicative amount. If anyone could explain this to me, that would be great!