Volatility (in Finance) Is Sigma, Not Sigma Squared
If you are asking this question I assume that you know that sigma denotes standard deviation, while sigma squared denotes variance in statistics. Variance is standard deviation squared. Standard deviation is the square root of variance.
Volatility, as it is commonly understood, calculated, and interpreted in finance, is the standard deviation of returns. Therefore, the answer to the question is that volatility is sigma and it is not sigma squared.
Here you can see more information about volatility, standard deviation, and variance, and their various relationships:
Historical Volatility Calculation – Here you can find how volatility is usually calculated as the standard deviation of logarithmic returns, step-by-step. The parameters of the calculation (for example the period length or annualization factor) are also explained here.
Historical Volatility Calculator + Guide – This is an Excel calculator of volatility based on historical data. The guide explains not only the various functions and settings of the calculator, but also the calculation of volatility using the two most popular methods (the standard deviation of logarithmic returns and the zero mean or non-centered method).
Is Volatility and Standard Deviation the Same? – Although volatility is usually understood as standard deviation, the two terms are of course not total equivalents. There are other alternative measures of volatility in finance. There are of course other uses of standard deviation besides volatility of investment returns.
Calculating Variance and Standard Deviation in 4 Easy Steps – This page explains the calculation of variance and standard deviation in detail, even for those who have no experience with statistics. It also explains the relationship between variance and standard deviation and the difference between calculating them for a population and for a sample.