Is Volatility Variance? Are They the Same Thing?
The answer is:
Sometimes (very rarely) they may be considered the same.
They are definitely closely related.
Volatility (at least in finance) is usually understood as standard deviation rather than variance.
Volatility and Standard Deviation
Standard deviation is one of the ways how to calculate and interpret volatility of securities and investment returns.
One of the ways – not the only way – although it is by far the most popular way (used especially, but not only, for anything related to option trading).
Besides standard deviation, there are other methods how to calculate and interpret volatility. Some are just slight modifications of the standard deviation approach (for example non-centered historical volatility), while others are completely different concepts (for example range or average true range).
One of these alternative methods is variance, but its use as a measure of volatility is much less frequent than the use of standard deviation, at least for the average investor (it can be more useful in more complex quantitative finance).
Here you can see more details: Is Volatility and Standard Deviation the Same?
Variance and Standard Deviation
The relationship between variance and standard deviation is very close and very simple:
Standard deviation is the square root of variance.
Variance is standard deviation squared.
You can see the calculation of both variance and standard deviation explained here: