I've heard from different sources that, standard deviation measures how spread out the numbers are. But I've also heard the same for variance.
Is it technically correct to say this statement for both std and var?
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Yes it is. Standard deviation is a square root of variance. Square root is a monotonic transformation, meaning that it preserves the order, e.g, if
a > b then
sqrt(a) > sqrt(b), assuming a and b are non-negative and they always are for variance.
Standard deviation is easier to interpret and is more commonly used, when we calculate variance we square every term, taking a square root in standard deviation, we undo that.