>>13652572Really we could use any dispersion - a nonnegative real number that is zero if all the data are the same and increases as the data become more diverse - to measure how much our data "varies".
There are tons of measures of dispersion
Standard deviation
Interquartile range
Range
Mean absolute difference (Gini mean absolute difference)
Median absolute deviation
Average absolute deviation (or simply average deviation)
Distance standard deviation
Coefficient of variation
Quartile coefficient of dispersion
Relative mean difference, equal to twice the Gini coefficient
Entropy: While the entropy of a discrete variable is location-invariant and scale-independent, and therefore not a measure of dispersion in the above sense, the entropy of a continuous variable is location invariant and additive in scale: If Hz is the entropy of continuous variable z and z=ax+b, then Hz=Hx+log(a).
Variance
Variance-to-mean ratio
Berger–Parker index
Brillouin index of diversity
Hill's diversity numbers
Margalef's index
Menhinick's index
Q statistic
Shannon–Wiener index
Rényi entropy
McIntosh's D and E
Fisher's alpha
Strong's index
Simpson's E
Smith & Wilson's indices
Heip's index
Camargo's index
Smith and Wilson's B
Nee, Harvey, and Cotgreave's index
Bulla's E
Horn's information theory index
Rarefaction index
Caswell's V
Lloyd & Ghelardi's index
Average taxonomic distinctness index
Index of qualitative variation
and you can come up infinity more of them.
So why do we use variance as calculated? It's pretty much because Var(X+Y) = Var(X) + Var(Y) when X and Y are not correlated. It just makes the math easier. I don't think there is any other reason.
