Variance
The variance of a random variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{Var}(X) := \mathbf E[(X - \mathbf EX)^2]} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf EX} is the expectation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} .
Notation
Since the square root of the variance is the standard deviation, if we have a simple notation for the standard deviation, such as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} , then we can denote the variance as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2} .
Motivation
In several books I have seen the following three-step motivation for the variance. I'm not sure I'm convinced this is all that can be said to motivate the variance, but it seems to be a start.
- We want to measure the spread of the data. For each data point, we can subtract the mean from it to see how "deviant" it is. So one a priori reasonable approach is to calculate these deviations and find the average deviation. If we do this, however, we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf E[X - \mathbf E X] = \mathbf E X - \mathbf E X = 0} which is useless.
- We then have the idea to take the absolute values of these deviations, to prevent them from adding up to zero. So we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf E |X - \mathbf E X|} . But the absolute value is not smooth enough for us to be able to do all the things we would like (e.g. differentiation). However, note that this measure of the spread is also used.
- Finally, building off the previous idea, we decide to square things, and we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf E [(X - \mathbf E X)^2]} , which is the definition of variance.