Mean, Variance and Standard Deviation


The mean of a vector, usually denoted as \(\mu\) , is the mean of its elements, that is to say the sum of the components divided by the number of components

\[\bar{x} = \mu = \frac{1}{n} \sum_{i=1}^n x_i\]


The variance is the mean of the squared differences to the mean.

\[var(x) = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2\]

with \(var(x)\) being the variance of the variable \(x\), \(n\) the number of data samples, \(x_i\) the ith data sample and \(\bar{x}\) the mean of \(x\).

Standard Deviation

The standard deviation is simply the square root of the variance. It is usually denoted as \(\sigma\):

\[\sigma(x) = \sqrt{\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2}\]

We square root the variance to go back to the units of the observations.

Both the variance and the standard deviation are dispersion indicators: they tell you if the observations are clustered around the mean.

Note also that the variance and the standard deviation are always positive (it is like a distance, measuring how far away the data points are from the mean):

\[ \begin{align}\begin{aligned}var(x) \geq 0\\\sigma(x) \geq 0\end{aligned}\end{align} \]

Covariance and Correlation

\[cov(x, y) = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})\]


The correlation, usually refering to the Pearson’s correlation coefficient, is a normalized version of the covariance. It is scaled between -1 and 1

\[corr(x, y) = \frac{cov(x, y)}{\sigma_x \sigma_y}\]