# Statistics

## Mean, Variance and Standard Deviation

### Mean

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

### Variance

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

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 \(\sigma\):

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):

## Covariance and Correlation

### Correlation

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