# 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

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

### Variance¶

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¶

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

### 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

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