---                                                                                                 
myst:                                                                                           
    substitutions:                                                                              
        sentence: "When considering an existing set of points, it exists"
---                                                                                                 
                                                                                                    
```{include} /../core/dataserver/statistics/computing_elementary.md                                 
``` 

The minimum and maximum are trivially estimated by running over all the possible values. The average 
and standard deviation are estimated on the fly, using the following recursive formulae (where 
$\zeta_{i}$ represents the value of $\zeta$ using all data points up to $i$ for $i = 1,\ldots,n_{S}$):

- average: $\mu_{x_{0}}$ is set to 0 and then 

```{math}
\mu_{x_{i}} = \mu_{x_{i-1}} \times \frac{i}{i+1} + \frac{x_i}{i+1}
```

- standard deviation: $\sigma_{x_{0}}$ is set to 0 and then, for $i$ strictly greater than 0, 

```{math}
\sigma_{x_{i}} = \sigma_{x_{i-1}} \times \frac{i-1}{i} + \frac{i+1}{i} \times \frac{(x_i-\mu_{x_{i}})^2}{i}
```