> For the complete documentation index, see [llms.txt](https://blog.gabriele.pro/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://blog.gabriele.pro/a-robust-variance.md).

# A Robust Variance

The refoundation of the Data Science must do accounting with the *robust variance* *metric* here suggested. *Robustness,* a sample estimator property with respect to tone down hypothesis, here about the *deviation* squaring *metric,* is intended respect the variance abnormal effects already reported. The *variance* adopts a metric that has a superiority complex. At the same time, the *absolute deviation* suffers from the opposite problem: an inferiority complex: it certainly appears timid and haughty, while, in a certain sense, it fails to take a stand. With this idea in mind we suggests, for an *X* variable observed on *N*-unit collective, a class of *robust-variances*: , with, , where is an increasing function in the negative real segment and a decreasing one in the positive one thus producing opposite effects on the deviations with respect to the *variance*. This would amount to a re-match victory for the deviations which occupy *central* positions in order to snub the *lateral deviations,* as defined afterwards. In we prefer what can be called a *robust* *variance*: where *α* is a modulator parameter of the squaring defects using the ratio between the reciprocal of the *absolute deviation,* and the *absolute deviation,* .

First, let's see how two different types of *deviations* behave: *standardized*, ***zX*** , and *normalized, **tX**,* with *PnP* transformation (from *non-preferred* to *preferred*), in percentage, relative to the five parameters provided by Brussels and already considered previously.

<table data-header-hidden><thead><tr><th width="158"></th><th width="47"></th><th width="78"></th><th width="55"></th><th width="110"></th><th width="72"></th><th width="78"></th><th width="64"></th><th></th></tr></thead><tbody><tr><td><em><strong>zX</strong></em></td><td></td><td></td><td></td><td><strong>min</strong>≤<strong>max</strong></td><td>%||&#x3C;1</td><td></td><td></td><td></td></tr><tr><td><em>z</em>Pil</td><td>1</td><td>0.695</td><td><strong>V</strong></td><td>-1.4≤3.8</td><td>74</td><td>0.390</td><td><strong>V</strong></td><td>0.389</td></tr><tr><td><em>z</em>Deficit</td><td>1</td><td>0.782</td><td><strong>III</strong></td><td>-1.9≤2.0</td><td>67</td><td>0.564</td><td><strong>III</strong></td><td>0.228</td></tr><tr><td><em>z</em>Debt</td><td>1</td><td>0.816</td><td><strong>II</strong></td><td>-1.5≤2.3</td><td>67</td><td>0.632</td><td><strong>II</strong></td><td>0.210</td></tr><tr><td><em>z</em>Inflaction</td><td>1</td><td>0.752</td><td><strong>IV</strong></td><td>-1.2≤3.0</td><td>70</td><td>0.504</td><td><strong>IV</strong></td><td>0.778</td></tr><tr><td><em>z</em>Employment</td><td>1</td><td>0.830</td><td><strong>I</strong></td><td>-2.0≤1.9</td><td>63</td><td>0.660</td><td><strong>I</strong></td><td>0.091</td></tr></tbody></table>

We can call those *deviations* whose absolute value is less than its unity, here typically 66%, *central deviations*; those deviations which have an absolute value greater than one, here 34%, however, we will call *lateral deviations*. The *central deviations* are strongly scaled down by the *squaring* produced by the *variance* and the *lateral deviations* are overstated. This is the impact of *variance* as a metric. Therefore, the *robust* *variance* may be preferable, suggested as an alternative, or, if preferred, as a culmination. We note that *α* almost halved the squared defects *(0.43* precisel&#x79;*).* The European parameter with greater variability, like Inflection and Debt, needed priority interventions, rather than Employment and Debt, as absolute deviation indicates. In fact,*robust* *deviation*, do not agrees with the *absolute deviation*, *δ*.The overestimation effect can be summarized as follows: the *standard deviation*, σ, is on average 1.29 times the *absolute deviation*, *δ*, and 2.28 times the *robust* *deviation*,, while *δ* is, on average, 1.77 times bigger . When considering *deviations*, or rather *PnP* on a percentage scale, things are different on worthwhile variables such as classifications (a scale which ranges from *non-preferable* to *preferable*):

<table data-header-hidden><thead><tr><th width="157"></th><th width="86"></th><th width="82"></th><th width="62"></th><th width="111"></th><th width="95"></th><th></th><th></th><th></th></tr></thead><tbody><tr><td></td><td></td><td></td><td></td><td><strong>min</strong>≤<strong>max</strong></td><td><strong>%|</strong><em><strong>x</strong></em><strong>|&#x3C;|1|</strong></td><td></td><td></td><td></td></tr><tr><td><em>t</em>Pil</td><td>19.50</td><td>13.57</td><td><strong>V</strong></td><td>-27≤74</td><td>0</td><td>7.64</td><td><strong>V</strong></td><td>0.558</td></tr><tr><td><em>t</em>DeficitD*</td><td>30.46</td><td>25.06</td><td><strong>I</strong></td><td>-62≤38</td><td>0</td><td>19.66</td><td><strong>I</strong></td><td>0.290</td></tr><tr><td><em>t</em>Debt</td><td>26.66</td><td>21.76</td><td><strong>III</strong></td><td>-60≤40</td><td>3.7</td><td>16.86</td><td><strong>III</strong></td><td>0.219</td></tr><tr><td><em>t</em>Inflaction</td><td>24.65</td><td>18.53</td><td><strong>IV</strong></td><td>-72≤28</td><td>0</td><td>12.41</td><td><strong>IV</strong></td><td>0.782</td></tr><tr><td><em>t</em>Employment</td><td>26.65</td><td>22.13</td><td><strong>II</strong></td><td>-52≤49</td><td>3.7</td><td>17.60</td><td><strong>II</strong></td><td>0.090</td></tr></tbody></table>

*\* t*DeficitD = tDeficit distance from zero

In this second case it can be verified that the quadratic effect contracts 1.5% of *central deviations*, on average, while it typically dilates 98.5% of *lateral deviations*. The *robust* *deviation*, , do not agrees with the *absolute deviation*,, as in the previous case. The overestimation effect agrees with the previous one. The robust *Pnp* case requires priority interventions on Deficit and Inflation (not in Inflation and Pil like in *standardized case*).


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