Dispersion

The measure of relative position, the measures of dispersion, standard deviation, absolute deviation, and that of variability

We ended 2023 a bit hastily by looking at a multidimensional situation prematurely and thus risked losing most of you readers. Let's make a fresh start with our feet on the ground, one step at a time, trying to limit the formulas as much as we can. Previously we harshly criticized the classical measure of variability in use, variance, essentially due to an unforgivable flaw (if you will pardon my directness): Exaggerate the extreme gaps and compress the more central gaps and you overestimate their variability: by how much we will see later. Don't be alarmed beforehand, variance is capable of resurrecting itself like the mythical bird, the Arabian Phoenix, symbol of rebirth. We can continue to use σ by stating that it is an overestimation. How?

Comparing the standard deviation, σ, with the absolute deviation, δ, assuming that it is a question of measures of dispersion, not of variability exactly. Here a variability measure, ϒ, is added, the mean of the irregularities of the detachments.

As an example, let's consider the measure of relative position, E(X), the measures of dispersion, standard deviation, σ, absolute deviation, δ and that of variability γ, referring to the five series relating to the European parameters (transformed according to PnP):

	E(X)	σ	%	δ	%	sovrastima	ϒ	%
tPIL	26,6 I	19,5	15,2 V	13,6	13,5 V	43,4	2,012	39,0 I
t|DEF|	61,9 IV	30,5	23,8 I	25,1	24,9 I	21,5	1,042	20,2 II
tDEB	60,2 III	26,7	20,8 III	21,8	21,5 III	23,0	0,742	14,4 IV
tINF	72,1 V	24,7	19,3 IV	18,5	18,3 IV	33,5	0,900	17,4 III
tOCC	51,4 II	26,7	20,8 II	22,1	21,9 II	20,8	0,462	9,0 V

The position ranking, (E(X)), is innovative, the other two indicate agreement on the choices of any priority interventions presumably recommended/advised at the time. The relative position, E(X), evaluates Europe's margin for improvement in the year of reference of each quantity, suggesting the priorities of any interventions. These are, to be precise: GDP, Employment, Debt, Deficit, and Inflation, in that order.

As regards dispersion, σ and δ give information about the disparity between one country and another. The priorities identified derive from the sequence: Deficit, Employment, Debt, Inflation and GDP, in the order, which is in agreement with that which was presumably recommended at the time at European level. The over-estimations via the standard deviations, σ, on the information capacity of the five parameters do not seem at all negligible compared to the absolute deviations, δ. Note how the variability measure γ instead proposes a different sequence with respect to the intervention priorities, namely: GDP, Deficit, Inflation, Debt and Employment. This should make you think.