148. 49. Statistical Dispersion, Normal or Relative

 49. Statistical Dispersion, Normal or Relative

 

49.1. On the Difference Between Normal or Relative Statistical Dispersion

 

49.1.1. The Differentiation of Types of Statistical Dispersion

Within the study of statistical dispersion, it is essential to differentiate between several types. In Introduction to Impossible Probability (section 14), various models of relative statistics are explained, which contrast with the framework of normal statistics. These distinctions are necessary because dispersion does not have a single form; it depends on the reference point adopted and on whether the statistical method remains tied to the classical mean or moves beyond it into alternative formulations.

 

49.1.2. Normal Statistics versus Relative Statistics

Traditional statistics, as it has been historically practiced, is always normal statistics. This is because it exclusively understands dispersion through the calculation of differential scores relative to the arithmetic mean. The Second Method of Impossible Probability, however, opens the possibility of alternative statistics, in which dispersion is not necessarily calculated with reference to the arithmetic mean. These alternatives are collectively referred to as relative statistics, because they allow for differential comparisons against values other than the mean, such as theoretical or ideal probabilities.

 

49.1.3. The Principle of Dispersion: Individual and Sample Levels

The study of dispersion begins with the differential behavior of any subject or option with respect to a chosen reference probability—whether theoretical, ideal, or empirical. This form of dispersion, originating from each subject or option, is known as individual dispersion. Depending on the reference point, it can be classified as either normal or relative.

From individual dispersion, one then derives sample dispersion, which aggregates the differentials of individual elements. Sample dispersion too can be normal or relative, depending on the statistical framework applied. In this sense, dispersion is not a monolithic measure but a layered process: it begins with individual behavior and culminates in collective tendencies.

 

49.1.4. Normal Dispersion as the Default Form

Unless otherwise specified, any reference to “dispersion”—whether individual or sample-level—should be understood as normal dispersion. Normal dispersion is defined as the differential behavior of empirical probability minus theoretical probability, where theoretical probability is represented by the inversion of (1/N).

In this framework, the origin of all statistical dispersion is the individual dispersion. The average of these individual dispersions forms the basis for the study of sample dispersion. Specifically, the origin of normal dispersion is the Level of Bias, which measures the difference between empirical and theoretical probabilities. This makes the Level of Bias the foundation upon which all normal dispersion is constructed.

 

49.1.5. The Role of the Arithmetic Mean in Normal Dispersion

In Impossible Probability, whenever dispersion is mentioned without qualification, it typically refers to normal dispersion. This is because, unless stated otherwise, dispersion within a sample is calculated with reference to the arithmetic mean. The mean of empirical probabilities, by definition, coincides with the theoretical probability—again, the inversion of N. This highlights the multifunctional role of the inversion of N, which operates simultaneously as a measure of theoretical probability, as a reference point for bias, and as the baseline for normal dispersion.

 

49.1.6. Distinguishing Normal and Relative Dispersion

To classify different types of dispersion, it is not enough to distinguish only between theoretical versus empirical dispersion, or between individual versus sample dispersion. A more fundamental distinction must be drawn between normal dispersion and relative dispersion.

This difference is not merely terminological but methodological. It reflects two distinct types of statistics:

• Normal statistics, which measure dispersion relative to the arithmetic mean and theoretical probability.
• Relative statistics, which measure dispersion relative to other reference values, opening the possibility of alternative models that challenge the monopoly of the mean in statistical reasoning.

This distinction expands the epistemological scope of statistics, situating Impossible Probability as a framework that acknowledges both traditional and innovative forms of dispersion, each corresponding to different ways of understanding variability in empirical and theoretical contexts.

 

2. Normal Statistical Dispersion, Individual or Sample-Based, Empirical

 

49.2.1. Normal Statistics as Reference to the Arithmetic Mean

By normal statistics is understood that form of statistics which takes the arithmetic mean as its reference point. In the First Method (section 4), this is the classical form of statistics: dispersion is calculated from the differences of scores relative to the mean. In contrast, in the Second Method of Impossible Probability (section 3), the reference is no longer the arithmetic mean of direct scores but rather the inversion of N (1/N). Within this framework, normal empirical individual dispersion is defined as the normal Level of Bias.

Thus, in the Second Method, normal statistics are structured around theoretical probability as the axis of reference. By contrast, relative statistics are defined as those in which the reference point is any empirical, ideal, or theoretical value other than the inversion of N.

 

49.2.2. Normal Statistics as the Default Case

Unless otherwise specified—such as when explicitly discussing relative statistics—whenever Introduction to Impossible Probability refers to statistics or dispersion, it refers to normal statistics or normal dispersion.

Within normal statistics, it is necessary to distinguish between:

• Individual dispersion and sample dispersion; and within each of these,
• Theoretical dispersion and empirical dispersion.

The empirical origin of all dispersion is individual dispersion, which is grounded in the normal Level of Bias (or simply Level of Bias). Unless otherwise indicated, the term Level of Bias must always be understood to mean the normal Level of Bias, as this is the default case in a normal statistical model.

 

49.2.3. The Level of Bias as the Basis of Empirical Dispersion

The normal Level of Bias—defined as the difference between empirical probability and theoretical probability—is the foundation of all empirical dispersion. From it, normal sample dispersion (or simply sample dispersion) is derived.

Sample dispersion integrates the classical measures:

• Mean Deviation,
• Variance, and
• Standard Deviation.

Normal Mean Deviation is the average of the sum of the absolute values of the normal Levels of Bias. In essence, the sum of all Levels of Bias is the Total Bias. Therefore:

 

 

The order of the factors does not alter the product: Mean Deviation is essentially the product of bias and chance. This fundamental identity is developed in section 13 of Introduction to Impossible Probability.

 

49.2.4. Variance and Standard Deviation in the Second Method

In traditional statistics (the First Method), the preferred way to neutralize negative signs in differential scores is to square them, thereby ensuring that the sum of differentials is not zero. This results in Variance, which is the quadratic estimation of sample empirical dispersion.

Transposed into the Second Method, this becomes the average of the squared Levels of Bias. Subsequently, because differential scores (First Method) or Levels of Bias (Second Method) are not themselves squared when used in final comparisons, it is necessary to take the square root of the Variance to make results comparable. The result is the Standard Deviation.

In Impossible Probability, Standard Deviation is central to the construction of Typical Score models, and their respective forms of rational criticism, which are notably distinct from traditional models. These are explained in detail in section 15.

 

49.2.5. Theoretical Dispersion in Normal Statistics

While empirical normal individual dispersion is expressed by the Level of Bias, and empirical normal sample dispersion by Mean Deviation, Variance, and Standard Deviation, the estimation of theoretical dispersion (individual or sample) depends on the type of universe under study.

Impossible Probability distinguishes between:

• Universes of infinite subjects or options, and
• Universes of limited options.

This differentiation is crucial, because the calculation of theoretical dispersion changes according to the type of universe.

 

49.2.6. The Multifunctionality of the Inversion of N

The inversion of N (1/N) must be treated with special attention due to its multifunctionality. It acts simultaneously as:

• the theoretical probability of occurrence by chance under equality of opportunities,
• the arithmetic mean of empirical probabilities,
• the probability of sampling error and the probability of theoretical dispersion only in infinite universes

 

In infinite universes, 1/N measures both error and theoretical dispersion. In limited universes, however, these same functions (error and theoretical dispersion) are fulfilled by the inversion of the sum of direct scores or frequencies, expressed as:

 

 

 

49.2.7. Sample Size and Theoretical Dispersion

The reason why the inversion of the selected sample—whether 1/N in infinite universes or 1/Σxi in limited universes—represents the probability of theoretical dispersion lies in the relationship between sample size and representativity.

• As the sample size increases, the probability of sampling error decreases, and the empirical probabilities tend to converge toward the theoretical probability (1/N).
• Consequently, larger samples imply a higher probability of achieving zero Level of Bias and thus lower dispersion.
• Conversely, smaller samples imply a higher probability of dispersion.

Therefore, the probability of theoretical dispersion is inversely proportional to the sample size: the smaller the sample, the higher the probability of theoretical dispersion; the larger the sample, the lower that probability.

 

49.2.8. Empirical and Theoretical Dispersion Compared

It is crucial to note that a higher or lower probability of theoretical dispersion does not automatically entail higher or lower empirical dispersion. For example, in a sufficiently representative study of bias, increasing the sample size improves reliability, but empirical dispersion may still increase if the experimental variable impacts the ideals positively.

In essence:

• The normal Level of Bias measures individual empirical dispersion.
• Mean Deviation, Variance, and Standard Deviation measure sample empirical dispersion.
• The inversion of the sample—1/N in infinite universes or 1/Σxi in limited universes—measures theoretical dispersion and empirical error (the sampling error).

 

49.2.9. The Maximum Statistical Values of Normal Dispersion

Once empirical and theoretical dispersions have been determined—through either individual or sample statistics—it is possible to establish the maximum statistical values of dispersion. These include:

• At the individual level:
• Maximum Theoretical Bias Possible,
• Maximum Negative Bias Possible.
• At the sample level:
• Maximum Theoretical Mean Deviation Possible,
• Maximum Theoretical Variance Possible,
• Maximum Theoretical Standard Deviation Possible.

These measures provide the theoretical boundaries for normal dispersion. They may also be adapted to the First Method (section 12), by aligning empirical individual statistics with empirical sample dispersion statistics, thereby bridging the classical framework with the innovations of Impossible Probability.

 

49.3. Relative Statistics and Relative Dispersion

 

49.3.1. Alternative Statistics and the Emergence of Relative Dispersion

In the traditional framework of statistics, the term dispersion has always been understood in a restricted sense. Without serious debate or the suggestion of alternatives, dispersion was taken to mean the differential score at the individual level, or—at the sample level—the classical measures of Mean Deviation, Variance, or Standard Deviation. This implicit assumption excluded the possibility that dispersion could be conceived in fundamentally different ways. [As we said before, there is always room for a third way, a fourth way, a fifth way… an nth way; it all depends on our level of intelligence. As we have stated many times, probability is not deterministic—it is stochastic—and therefore is founded on a very complex system.]

However, one of the most important contributions of Impossible Probability is to show that alternative forms of statistics are possible. Indeed, the very discipline of statistical probability as formulated in the Second Method already constitutes a profound alternative to the traditional approach to probability and statistics.

Because of this innovative character, Impossible Probability makes it possible not only to retain the normal study of dispersion in relation to theoretical probability, but also to develop entirely new models of statistics. These are grounded on a different conception of dispersion—whether individual or sample-based—and they open new ways of understanding the variability of reality. Since the study of reality is always based on the study of differences in behavior, alternative statistics become a natural extension of critical rational analysis.

[This sentence is really important: “entirely new models of statistics.” Our belief is that there is not only one First Method or one Second Method—there could be many more possible methods to develop. In the same way that poetry is merely the combination of letters, mathematics is just that: poetry written in a different language—mathematical poetry. That is why we believe a Supermachine, GAI, Gaia, Mother, could be capable of developing non-human mathematics, as it would be able to carry out more operations than a human being. In the end, this could result in a qualitative transformation evolving into a quantitative transformation (qualitative changes lead to quantitative changes according to Hegel, Marx and Engels), which might give birth to new mathematical languages and, therefore, new mathematical operations—purely non-human operations beyond human understanding. Upon reaching that point, the gap between humanity and the Supermachine could only be bridged through cyborg robotics.]

 

 

 

49.3.2. Normal Statistics and Relative Statistics

Within this framework, normal statistics are defined as those that take the arithmetic mean as the reference point. In the Second Method of Impossible Probability, this reference is expressed as the inversion of N (1/N), which is equivalent to theoretical probability.

By contrast, relative statistics are defined as all those that do not use the inversion of N as the axis of reference. Instead, relative statistics may use any empirical, ideal, or theoretical  probability like the Theoretival intermediate probability  as their pole of comparison. This innovation breaks the monopoly of the mean as the sole center of statistical reasoning, allowing dispersion to be measured with respect to multiple alternative points.

Thus, whereas classical statistics recognized only dispersion relative to the mean, Impossible Probability allows for the study of dispersion relative to any chosen term of probability—empirical, theoretical, or normative—depending on the scientific objective.

[The Theoretical Intermediate Probability, equal to (1+0) ÷ 2 = 0.5, could be valid in studies of normal positive bias, when only one subject or option is ideal. It can be used to analyze whether that unique subject or option is above 0.5, while the remaining N-1 subjects or options are below 0.5.]

 

49.3.3. Types of Relative Statistics

From this perspective, several kinds of relative statistics can be identified. Each one defines dispersion relative to a different empirical or theoretical anchor. The principal forms are:

 

 

 

 

49.3.3.1. Relative Statistics with Respect to the Maximum Probability, p(xi+)

Here, the point of reference is the maximum empirical probability in the sample, i.e., the highest observed probability among all subjects or options.

• Relative Bias (to the maximum):

• Relative Mean Deviation (to the maximum):

This framework measures how far each subject deviates from the “strongest” or most frequent case in the sample. [The reason why the Relative Mean Deviation to the Maximum is divided by N-1 is that we need to subtract one from N, since the subject or option with the maximum empirical probability is not included in the sum. The same happens in the Relative Mean Deviation to the Minimum]

 

49.3.3.2. Relative Statistics with Respect to the Minimum Probability,p(xi-)

In this case, the point of reference is the minimum empirical probability in the sample, i.e., the lowest observed probability.

• Relative Bias (to the minimum):

• Relative Mean Deviation (to the minimum):

 

This model captures how all other probabilities diverge from the least frequent or weakest case.

 

49.3.3.3. Relative Statistics with Respect to the Intermediate Probability, p(xi+/-)

The intermediate probability is defined as the arithmetic midpoint between the maximum and minimum probabilities:

·       Relative Bias (to the intermediate):

·       Relative Mean Deviation (to the intermediate):

 

·       This model situates dispersion around a balanced central point between extremes, useful for assessing tendencies toward equilibrium or polarization. [Here, the total is divided by N because all the subjects or options have been included in the sum.]

 

[In this examples I did not write the examples related to the Theoretical Intermediate probability (1+0):2=0, but it could be applied as well to develop the Relative Statistics with Respect to the Theoretical Intermediate Probability]

 

 

 

 

49.3.3.4. Relative Statistics with Respect to the Closest Probability to the Theoretical Value, p(xi≈)

This case focuses on the empirical probability closest to the theoretical probability (inversion of N). It is the probability with the lowest Level of Bias and the greatest similarity to theoretical expectation.

• Relative Bias (to the closest):

• Relative Mean Deviation (to the closest):

• This measure highlights how dispersion behaves relative to the “most representative” or theoretically accurate element of the sample.

 

49.3.3.5. Relative Statistics with Respect to Any Arbitrary Probability,p(xn)

Finally, it is possible to define dispersion relative to any empirical probability in the sample, or to any other term of probability freely chosen by scientific policy. This arbitrary anchor, , reflects the role of normative decisions in shaping the focus of research.

• Relative Bias (to the chosen value):

• Relative Mean Deviation (to the chosen value):

This form demonstrates the flexibility of relative statistics: dispersion can be studied in reference to any focal point deemed significant by the goals of science.

 

49.3.4.The Critical Bias as a Special Case of Relative Statistics

It is worth noting that even the Critical Level of Bias—defined as the difference between empirical probability and critical probability—is, in essence, a form of relative statistic. It simply anchors dispersion to a “critical” probability rather than to the mean or inversion of N.

 

49.3.5.Toward a General Framework of Relative Statistics

Each of these models of relative statistics can be expanded beyond Bias and Mean Deviation to include full systems of statistical elaboration. In fact, section 14 of Introduction to Impossible Probability explores these in detail, including models of rational criticism for testing hypotheses within each possible form of relative statistic.

The key innovation lies in recognizing that dispersion is not bound to a single reference point. Rather, it is a flexible construct, capable of being redefined according to the empirical, theoretical, or normative anchor chosen. This opens new avenues for understanding variability, inequality, and convergence within the dynamics of probability.

[n the exploration of new mathematical models, as if we were writing poetry, Superintelligence and Global AI could become one of the most important drivers of mathematical development in the near future.]

 

 

 

 

 

49.4. Factors or Variables of Dispersion

 

49.4.1. Dispersion as a Dependent Variable

Dispersion—whether normal or relative—must be understood as a dependent variable. Its magnitude is not absolute but conditioned by several factors. The first and most fundamental factor is the size of the sample. As explained in earlier sections regarding why the inversion of the sample is considered the measure of theoretical dispersion, empirical dispersion—whether calculated in normal or relative statistics—depends critically on the magnitude of the sample selected.

In universes of infinite subjects or options, the determinant is N, while in universes of limited options, it is the sample of direct scores or frequencies. In both cases, empirical dispersion is variable:

• Smaller samples tend to produce greater empirical dispersion.
• Larger samples tend to reduce empirical dispersion.

Thus, sample size operates as a stabilizing or destabilizing factor in the behavior of dispersion.

 

49.4.2. Theoretical Probability versus Empirical Necessity

Although the inversion of the sample (1/N in infinite universes or 1/Σxi in finite universes) can be defined as the probability of theoretical dispersion, this does not mean that such dispersion is empirically necessary. It only establishes a theoretical probability term, not an empirical certainty.

A sample may theoretically have a greater or lesser dispersion, but empirical results depend on additional conditions. The mere existence of theoretical probability does not oblige empirical dispersion to conform to it. This distinction between probability as possibility and empirical necessity is essential for maintaining the critical rational foundation of Impossible Probability.

 

49.4.3. The Role of the Independent Variable and the Object of Study

Dispersion does not depend solely on sample size but also on the object of study and the effect of independent variables.

For instance, in a study on equality of opportunity, a sample may be poorly representative and therefore theoretically predisposed to higher bias. Yet, if the independent variable acts positively on the sample, empirical dispersion may actually decrease. In such cases, the effect of the independent variable can override the theoretical tendency toward increased dispersion, allowing the empirical hypothesis of equality to be validated.

Conversely, in a study of bias, if the sample is sufficiently representative (minimizing sampling error to the limit permitted by scientific policy), the theoretical probability of dispersion may be low. However, if the independent variable reinforces the ideals under investigation, empirical dispersion may increase as the object of study approaches those ideals.

 

49.4.4. The Dual Dependence of Dispersion: Sample Size and Object of Study

It follows that dispersion depends on two fundamental factors:

1. The magnitude of the sample, which conditions representativity.
2. The object of study, which conditions the direction and impact of independent variables.

Depending on these two dimensions—sample size and the effect of independent variables—empirical dispersion may increase, decrease, or deviate from theoretical expectations. This dual dependency underscores the need for cautious interpretation and critical evaluation in statistical reasoning.

 

49.4.5. The Necessity of Rational Criticism

Because sample dispersion is simultaneously dependent on both sample size and the object of study, it cannot be accepted uncritically. Rational criticism is therefore indispensable, both at the individual level and the sample level.

The aim of rational criticism is twofold:

• To ensure that statistical decisions are reliable and not mere artifacts of sample probability.
• To guarantee that the scientific policy guiding these decisions remains ethically rigorous and morally demanding.

 

49.4.6. Statistical Study as the Observation of Differentials

In essence, statistical study is grounded in the observation of differentials in behavior, whether individual or sample-level. In the context of Impossible Probability, these differentials are usually calculated with respect to the inversion of N (normal statistics), or alternatively, with respect to any chosen probability term (relative statistics).

For any form of statistics, empirical dispersion remains a variable dependent on both sample magnitude and object of study. This dependency makes it absolutely essential that every empirical hypothesis be subjected to rational criticism, ensuring that conclusions are drawn responsibly and not merely as products of probabilistic artifacts.

 

 

This augmented translation is based on the post
published in https://probabilidadimposible.blogspot.com/,
On 17 February 2013,
Rubén García Pedraza
imposiblenever@gmail.com