Neutrosophic Statistics is an extension of Interval Statistics, while Plithogenic Statistics is the most general form of statistics

In this paper, we prove that Neutrosophic Statistics is more general than Interval Statistics, since it may deal with all types of indeterminacies (with respect to the data, inferential procedures, probability distributions, graphical representations, etc.), it allows the reduction of indeterminacy, and it uses the neutrosophic probability that is more general than imprecise and classical probabilities and has more detailed corresponding probability density functions. While Interval Statistics only deals with indeterminacy that can be represented by intervals. And we respond to the arguments by Woodall et al. [1]. We show that not all indeterminacies (uncertainties) may be represented by intervals. Also, in some cases, we should better use hesitant sets (that have less indeterminacy) instead of intervals. We redirect the authors to the Plithogenic Probability and Plithogenic Statistics which are the most general forms of MultiVariate Probability and Multivariate Statistics respectively (including, of course, the Imprecise Probability and Interval Statistics as subclasses).


Introduction
First, we present the distinctions between Neutrosophic Statistics and Interval Statistics and give conclusive examples of neutrosophic algebra that provide more accuracy than interval algebra.Afterward, we respond to the criticisms presented by Woodall et al. Neutrosophic Statistics was first defined (book [1]) in 1998, developed (Book [3]) in 2014, related with Neutrosophic Probability (Book [9]), connected and extended to other fields (Books 2, 4-8), a PhD Thesis on Neutrosophic Statistics in 2019 (PhD Thesis [1]), that resulted in an explosion of articles about its applications (Articles & Seminars [1 -48]) to many fields such as medicine, biology, economics, administration, computer science, engineering, etc., regarding the decision making, rock joint roughness coefficient, repetitive sampling, indeterminate similarity coefficient, indeterminate sample/population size, individuals that only partially belong to a sample/population, indeterminate mean/variance/standard deviation, control charts, probability distributions of indeterminate functions, measurement errors, tests or hypotheses under uncertainty/indeterminacy, etc.

Neutrosophic Statistics vs. Interval Statistics
In this paper, we make a comparison between Neutrosophic Statistics (NS) and Interval Statistics (IS).We show that they are different and in many cases the NS is more general than IS.
NS is not reduced to only using neutrosophic numbers in statistical applications, as Woodall et al. assert, but it is much broader.NS deals with all types of indeterminacy, while IS deals only with indeterminacy that may be represented by intervals.
Below we present several advantages of applying NS over IS: -Neutrosophic Statistics is based on Set Analysis, while Interval Statistics is on Interval Analysis, therefore Interval Statistics is a particular case of Neutrosophic Statistics (that uses all types of sets, not only intervals).-The numerical neutrosophic numbers permit the reduction of indeterminacy through operations, while the intervals increase the indeterminacy (see examples below).-Not all uncertain (indeterminate) data can be represented by intervals as in IS, while NS deals with all types of indeterminacy.-NS deals with a sample or population whose size is not well-known.
-NS deals with a sample or population which contains individuals that only partially belong to the sample/population and others whose appurtenance is unknown.-NS deals with sample or population individuals whose degree of appurtenance to the sample or population may be outside of the interval [0, 1], as in neutrosophic overset (degree > 1), underset (degree < 0), and in general neutrosophic offset (both appurtenance degrees, > 1 and < 0, for various individuals) -Neutrosophic (or Indeterminate) Data is vague, unclear, incomplete, partially unknown, and conflicting indeterminate data.-NS also deals with refined neutrosophic data used in Big Data.
-Partially indeterminate curves.-Neutrosophic Random Variable, which may not be represented as an interval sequence.
-NS also uses Thick Functions (as intersections of curves, that may not be represented by intervals) as probability distributions.-Neutrosophic Probability Distribution (NPD) of an event (x) to occur is represented by three curves: NPD(x) = (T(x), I(x), F(x)), where T(x) represents the chance that the event x occurs, I(x) the indeterminate-chance that the event x occurs or not, and F(x) the chance that the event x does not occur.With T(x), I(x), and F(x) being classical or neutrosophic (unclear, approximate, thick) functions depending on each application, and 0 ( ) ( ) ( ) 3  for all x in the given neutrosophic probability space.
-NPD is better than the classical or imprecise probability distributions since it is a MultiVariate Probability Distribution that presents more details about the event.-Diagrams, histograms, pictographs, line/bar/cylinder graphs, and plots with neutrosophic data (not represented by intervals).-Not well-known (or completely unknown): the mean, variance, standard deviation, probability distribution function, and another statistic -For example, it is no need to increase the uncertainty by extending the set of possible values, for example, {0.
We deal with indeterminacy with respect to the probability or statistics space (either the surface, the die, or the coin), indeterminacy with respect to the observer that evaluates the event, and indeterminacy with respect to the event [4].
You cannot approximate the indeterminacy from these examples by using some interval, so you need neutrosophic probability and statistics that deal with all types of indeterminacies.
In conclusion: we cannot represent all types of indeterminacies by intervals.
For the sake of the truth, we'll respond below to the critics [1].

Woodall et al. [1] in their section on Neutrosophic Mathematics:
The basic rules for arithmetic given by Smarandache [42, pp. 31-33] do not match the rules given by Zhang et al. [37].
Smarandache [42] expressed neutrosophic numbers in the form a + bI, where a and b are real numbers, and I represent the indeterminacy interval such that I^2 = I and 0 • I = 0.

Response:
This is false since although the book [reference 2 in this paper] contains the literal neutrosophic numbers, they were never used in the applications of neutrosophic statistics.Instead, all the times there were used the numerical neutrosophic numbers.
The authors should learn that there are two types of neutrosophic numbers of the form a + bI, where a, and b are real (or complex) numbers, while "I" = indeterminacy.
i. Literal Neutrosophic Numbers, when "I" is just a letter, where I 2 = I (because indeterminacy × indeterminacy = indeterminacy) and 0•I = 0, are used in the neutrosophic algebraic structures, but not in no paper on applications of the neutrosophic statistics -upon the best of my knowledge.ii.Numerical Neutrosophic Numbers, where the indeterminacy "I" is a real subset, in order to approximate the imprecise data.This is more general than the interval, since "I" may be any subset.
For example, N = 3 + 2I, where "I" is in the discrete hesitant subset {0.3, 0.9, 6.4, 45.6} of only four elements, which is not part of interval analysis (statistics).On the interval statistics, you take the interval [0.3, 45.6] in order to include the above numbers, but this increases very much the uncertainty.Of course, there are particular cases when the "I" is an interval I = [I 1 , I 2 ],

Woodall et al. [1]:
Using the approach of Zhang et al. [37] and interval arithmetic, however, the interval for the average would be [3,5].We consider the interval arithmetic approach to lead to much more useful and realistic results.Therefore, the comparison with Zhang et al. interval arithmetic is irrelevant with respect to the neutrosophic statistics, since Zhang et al.only used their arithmetic on the neutrosophic sets.
Then using the approach of Smarandache [42], the average of these two neutrosophic numbers would be 4 + 0I, or simply the precise value 4.This result does not seem reasonable.

Response:
This just shows the advantage of the numerical neutrosophic numbers over the intervals, since they allow for the reduction of indeterminacy while using intervals the indeterminacy increases.
For example: N 1 = 4 + 2I, where I  [0, 1], shows that 2I is the indeterminate part of the number N 1 , similarly to N 2 = 4 -2I.If we add them, the indeterminacies of N 1 and N 2 cancel out, and the average is: , therefore the indeterminacy is between [0, 2].

Woodall et al. [1]:
We note that Smarandache [42] and others do not refer to interval statistical methods despite their very strong similarities with neutrosophic statistical methods.

Response:
In the beginning, in the book [2], page 5, there is no reference to the interval analysis/statistics, but it is to the set analysis/statistics that are more general than the interval analysis/statistics: Also, when computing the mean, variance, standard deviation, probability distributions, and other statistics concepts in classical and interval statistics it is automatically assumed that all individuals belong 100% to the respective sample or population, but in our world, one often meets individuals that only partially belong, partially do not belong, and partially their belongs is indeterminate.The neutrosophic statistics results are more accurate/real than the classical and interval statistics since the individuals who only partially belong do not have to be considered at the same level as those that fully belong.
The Neutrosophic Probability Distributions may be represented by three curves: one representing the chance of the event to occur, the other the chance of the event not to occur, and a third one the indeterminate chance of the event to occur or not."They provide more details than classical and interval statistics.
"Neutrosophic Statistics is the analysis of events described by the Neutrosophic Probability.
Neutrosophic Probability is a generalization of the classical probability and imprecise probability in which the chance that an event A occurs is t% true -where t varies in the subset T, i% indeterminate -where i varies in the subset I, and f% false -where f varies in the subset F. In classical probability, the sum of all space probabilities is equal to 1, while in Neutrosophic Probability it is equal to 3.
In Imprecise Probability: the probability of an event is a subset T in [0 The examples involving imprecise sample sizes given in Smarandache [42] all involve attribute data without carefully expressed operational definitions.It seems impossible to have a sample of variables data without knowing the sample size.(p.4)

Response:
We disagree.There are many frequent examples of populations and samples from our everyday life: such as a school of fish in a river, a flock of migratory birds, trees in a forest, plants on a given field, a herd of cattle, etc.More examples are below:

Indeterminate Sample Size
"A statistician wants to analyze the reaction of the spectators to a handball match, where team A plays against team B. Suppose that about 4,000 tickets have been sold.Spectators who attend the match form a sample, whose size cannot be exactly determined, because there are also spectators who entered without tickets (as guests, or

Comment by Woodall et al. [127]:
There is no reason to treat the sample sizes as indeterminate. Answer: A set of individuals may be considered a population with respect to a reference, but a sample with respect to the larger reference.
A simple example is when a population's size is indeterminate, but that population becomes a sample with respect to a super-population.
So, there are many cases when the sample size may not be well known.
Let's consider the population P of trees, whose size is indeterminate (between 100-120 trees), in a given park in a city.But, with respect to the trees in all 10 parks of the city, the population P is a sample ( of indeterminate size: {100, 101, ..., 120} ).
Notice that the sample's size is not an interval, but a discrete finite set.
Therefore, most times in the real world it is not possible to exactly estimate a sample or population size.

Woodall et al.
By the way, we spent several years studying fuzzy logic methods, finding no advantages over the use of probability and statistics.

Answer:
You have used or tried to use the fuzzy logic in statistics, I understand.
But the main distinction between fuzzy and neutrosophic logic is that neutrosophic logic has introduced indeterminacy as an independent component.

Woodall et al.:
The repetitive sampling approach provides for the possibility of more than n observations to be collected at any sampling time.

Answer:
This one better falls under the Plithogenic Probability and Statistics that consider Multivariate Analysis of events and their statistics.And consequently: the variance, standard deviation, probability distribution function, and other statistics depending on them will be different as well.
But, the neutrosophic mean is more accurate since it reflects the real (not idealistic) mean because it takes into account the degree of membership of each element with respect to the set."[5] And consequently, the other statistics depending on them are more accurate.

2.10
The Thick Function (Distribution), from the neutrosophic statistics, is defined as: The thick curve as the graph of a thick function [2] was introduced in 2014, and it is different from the interval functions, because we may have a probability distribution in between two curves, For example, let which is a thick function, i.e. the zone between two below curves.I have one question about neutrosophic arithmetic.Suppose one has, as in your example, 4+2I and 4-2I.You give the sum as the constant 8. Suppose one writes the numbers equivalently as 4+2I and 2+2I, then the sum is 6+4I.why should the sum depend on how the numbers are expressed?

Answer:
The numerical neutrosophic numbers (N) are chosen by the researcher upon the parts which are considered determinate (a) and indeterminate (bI), so N = a+bI.

d. Division
Interval Statistics (IS) The Plithogenic Probability of an event to occur is composed of the chances that the event occurs with respect to all random variables (parameters) that determine it.The Plithogenic Probability, based on Plithogenic Variate Analysis, is a multi-dimensional probability ("plitho" means "many", synonym with "multi").We may say that it is a probability of sub-probabilities, where each subprobability describes the behavior of one variable.We assume that the event we study is produced by one or more variables.Each variable is represented by a Probability Distribution (Density) Function (PDF).
Plithogenic Statistics (PS) encompasses the analysis and observations of the events studied by the Plithogenic Probability.Plithogenic Statistics is a generalization of classical Multivariate Statistics, and it is a simultaneous analysis of many outcome neutrosophic/indeterminate variables, and it as well as a multi-indeterminate statistic.

Conclusion
In this paper, we made a comparison between Neutrosophic Statistics (NS) and Interval Statistics (IS).We showed that they are different and in many cases, the NS is more general than IS.
NS is not reduced to only using neutrosophic numbers in statistical applications, as Woodall et al. assert, but it is much broader.NS deals with all types of indeterminacy, while IS deals only with indeterminacy that may be represented by intervals.
And we responded to the arguments by Woodall et al. [1].

154 2 . 9
Mean of a Sample with partially belonging individuals: Let S = {a, b, c, d} be a sample set of four elements, such that a = 2, b = 8, c = 5, and d = 11.In classical statistics it is assumed that all elements belong 100% to the sample, therefore S = {a(1), b(1), c(1), d(1the real world, not all elements may totally (100%) belong to the sample, for example, let's assume the neutrosophic sample be: NS = {a(1.1),b(0.4), c(0.6), d(0.3)}, which means that: the element a belongs to 110% (someone who works overtime, for example, as in the neutrosophic overset (see [B4]), b belongs only 40% to the sample, c belongs to 60%, and d belongs to 30%.classical mean and the neutrosophic mean are different, CM = 6.5  4.875 = NM.
Fuzzy Statistics -Plithogenic Spherical Fuzzy Statistics -and in general: Plithogenic (fuzzy-extension) Statistics -and Plithogenic Hybrid Statistics.Plithogenic Refined Statistics are, similarly, the most general form of statistics that studies the analysis and observations of the events described by the Plithogenic Refined Probability.See more development, and extension of IntervalStatistics and Neutrosophic Statistics to Plithogenic Probability & Plithogenic Statistics that are generalizations of MultiVariate Probability & Statistics: [6].
Woodall et al. made confusion since Zhang et al. [reference 3, in this paper] paper deals with the Interval Neutrosophic Set (not within the frame of Neutrosophic Statistics), where an element Doi : https://doi.org/10.54216/IJNS.190111Received: April 03, 2022 Accepted: August 08, 2022 151 Response: Later on, more citations and comparisons have been presented between neutrosophic statistics vs. classical and interval statistics, watch this: http://fs.unm.edu/NS/NeutrosophicStatistics.htm"The Neutrosophic Statistics is also a generalization of Interval Statistics, because, among others, while Interval Statistics is based on Interval Analysis, Neutrosophic Statistics is based on Set Analysis (meaning all kinds of sets, not only intervals, for example, finite discrete sets).
"In most of the classical statistics equations and formulas, one simply replaces several numbers with sets.And consequently, instead of operations with numbers, one uses operations with sets.One normally replaces the parameters that are indeterminate (imprecise, unsure, and even completely unknown)."Doi : https://doi.org/10.54216/IJNS.190111Received: April 03, 2022 Accepted: August 08, 2022 152

5 Woodall et al. [1]:
, 1], not a number p in [0, 1], what's left is supposed to be the opposite, subset F (also from the unit interval [0, 1]); there is no indeterminate subset I in imprecise probability [see B9].The function that models the Neutrosophic Probability of a random variable x is called Neutrosophic distribution: NP(x) = ( T(x), I(x), F(x) ), where T(x) represents the probability that value x occurs, F(x) represents the probability that value x does not occur, and I(x) represents the indeterminate / unknown probability of value x [see B3]."Therefore, a more detailed characterization of a neutrosophic random variable is not done in classical and interval statistics.See this book: F. Smarandache, Introduction to Neutrosophic Measure, Neutrosophic Integral, and Neutrosophic Probability, Sitech Publishing House, Craiova, 2013, http://fs.unm.edu/NeutrosophicMeasureIntegralProbability.pdf 2.
while others who had bought tickets could not come for various reasons.Therefore, the sample size could be estimated, for example, between for example between 3,900 and 4,200.""To estimate how many people watched the game on TV is even vaguer.Electronically one finds out that about 3 million people have watched it.But this is ambiguous as well since many people could have been watching on the same TV set, while some TVs would have been left on without anyone watching because the owners would have been busy with other things.The sample size was estimated, for example, between 2.

14 Refined Neutrosophic Statistics used in the Big Data In
this Big Data world, we are facing this kind of situation with more uncertainties resulting from multiple variables, leading to Refined Neutrosophy.Thus, we may use the Refined Neutrosophic Statistics, i.e. when the indeterminacy "I" is split into many types of uncertainties I 1 , I 2 , ..., I s , where s ≥ 2, as many as needed into the application.T 0 means that T is discarded, and similarly for I 0 and F 0 .This leaves room for defining the Refined Fuzzy Set, under the form T 1 , T 2 , ..., T p , where p is an integer ≥ 2, and I 0 and F 0 are discarded.And for Refined Intuitionistic Fuzzy Set, under the form T 1 , T 2 , ..., T p , and F 1 , F 2 , ..., F s , for integers p, s ≥ 1, and at least one of p or s is ≥ 2 (in order to assure the refinement of a least one component T or F).
Doi : https://doi.org/10.54216/IJNS.190111Received:April03,2022Accepted: August 08, 2022 Refined Neutrosophic Statistics followed the steps of the Refined Neutrosophic Set(Smarandache,   2013).Therefore, an element from Big Data that belongs to a refined neutrosophic set, xM  , may have refined neutrosophic coordinates, for example, x(T, I 1 , I 2 , I 3 , F) if there are only 3 types of uncertainties.We may have as many types of uncertainties as needed in the problem.wherep,r, s are integers ≥ 0, and at least one of p, r, s is ≥ 2 (to ensure the existence of refinement of at least one of the three neutrosophic components T, I, and F).When p, r, or s is equal to 0, that component is discarded.For example,

15 Plithogenic Probability & Plithogenic Statistics are generalizations of MultiVariate Probability & Statistics The
Therefore the Plithogenic Variate Analysis studies a neutrosophic/indeterminate system as a whole, characterized by many neutrosophic/indeterminate variables (i.e.neutrosophic/indeterminate subsystems), and many neutrosophic/indeterminate relationships.Hence many neutrosophic measurements and observations are needed.