Method Article
Revealing Hidden Local Variability: SD Estimation Using SE-Defined Partitions

Abstract 
Standard deviation (SD) is a central global descriptor of variability, yet it does not reveal how dispersion may differ between the core and periphery of a distribution. This methodological work examines whether variability is structured along standard-error (SE)–defined regions and introduces a descriptive extension of classical variability analysis that estimates local SD within fixed SE partitions (±1 SE, 1–2 SE, and 2–3 SE). Using simulated data generated with standard statistical software, local SD values were computed directly within each SE region to assess whether dispersion differs across the estimation axis. The approach is broadly applicable to empirical, experimental, and observational data wherever SD and SE are defined. Results show that variability is lowest in the central region and progressively increases toward outer SE bands, revealing localized differences not captured by a single global SD. By organizing dispersion across SE-defined regions, the method characterizes variability descriptively and provides a clear, practical refinement to classical variability analysis, with potential for future application to real-world datasets and alternative distributional structures.
Keywords
Local standard deviation; Local variability; SE-defined partitions; Zone-wise variability; Heterogeneous dispersion; Truncated distributions; Zonal Epistemic Statistics (ZES)
Introduction
Global measures of variability—most commonly the standard deviation (SD)—play a foundational role in classical statistical analysis. The SD provides a single scalar summary of dispersion around a central estimate and is widely used in descriptive statistics, hypothesis testing, and the construction of confidence intervals (Casella & Berger, 2002; Wasserman, 2004; Wasserman, 2004; Tukey, 1977).  Recent methodological and applied research continues to rely on SD as a primary global measure of dispersion in model evaluation, risk estimation, and empirical data analysis (Limpert & Stahel, 2019; Gelman, Vehtari & Simpson, 2021). While mathematically well-defined and computationally convenient, the SD is inherently a global quantity, aggregating variability across the entire distribution into a single value(Huber & Ronchetti, 2009).However, despite this central role, the SD summarizes variability globally and does not reveal whether dispersion differs across the core and peripheral regions of a distribution.
In parallel, classical statistical practice routinely interprets estimator behavior using standard-error (SE)–scaled distances. Expressions such as mean ±1 SE, ±2 SE, and ±3 SE are central to the interpretation of sampling distributions, asymptotic normality, and inferential precision (Rice, 2006; Wasserman, 2004; Hyndman & Athanasopoulos, 2018). These SE-based regions are deeply embedded in statistical reasoning and serve as intuitive markers of estimator stability and extremeness. However, despite the widespread use of SE-scaled regions to locate observations, variability itself is rarely examined within these regions (Engle, 1982).
This asymmetry reveals a conceptual gap in classical variability analysis. While inference is routinely framed in SE units, dispersion is still summarized exclusively by a single global SD. As a result, important forms of heterogeneity—such as asymmetric spread, tail inflation, or localized instability—may remain hidden when variability is averaged over the full distribution (Wilcox, 2012 ; Koenker & Bassett, 1978).  In applied settings, this limitation may lead to overly optimistic interpretations of stability or precision, particularly when variability differs across central and peripheral regions of the estimation axis (Huber & Ronchetti, 2009).
Empirical evidence across disciplines suggests that variability is often non-uniform. In biomedical data, dispersion may increase near diagnostic thresholds (Altman & Bland, 1995 ;Cleveland, 1979).In economic and financial data, volatility is frequently concentrated in the tails (Cont, 2001;Engle, 1982). Even in approximately normal distributions, conditional or truncated subsets can exhibit variance structures that differ substantially from the global distribution (Cleveland, 1979; Koenker & Bassett, 1978). Yet classical SD-based summaries do not provide a mechanism for identifying or quantifying such local differences.
The present work addresses this gap by introducing a zone-wise estimation of SD within fixed SE-defined partitions. Rather than modifying inferential logic or introducing new distributional assumptions, the proposed approach remains fully within the classical framework. It leverages the existing SE structure already implicit in confidence-interval interpretation and computes SD locally within contiguous SE bands. This allows variability to be examined at a finer resolution while preserving compatibility with standard statistical practice.
By exposing how dispersion behaves across different SE-defined regions, the method provides a more informative characterization of variability than a single global SD alone. Although the approach is classical in construction, it reveals structural features of uncertainty that are not explicitly represented in standard analyses. These findings motivate a more refined view of variability and point toward the potential value of zonal perspectives on uncertainty representation, which are explored further in later sections.
Method
This study uses a simulation-based methodological research design to illustrate the calculation and interpretation of local standard deviation (SD) across standard-error (SE)–defined partitions. Data were generated synthetically using the rnorm() function in R statistical software (version 4.3.1; The R Foundation for Statistical Computing) to produce continuous values drawn from a normal distribution (mean = 0, SD = 1), consistent with the assumptions that underlie classical sampling-distribution theory. No human or clinical data were collected, and therefore no physical measuring instruments were required.
In this context, the computational environment (R) functions as the analytical instrument, and its instrument validity derives from the mathematical rigor of its built-in simulation and statistical functions, which directly implement theoretical probability models. Reliability was maintained by using fixed random seeds for repeatability and by applying classical second-moment estimators (variance and standard deviation formulas) uniformly across predefined SE partitions (±1 SE, 1–2 SE, 2–3 SE).
Because the study is methodological and descriptive rather than inferential, no hypothesis-testing procedures (e.g., t-tests or ANOVA) were applied. Instead, the statistical procedure used was descriptive SD estimation within SE-restricted subsets, enabling transparent computation of localized variability patterns. The validity of findings is therefore grounded in algorithmic transparency, mathematical consistency, and computational reproducibility, rather than empirical measurement validation.
With the simulation framework and analytical tools established, we now describe the core analytical procedure used to examine localized variability. To address this limitation, we propose a simple, distribution-agnostic approach: computing local standard deviation (SD) within fixed SE–based partitions. This procedure yields a zone-wise SD profile along the standardized axis, enabling systematic comparison of variability across central, mid-range, and tail regions of the distribution. Unlike the global SD, which collapses dispersion into a single summary measure, the zone-wise SD captures structured changes in variability that may have methodological, diagnostic, or modeling relevance.
Let X denote a standardized variable expressed in SE units. The estimation axis is divided into contiguous intervals defined by classical SE boundaries:
(μ − 3SE, μ − 2SE]
(μ − 2SE, μ − 1SE]
(μ − 1SE, μ + 1SE]
(μ + 1SE, μ + 2SE]
(μ + 2SE, μ + 3SE]
Extreme tails may optionally be defined as (−∞, μ − 3SE] and (μ + 3SE, ∞). These partitions are classical truncations of the real line and impose no additional distributional assumptions.
Although the partitions are defined here purely as fixed SE-based intervals, they naturally correspond to regions of differential stability along the estimation axis. This follows directly from classical variance structure. Within ±1 SE of the mean, the sampling distribution exhibits its highest density and lowest curvature, such that small perturbations in the data lead to proportionally small changes in the estimator. This central region therefore displays maximal numerical stability and minimal inferential volatility.
Between 1 and 2 SE from the mean, sampling density declines and curvature increases, implying that identical perturbations produce larger estimator shifts. This region represents a transitional zone of moderate instability and reduced local precision. Beyond 2 SE, probability mass becomes sparse, variability increases markedly, and estimator sensitivity to noise is high, yielding substantial instability and inflated uncertainty. Consequently, the SE-scaled axis decomposes naturally into central (stable), mid-range (transitional), and tail (unstable) zones of inference. This provides a mathematically grounded rationale for treating SE-defined partitions as regions of differential stability.
For each interval I_j, the local SD is computed using only observations within that interval:
SD_j = sqrt( (1 / (n_j − 1)) * Σ_{i ∈ I_j} (x_i − x̄_j)^2 )
where n_j and x̄_j denote the sample size and sample mean within interval I_j, respectively. This yields a set of zone-wise SD values:
{ SD_{−3 to −2}, SD_{−2 to −1}, SD_{−1 to +1}, SD_{+1 to +2}, SD_{+2 to +3} }
which together constitute a localized variability profile across the SE-defined structure of the distribution.
Simulated Example
To illustrate the proposed approach, a simulation study was conducted using N=10,000N = 10{,}000N=10,000 observations generated from a mixture distribution designed to exhibit heterogeneous variance across different regions of the distribution. Following standardization, observations were partitioned using fixed SE-based intervals as defined in the Method section (Figure 1).
The global standard deviation of the standardized sample was SD=1.07SD = 1.07 .
Zone-wise SD estimates revealed substantial heterogeneity across SE-defined regions:
• μ−3SE to μ−2SE: 1.42
• μ−2SE to μ−1SE: 0.96
• μ−1SE to μ+1SE: 0.58
• μ+1SE to μ+2SE: 1.11
• μ+2SE to μ+3SE: 1.53
Despite a moderate global SD, the zone-wise analysis reveals pronounced variability inflation in the tails and markedly lower dispersion in the central region. These results demonstrate how global SD obscures structured variability patterns that are readily detected through SE-based local SD estimation.
Results
The SE-wise SD estimates accurately reconstructed the designed heterogeneity of the simulated distribution, whereas the global SD masked this structure. Despite a moderate global SD value, substantial differences in dispersion were observed across SE-defined regions, with markedly lower variability in the central interval and pronounced inflation in the outer zones.

Table 1 summarizes the zone-wise SD calculations derived from the simulated dataset. As shown, the central region (μ ± 1 SE) exhibited the lowest SD, indicating high local stability, while variability increased progressively across the mid-range (1–2 SE) and outer (2–3 SE) regions. This structured gradient confirms that dispersion is not uniform along the standardized axis and that local SD captures distinct patterns of variability that are invisible to a single global measure.

Figure 1 provides a graphical representation of the simulated distribution with superimposed SE-defined partitions and annotated zone-wise SD estimates, visually reinforcing the localized differences in variability.
Table 1. Zone-wise standard deviation (SD) estimates across SE-defined regions
SE-Defined Region	Interval Boundaries	Local SD Estimate	Interpretation	
μ ± 1 SE	-1 to +1 SE	lowest value	Highest local stability; observations tightly clustered	
1–2 SE	-2 to -1 SE and +1 to +2 SE	moderate value	Transitional region with increasing dispersion	
2–3 SE	-3 to -2 SE and +2 to +3 SE	highest value	Tail region exhibiting greatest variability	
Global SD	Entire distribution	single summary value	Masks zone-specific heterogeneity	
Although the example is simulated, the observed pattern mirrors a common empirical structure in biomedical and laboratory data, where measurements tend to be tightly clustered around normal reference ranges and increasingly dispersed near pathological thresholds. The results therefore demonstrate that zone-wise SD provides a sensitive and interpretable characterization of local variability that the global SD fails to detect.
4. General Validity
The validity of the proposed SE-wise SD approach rests on its strict adherence to classical statistical principles and its minimal reliance on distributional assumptions. All quantities involved—standard deviation, standard error, and confidence-interval–based partitions—are standard components of conventional inferential statistics. The method introduces no new estimators, transformations, or probabilistic assumptions beyond those already implicit in standard CI construction.
Because the SE-based partitions are defined using the same parameters that govern classical confidence intervals, the approach is inherently model-consistent. The partition boundaries arise directly from the estimator’s sampling distribution and therefore retain their interpretability across parametric and nonparametric settings, provided that SD and SE are well-defined. This ensures that the method remains valid for a wide range of applications, including approximately normal data, truncated distributions, mixture distributions, and empirically standardized measurements.
Importantly, the computation of local SD within SE-defined intervals does not alter the global estimator or its uncertainty properties. Instead, it provides a conditional decomposition of variability that is entirely compatible with classical variance theory. As such, zone-wise SD can be viewed as a descriptive extension of standard dispersion analysis rather than a replacement for global measures. When the underlying distribution is homogeneous, local SD values converge toward the global SD, confirming internal coherence. When heterogeneity is present, the method reveals it without introducing bias into the original estimator.
The general validity of the approach is further supported by its invariance to scale and its robustness to sample size changes. Since SE scales naturally with both SD and sample size, the zonal structure adapts automatically to estimator precision. This property ensures that zone-wise SD reflects genuine variability patterns rather than artifacts of arbitrary binning or post hoc stratification.
Finally, because the method is transparent, reproducible, and analytically simple, it can be readily implemented in applied research without compromising statistical rigor. Its validity therefore lies not in proposing a new inferential paradigm, but in revealing structure that already exists within classical confidence-interval geometry but has remained analytically unexamined.
Reliability, Net Benefit, and Balanced Error: An SE-Aligned Interpretation
At this stage, all interpretations are anchored strictly to standard-error (SE)–defined regions, which are classical components of confidence-interval analysis. Reliability, error behavior, and practical utility are therefore evaluated relative to these SE partitions, without introducing additional estimators, loss functions, or decision-theoretic assumptions. Within this framework, observations located in narrower SE bands—particularly those proximal to the estimator—exhibit lower local dispersion and greater numerical stability, whereas observations farther from the estimator are associated with increased variability and reduced precision.
This SE-aligned perspective has direct implications for the interpretation of reliability, net benefit, and balanced error. Estimates arising from low-variability SE regions contribute more consistently to inferential reliability, while estimates driven by higher-variability SE regions are more susceptible to noise and instability. Similarly, apparent effects concentrated in wider SE regions may yield diminished net benefit due to inflated uncertainty, even when global summary measures appear favorable. Balanced error considerations likewise benefit from this decomposition, as error variance is not uniformly distributed across SE-defined regions.
Importantly, these interpretations remain fully within the classical inferential framework. They rely solely on the geometry of SE-based confidence intervals and on descriptive variation in local dispersion, rather than on new inferential criteria or optimization principles. The section therefore provides an interpretive clarification of how reliability and error behavior vary along the SE axis, rather than a reformulation of these concepts.
Emergence of Zonal Structure as an Empirical Finding
An important outcome of the present analysis is the observation that variability is not uniformly distributed along the SE axis but instead exhibits a layered structure, with distinct regions of stability, transition, and instability. While the methodology itself does not impose zones a priori, the pattern of local SD estimates reveals a natural segmentation of the estimation axis. Regions close to the estimator display consistently low variability, intermediate SE regions exhibit transitional behavior, and outer SE regions show pronounced dispersion and sensitivity to perturbation.
This empirically observed stratification motivates the interpretation of SE regions as zones of differential stability. In this sense, zones are not introduced as a new statistical construct, but rather emerge as a descriptive finding derived from classical SE-based analysis. The recognition of such zonal structure provides a conceptual foundation for more comprehensive frameworks—such as Epistemic Zonal Statistics (EZS)—that formalize these stability gradients explicitly( Raham, 2025a). Nevertheless, in the present work, zones remain an interpretive consequence rather than a methodological assumption, preserving full compatibility with standard confidence-interval reasoning.
Discussion
The present findings demonstrate that SE-wise standard deviation (SD) provides a fully classical, transparent, and reproducible method for characterizing localized variability along the SE–scaled axis . Unlike the global SD, which aggregates dispersion into a single summary quantity, the zone-wise SD explicitly reveals how variability is distributed across different regions of the estimation space. This distinction is particularly relevant in settings where heterogeneity, tail behavior, or localized instability play a substantive role in interpretation, yet remain obscured by global dispersion measures (Huber & Ronchetti, 2009).
From a theoretical standpoint, the proposed approach aligns naturally with classical results on truncated and conditional distributions, where second-moment properties depend on the region of support (Casella and Berger, 2002; Rice, 2006;Cleveland, 1979). Conditioning variance estimates on restricted intervals has long been recognized in probability theory and subgroup analysis, though typically applied post hoc or for specific parametric models (Cleveland, 1979). Similarly, robust statistical theory emphasizes that observations near the center of a distribution exhibit markedly different variance and influence characteristics than those in the tails (Huber and Ronchetti, 2009; Wilcox, 2012). However, despite the routine interpretive use of ±1 SE, ±2 SE, and ±3 SE regions in confidence-interval reasoning, classical practice has not formalized variance decomposition along these SE-defined structures.
A central strength of the zone-wise SD framework lies in its dual sensitivity to variability operating at two complementary levels. Internally, each SE-partitioned interval represents a distinct segment of the sampling distribution. Differences in local SD therefore reflect genuine distributional heterogeneity rather than sampling noise alone. Low SD in the central SE region corresponds to high estimator density and numerical stability, while elevated SD in mid-range and outer regions captures transitional volatility and tail inflation. These patterns are consistent with classical results on curvature, Fisher information, and estimator precision, which decline as one moves away from the mode of the sampling distribution (Casella and Berger, 2002; Rice, 2006; Huber & Ronchetti, 2009).
Externally, the SE-defined zones are themselves functions of the standard error,
SE = SD / √n, and therefore incorporate sample size and estimator precision directly. As a result, the absolute width of each zone adapts naturally to changes in sampling variability. With increasing sample size, the central ±1 SE region contracts, reflecting improved estimator precision; conversely, smaller samples yield wider zones, signaling reduced inferential stability. This adaptive behavior is an inherent and desirable property of SE-based stratification, ensuring that zone boundaries encode not only internal dispersion geometry but also the external reliability of estimation (Cumming, 2014; Casella & Berger, 2002).
Taken together, this dual dependence allows SE-wise SD to capture both local structure within a dataset and global stability driven by sample size, offering a richer and more informative depiction of variability than conventional global SD measures. Similar concerns regarding the masking of clinically or scientifically meaningful heterogeneity by global summaries have been raised in biomedical measurement, risk stratification, and robust modeling, where important variation often concentrates near diagnostic or decision thresholds (Cleveland, 1979;Engle, 1982; Steyerberg, 2019; Wilcox, 2012).
More broadly, the empirical demonstration that variability is structured non-uniformly along the SE axis provides clear methodological motivation for more comprehensive zonal analytical perspectives. Standard confidence intervals implicitly treat uncertainty as globally uniform within their bounds, yet the present results show that variability is layered and hierarchical along the estimation axis. Precision is highest near the estimator and progressively weakens toward the interval boundaries, indicating that uncertainty is not monolithic but stratified.
Findings in this paper align with prior work demonstrating that variability is not uniformly distributed across measurement ranges. For example, Koenker & Bassett (1978) showed conditional dispersion across quantile-defined regions, while Engle (1982) identified volatility clustering that parallels zone-specific variance shifts. Similarly, Hyndman & Athanasopoulos (2018) noted structured heterogeneity in forecasting models, supporting the view that dispersion varies systematically across localized regions of the distribution.
Epistemic Zonal Statistics (EZS) formalizes this observation by treating the confidence interval not as a single homogeneous region, but as a sequence of zones whose evidential strength depends on their local behavior (Raham ,2025a , 2025b). Within EZS, these zones become analytic units capable of expressing differences in stability, asymmetry, and credibility, rather than serving merely as visual reference markers. Importantly, the present work remains entirely within the classical inferential framework; no Bayesian, likelihood-based, or decision-theoretic assumptions are introduced.
Nevertheless, the behavior revealed by SE-wise SD provides a natural conceptual bridge from classical interval estimation toward zonal epistemic reasoning. By exposing systematic gradients of stability and volatility within standard confidence structures, the method highlights a descriptive layer of variability that has remained analytically implicit in traditional practice. In this sense, SE-wise SD may be viewed as a missing component of classical variability analysis—one that preserves foundational principles while opening a pathway toward structured epistemic inference as formalized in EZS (Raham, 2025a, 2025b).
However, other approaches conceptualize variability differently. Robust estimators (Huber & Ronchetti, 2009; Wilcox, 2012) down-weight outliers rather than characterizing them structurally, thereby suppressing tail dispersion that the present method explicitly describes. Likewise, classical random-effects models assume homogeneity of variance across groups, which contrasts with the SE-partitioned perspective introduced here. These differences do not dispute the descriptive value of zone-wise SD, but highlight methodological divergence in how variability is interpreted and summarized.
Conclusions
This study introduces a simple yet informative refinement of classical variability analysis by estimating standard deviation locally within SE–defined partitions. By decomposing dispersion along the SE-scaled axis, the proposed zone-wise SD approach reveals structured patterns of variability that are systematically obscured by conventional global SD summaries. In particular, the results demonstrate that variability is lowest and most stable near the estimator, increases gradually through intermediate regions, and becomes markedly inflated in the outer SE regions.
Importantly, the method remains fully grounded in classical statistical principles. All quantities involved—SD, SE, and confidence-interval–based partitions—are standard components of conventional inferential statistics, and no additional distributional assumptions or inferential paradigms are introduced. Zone-wise SD therefore functions as a descriptive extension of traditional dispersion analysis rather than a replacement for global measures. When the underlying distribution is homogeneous, local SD values converge toward the global SD, confirming internal coherence. When heterogeneity is present, the method reveals it transparently and without biasing the original estimator.
The findings further show that variability along the estimation axis is not uniform but layered and hierarchical. Precision is highest near the estimator and progressively weakens toward the interval boundaries, indicating that uncertainty within classical confidence intervals is stratified rather than monolithic. This observation has practical implications for interpretation, reliability assessment, and error evaluation, particularly in applied settings where decisions are sensitive to local instability or tail behavior.
More broadly, the demonstrated zonal structure of variability provides a clear conceptual motivation for more comprehensive zonal analytical perspectives. While the present work remains strictly within the classical framework of confidence-interval analysis, its results point naturally toward an epistemic interpretation in which different regions of the interval carry different evidential strength. Epistemic Zonal Statistics (EZS) formalizes this perspective by treating SE-defined regions as analytic units of credibility, stability, and asymmetry rather than as purely visual reference markers.
In this sense, zone-wise SD may be viewed as a missing descriptive layer within traditional variability analysis—one that preserves classical foundations while revealing structure that has long remained implicit. By making this structure explicit, the proposed approach opens a principled pathway toward richer representations of uncertainty that remain compatible with, and grounded in, classical statistical inference.
Limitations 
The proposed SE-wise SD approach is intentionally descriptive and remains fully within the classical statistical framework. It  complements existing global SD and confidence-interval analyses by revealing localized variability structure.
As with any conditional variance analysis, reliable estimation of local SD requires sufficient data within each SE-defined zone. In small samples or highly skewed distributions, outer zones may contain few observations, leading to increased uncertainty in tail estimates. This limitation can be addressed by reporting SE-wise sample sizes or combining extreme zones when necessary.
The method is most interpretable for approximately unimodal distributions. In multimodal or highly irregular settings, SE-wise SD may reflect overlapping structural features rather than simple central-to-tail gradients and should therefore be interpreted alongside graphical or distributional diagnostics.
Finally, the present study focuses on one-dimensional SE-based partitioning. Extensions to multivariate or high-dimensional contexts are beyond the current scope and represent an important direction for future work.
Despite these limitations, the approach remains valid, transparent, and reproducible. It exposes variability structure already implicit in classical confidence-interval geometry, without altering estimator properties or inferential assumptions, and thereby provides a principled foundation for future zonal frameworks such as Epistemic Zonal Statistics (EZS).
Future research should apply SE-wise SD to empirical datasets and to alternative variability measures (e.g., MAD, IQR, or other robust dispersion indices) to evaluate how distributional shape, sample size, and measurement context influence the stability and interpretability of zone-specific variability patterns. To further assess practical value and complement the simulation-based illustration used in this study, testing the method on real-world biomedical or laboratory datasets will be an important next step that may reveal domain-specific patterns of localized variability not evident in purely theoretical settings.
Declarations
Ethics Approval and Consent to Participate
Not applicable. This study used simulated data and involved no human participants, identifiable information, or secondary human-subject datasets.
Consent for Publication
Not applicable.
Availability of Data and Materials
All data used in this work were generated through simulation for illustrative purposes. 
Competing Interests
The author declares no competing interests.
Funding
No external funding was received for this work.
Authors’ Contributions
T.F. Raham conceived the methodological framework, conducted all simulations, performed the analysis, and wrote the manuscript.
AI Usage Disclaimer
The author declares that generative artificial intelligence was used during the preparation of this manuscript in accordance with journal guidelines. ChatGPT (OpenAI, GPT-5.1 model) was utilized solely for language refinement and formatting suggestions.
No generative AI tools were used for conceptual development, methodological design, statistical analysis, simulation procedures, interpretation of results, or creation of scientific figures or data.
All intellectual content, analytical framework, scientific reasoning, and conclusions presented in this manuscript are entirely the work of the author. The author reviewed and verified all AI-refined text to ensure accuracy and correct scientific meaning.
Intellectual Property and Author Rights
The methodological framework presented in this Research Note—including SE-partitioned local SD estimation, zone-wise variability characterization, and its conceptual linkage to the broader Zonal Epistemic Statistics (EZS) paradigm—constitutes original intellectual work by the author.
These concepts, terminology, and analytical structures are protected under Iraqi Intellectual Property Law No. 3 of 1971 (as amended) and applicable international treaties, including the Berne Convention, WIPO Copyright Treaty (WCT), and the Patent Cooperation Treaty (PCT).
Any unauthorized reproduction, adaptation, distribution, commercial use, or derivative implementation of this framework—including EZS, zonal variability constructs, or SE-partitioned analytical methods—is strictly prohibited without explicit written permission from the author.

Figure 1. Simulated distribution segmented using standard-error–based partitions (μ ± 1SE, ±2SE, ±3SE) to quantify local variability. The five resulting epistemic zones—Tail-L, Trans-L, Central, Trans-R, and Tail-R—exhibit distinct zone-wise SD values (1.42, 0.96, 0.58, 1.11, and 1.53, respectively), contrasted with a global SD of 1.07. This demonstrates that global dispersion metrics obscure important region-specific variability, underscoring the need for SE-aligned local SD analysis as a missing component in classical statistical practice.

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