Percentage Theory of Measurement Indices

• Percentage Theory of Measurement Indices is a framework that explains how expressing numerical indicators as parts of a whole enhances comprehension and comparability.
• It formalizes techniques like min-max normalization, percentization, and fractional scoring to produce equitable and meaningful measurement units.
• The theory is applied in effect-size analysis and bibliometrics, addressing challenges like ties, thresholds, and aggregation issues with precise methodological strategies.

Searching arXiv for the cited papers and closely related work on percentage-based measurement indices and percentile indicators. Percentage Theory of Measurement Indices denotes a family of arguments about how numerical indicators acquire meaning and comparability when they are expressed as parts of a whole, especially on $0\!\sim\!1$ or $0\!\sim\!100$ scales. In one strand, developed for effect-size analysis, the theory holds that an estimator should first support comprehension of an estimand and then support comparison across estimands, with percentage scaling supplying uniformly meaningful and equitable units (Zhao et al., 2024, Zhao et al., 18 Jul 2025). In another strand, developed in bibliometrics, percentage and percentile indicators such as $PP(\text{top }x\%)$, $PP_{\text{top }10\%}$, $R(6)$, and $I3(6)$ are analyzed as rank-based measures whose interpretation depends on exact handling of ties, thresholds, and database baselines [(Bornmann et al., 2016); (Waltman et al., 2012); (Schreiber, 2012)]. Across these strands, the central issue is not merely numerical rescaling, but the relation between theoretical percentage meaning and empirical implementation.

1. Functionalist foundations and the meaning of percentage-based indices

A central formulation appears in the theory of the percentage coefficient $b_p$. This work proposes a functionalist theory of measurement indices, according to which statistical indicators have two primary functions: Comprehension and Comparison (Zhao et al., 2024). Comprehension means that an indicator should help readers understand the estimand; comparison means that it should help readers compare two or more estimands meaningfully. The ordering is explicit: comprehension comes before comparison, because a coefficient that cannot be interpreted in its own right cannot serve as a stable basis for cross-estimand contrast (Zhao et al., 2024).

Within this framework, “effect” is not treated as monolithic. Rather, effect is described as a conglomerate of components, and different indicators measure different subfunctions or components of effect. Regression coefficients may measure unit effect, also called efficiency, rather than an all-encompassing effect (Zhao et al., 2024). The percentage coefficient $b_p$ is therefore not presented as a universal effect-size index, but as a generic indicator of efficiency/unit effect.

The same theoretical line is extended historically and conceptually in the account of percentage scale $ps$ and percentage coefficient $b_p$. There the key proposition is stated in stronger measurement-theoretic terms: “Equitable units are necessary and sufficient for comparability of two indices” (Zhao et al., 18 Jul 2025). On this view, percentage thinking is the practice of representing quantities and effects as parts of a whole; percentization standardizes scale ranges, and this equalization of scale ranges is taken to equalize units as well. This links interpretability to meaningful units and comparability to equitable units (Zhao et al., 18 Jul 2025).

A plausible implication is that the “percentage theory” is best understood not as a single formal doctrine, but as a shared measurement claim: indicators become more intelligible when their units are anchored to an explicit whole, and more comparable when different variables are expressed relative to commensurate wholes.

2. Percentage scales, percentization, and the definition of $0\!\sim\!100$0

The percentage-theoretic literature defines a percentage scale as a scale anchored by conceptual minimum coded $0\!\sim\!100$1 and conceptual maximum coded $0\!\sim\!100$2 (Zhao et al., 2024). Closely related formulations describe percentage scales as ranging conceptually $0\!\sim\!100$3, $0\!\sim\!100$4, or sometimes $0\!\sim\!100$5, with the explicit remark that moving the decimal point two positions to the right turns a $0\!\sim\!100$6 scale into a $0\!\sim\!100$7 scale (Zhao et al., 18 Jul 2025). The defining transformation is percentization.

The core formulas given for percentage scaling are: $0\!\sim\!100$8 and

$0\!\sim\!100$9

where $PP(\text{top }x\%)$0 is the transformed percentage score, $PP(\text{top }x\%)$1 or $PP(\text{top }x\%)$2 is the original score, and $PP(\text{top }x\%)$3 or $PP(\text{top }x\%)$4 are the minimum and maximum anchors (Zhao et al., 2024). A general min-max normalization form is also given: $PP(\text{top }x\%)$5 (Zhao et al., 2024). A later exposition gives the corresponding $PP(\text{top }x\%)$6 and $PP(\text{top }x\%)$7 forms as

$PP(\text{top }x\%)$8

and

$PP(\text{top }x\%)$9

(Zhao et al., 18 Jul 2025).

Within this formalism, percentage coefficient $PP_{\text{top }10\%}$0 is defined simply as the regression coefficient when both the dependent variable and independent variable are on percentage scales (Zhao et al., 2024, Zhao et al., 18 Jul 2025). It is therefore not a new model, but a re-expression of the ordinary regression coefficient after variables are transformed onto conceptual $PP_{\text{top }10\%}$1 percentage scales (Zhao et al., 2024). Its principal interpretations are stated as effect per unit, efficiency, and whole-scale effect: $PP_{\text{top }10\%}$2 is the change in the dependent variable associated with an increase in the independent variable from conceptual minimum to conceptual maximum (Zhao et al., 2024).

The theory emphasizes conceptual rather than merely observed anchoring. Percentization should be based on conceptual anchoring, defined as choosing scale maximum and scale minimum “based on conceptual legitimacy and appropriateness, but not necessarily on the appearance of the data in hand” (Zhao et al., 18 Jul 2025). The age example illustrates this distinction: observed values may run from 18 to 83, but conceptual anchors may be set to 0 and 100 for interpretive and comparative reasons (Zhao et al., 18 Jul 2025). The related notion of neighborhood rounding allows convenient round numbers near reasonable conceptual boundaries (Zhao et al., 18 Jul 2025).

3. Historical lineage: from percentage thinking to normalization

The historical reconstruction of the theory traces percentage thinking through several institutional and intellectual settings. It begins with the statement that traces of percentage thinking date back to Roman taxation and fiscal records, where quantities were already assessed “per hundred” (Zhao et al., 18 Jul 2025). It then identifies Simon Stevin and La Thiende (1585) as a major moment in decimalization, followed by the nineteenth-century metrication movement that institutionalized base-10 measurement worldwide (Zhao et al., 18 Jul 2025).

The same account links percentage-style bounded measurement to classic statistical indices, especially Pearson’s correlation coefficient $PP_{\text{top }10\%}$3, ranging from $PP_{\text{top }10\%}$4 to $PP_{\text{top }10\%}$5, the coefficient of determination $PP_{\text{top }10\%}$6, ranging from $PP_{\text{top }10\%}$7 to $PP_{\text{top }10\%}$8, and the Percent of Maximum Possible (POMP) associated with Cohen et al. (1999) (Zhao et al., 18 Jul 2025). The connection is not that these indices are identical, but that they share bounded, normalized scale logic.

Modern data mining and machine learning are presented as a further stage in this lineage. The relevant claim is that min-max normalization maps any feature to $PP_{\text{top }10\%}$9, equalizing the scale ranges (Zhao et al., 18 Jul 2025). The historical argument then becomes a measurement-theoretic one: because $R(6)$0 percentage scale assigns the entire scale to be the unit, equalizing scales is said to equalize units, and thereby support comparability (Zhao et al., 18 Jul 2025).

This suggests a broad continuity between older percentage representations, bounded statistical coefficients, and contemporary normalization practice. However, the stronger claim that the success of AI serves as a large-scale confirmation of the comparability of percentage-based indices is specific to that theoretical paper and should be read as part of its own argumentative framework rather than as a consensus across all measurement literatures (Zhao et al., 18 Jul 2025).

4. Percentile indicators in bibliometrics: theory, expectation, and empirical baseline

In bibliometrics, percentage theory takes the form of percentile-based indicators. A widely used example is $R(6)$1, defined as the proportion of papers of a unit that belongs to the $R(6)$2 most frequently cited papers in the corresponding fields and publication years (Bornmann et al., 2016). Thus $R(6)$3, $R(6)$4, and $R(6)$5 are field- and year-normalized indicators used, for example, in the Leiden Ranking (Bornmann et al., 2016).

The common expectation is that $R(6)$6 of papers can be expected to belong to the top $R(6)$7 most frequently cited papers. The central question raised in the empirical study of expected values is whether this claim is literally true in real databases and how strongly random samples deviate from it (Bornmann et al., 2016). Using an in-house Max Planck database derived from SCI-E, SSCI, and AHCI for papers from 1980–2010, the reported population values are:

• $R(6)$8
• $R(6)$9
• $I3(6)$0

These values are not exactly $I3(6)$1, $I3(6)$2, and $I3(6)$3 because “the impact of the papers in our database is not fractionally assigned to subject categories. Instead, an average citation impact is calculated for papers assigned to more than one subject category” (Bornmann et al., 2016). The distinction between nominal and empirical baseline is therefore fundamental.

The empirical test draws 1000 random samples for different sample sizes and computes the three indicators for each sample (Bornmann et al., 2016). For sample size 500 and $I3(6)$4, the mean across 1000 samples is $I3(6)$5, the population value is $I3(6)$6, the minimum is $I3(6)$7, and the maximum is $I3(6)$8 (Bornmann et al., 2016). More generally, deviations are larger for smaller samples and smaller for larger samples. The reported ranges are especially wide for small samples: $I3(6)$9 can range from $b_p$0 to $b_p$1, $b_p$2 from $b_p$3 to $b_p$4, and $b_p$5 from $b_p$6 to $b_p$7 (Bornmann et al., 2016).

The conclusion is twofold. First, the expected value from the indicator definition is not necessarily the same as the empirical expected value of a given database; one should know the population value of the database rather than assume the nominal $b_p$8 (Bornmann et al., 2016). Second, despite complex normalization procedures, the nominal values of $b_p$9, $b_p$0, and $b_p$1 can really be expected in repeated random sampling, provided samples are large enough (Bornmann et al., 2016). The theory is therefore statistically valid in expectation, but not as a guarantee for any single realized sample.

5. Fractional scoring, ties, and exact theoretical totals

A major branch of the literature addresses the fact that citation distributions are discrete and contain many ties. In such settings, percentile-based categories are not uniquely defined by naive thresholding. The proposed solution is fractional scoring, which treats a paper not as a single percentile point but as an interval of percentile ranks and assigns it fractionally across all percentile rank classes it overlaps (Schreiber, 2012).

The basic ambiguity can be seen in small examples. In a set of $b_p$2 papers, the third paper can correspond to a percentile between $b_p$3 and $b_p$4; taking the midpoint gives $b_p$5, but $b_p$6 lies exactly on the boundary between bottom-50% and top-50% classes (Schreiber, 2012). Fractional scoring avoids choosing a single point. For the third of five papers, the interval $b_p$7 belongs to the bottom 50% and $b_p$8 to the top 50%, so the paper is counted half in each (Schreiber, 2012).

The same logic is generalized to multiple percentile rank systems. The literature discusses a two-class system, the six-class system used for $b_p$9 and $ps$0, and 100 percentile rank classes (Schreiber, 2012). For six classes, the scheme is:

1. bottom 50%
2. 50%–75%
3. 75%–90%
4. 90%–95%
5. 95%–99%
6. top 1%

with weights $ps$1 in the Leydesdorff–Bornmann scheme (Schreiber, 2012).

The central guarantee is exact reproduction of the theoretical total. When all papers are fractionally assigned and weighted, the aggregation yields theoretical totals of $ps$2 for 2 percentile-rank classes, $ps$3 for 6 classes, and $ps$4 for 100 classes (Schreiber, 2012). In the more general bibliometric framework, the overlap between citation-count blocks and percentile intervals is formalized by

$ps$5

and the resulting score for publications with $ps$6 citations is

$ps$7

(Waltman et al., 2012). For a research group with $ps$8 publications having $ps$9 citations, the percentile-based indicator is the average score: $b_p$0 (Waltman et al., 2012).

The field-independence result follows by setting the evaluated set equal to the field distribution, yielding

$b_p$1

which depends only on percentile boundaries and scores, not on the underlying citation distribution (Waltman et al., 2012). For $b_p$2, this implies an exact field total of $b_p$3 (Waltman et al., 2012).

Empirical evidence reinforces the theoretical point. In four large datasets, including 2373 physics papers from Chemnitz, 3354 highly cited physicists’ papers, and two EPL datasets, non-fractional rules produce divergent values of $b_p$4, while fractional scoring reproduces exactly the theoretical benchmark $b_p$5 (Schreiber, 2013). For the Chemnitz dataset, lower-counting yields $b_p$6, higher-counting $b_p$7, midpoint $b_p$8, and fractional scoring exactly $b_p$9 (Schreiber, 2013).

6. Nominal percentages, empirical contingencies, and aggregation problems

A recurrent theme in the literature is that percentage-like labels need not retain their literal numerical meaning under empirical bibliometric rules. One study states the issue directly: in many empirical cases, quartiles are not quarters, medians are not halves, world baselines are not unity, and integer thresholds lead to inequality of performance evaluation in different science fields (Pislyakov, 2022).

For journals ranked within a discipline, the percentile formula

$0\!\sim\!100$00

is given, and JCR/InCites quartiles are assigned by floor division (Pislyakov, 2022). Because category sizes are often not multiples of four, quartiles are not exactly 25%-25%-25%-25%. Across 236 categories in JCR-2020, the paper reports approximately $0\!\sim\!100$01 journals in Q2 over Q1, $0\!\sim\!100$02 in Q3 over Q1, and $0\!\sim\!100$03 in Q4 over Q1, corresponding to roughly a 2.9% deviation from an even 25% distribution (Pislyakov, 2022). At the database level, multiple category assignment and the rule that InCites uses the best quartile for journals appearing in multiple research areas further inflate Q1 representation; the paper reports that about 27.1% of journals are Q1, but nearly half of all papers are published in Q1 journals, and 73.3% of papers are in Q1+Q2 (Pislyakov, 2022).

A similar nominal-versus-empirical discrepancy arises in subject normalization. For CNCI, the theoretical principle is that the world baseline should equal 1, but under whole counting and averaging of ratios with multiple field assignment, the global CNCI can deviate from unity (Pislyakov, 2022). The toy example with two papers yields $0\!\sim\!100$04 in one arrangement and $0\!\sim\!100$05 in another (Pislyakov, 2022). The paper identifies two remedies: ratio of sums instead of average of ratios, and especially fractional counting, which restores $0\!\sim\!100$06 in a closed universe (Pislyakov, 2022).

Threshold indicators for highly cited papers exhibit the same difficulty. Under inclusive rules, all papers with citations at least equal to the threshold are labeled highly cited, so the actual share can exceed the nominal top 1% when many papers are tied at the boundary (Pislyakov, 2022). The literature therefore distinguishes inclusive, exclusive, and fractional/fuzzy treatment of ties, and describes the fractional approach of Waltman and Schreiber as the cleanest arithmetic solution, even while noting that it may appear somewhat artificial (Pislyakov, 2022).

A different aggregation issue appears in the use of percentile indicators as proxies for scientific breakthroughs. One account argues that $0\!\sim\!100$07 and $0\!\sim\!100$08 are useful because very rare breakthrough publications cannot usually be counted with statistical reliability, whereas more frequent percentile events can be counted and then used to estimate rarer ones (Rodriguez-Navarro et al., 2022). The relevant scaling relations are given as

$0\!\sim\!100$09

$0\!\sim\!100$10

and

$0\!\sim\!100$11

(Rodriguez-Navarro et al., 2022). However, the paper holds that such inference is valid only for homogeneous research units. If countries or institutions are heterogeneous mixtures of groups with different aims and efficiencies, then aggregating first and calculating later (AFCL) underestimates rare-event output relative to calculating first and adding later (CFAL). This is formalized by the inequality

$0\!\sim\!100$12

when $0\!\sim\!100$13 (Rodriguez-Navarro et al., 2022).

Taken together, these results show that the percentage theory of measurement indices has two persistent constraints. First, theoretical percentage meanings can be distorted by discrete data, ties, multidisciplinarity, and database conventions. Second, aggregate-level percentage indicators can lose validity when the evaluated unit is internally heterogeneous. The literature therefore repeatedly returns to the same methodological demand: exact handling of boundaries and explicit attention to the unit of analysis.

7. Significance, scope, and recurring points of dispute

The significance of the percentage-theoretic literature lies in its attempt to connect interpretability, comparability, and formal exactness. In the effect-size strand, percentage scaling is advanced as a way to make coefficients both comprehensible and comparable, in contrast to raw coefficients, which may rely on non-uniform substantive units, and standardized beta, which is criticized as mathematically interpretable but substantively unhelpful (Zhao et al., 2024). In the bibliometric strand, fractional methods are advanced because they remove uncertainty in percentile assignment, ambiguity at class borders, deviation from ideal class proportions, rounding inconsistencies, and dependence on counting rule (Schreiber, 2012).

At the same time, the literature contains recurring disputes about what counts as exactness or fairness. One dispute concerns whether midpoint rules, average-percentile rules, or averaged weights are sufficient approximations. The fractional-scoring papers maintain that midpoint rounding is not equivalent to true fractional scoring and that only fractional scoring reproduces the theoretical total perfectly [(Schreiber, 2012); (Schreiber, 2013)]. Another dispute concerns whether percentage labels should be interpreted literally. The bibliometric analyses argue that rank categories such as quartile, percentile, baseline, and threshold are often only approximately aligned with their intuitive numerical names in actual database operations (Pislyakov, 2022, Bornmann et al., 2016).

A further tension concerns what kind of “whole” is appropriate. In the $0\!\sim\!100$14 literature, the preferred whole is a conceptual $0\!\sim\!100$15 range determined by substantive anchoring rather than by the observed sample (Zhao et al., 2024, Zhao et al., 18 Jul 2025). In bibliometrics, the relevant whole may be a field-year reference set, a citation-count block, or a homogeneous research unit [(Waltman et al., 2012); (Rodriguez-Navarro et al., 2022)]. This suggests that percentage-based measurement is not defined by one universal denominator, but by the requirement that the denominator be theoretically justified and consistently implemented.

The broadest common lesson is therefore methodological rather than merely notational. Percentage and percentile indices promise transparent interpretation because they express quantities relative to an explicit whole. But that promise is realized only under disciplined choices about anchors, ties, thresholds, field assignment, and aggregation. Where those choices are mishandled, nominal percentages cease to be exact percentages in practice; where they are handled carefully, percentage-based indices can recover theoretical expectations exactly or approximately, depending on the structure of the data and the design of the indicator [(Waltman et al., 2012); (Bornmann et al., 2016)].