Brookes' Dispersion Index (Δ)

• Brookes’ Dispersion Index (Δ) is a normalized quantitative measure that assesses how research outputs distribute across mutually exclusive categories.
• It calculates a weighted mean rank from frequency data and normalizes the result to indicate a continuum from perfect dispersion (0) to total concentration (1).
• The index is applied to evaluate field heterogeneity, compare bibliometric profiles, and inform research portfolio strategies despite potential classification challenges.

Brookes’ Measure of Categorical Dispersion (Δ) is a normalized quantitative index designed to capture the extent to which items—such as publications or research outputs—are distributed across a set of predefined, mutually exclusive categories. Anchored in bibliometric analysis, Δ offers a parsimonious scalar summary ranging from perfect thematic heterogeneity (maximum dispersion) to total concentration (maximum focus), facilitating the comparative assessment of disciplinary or topical breadth in real-world academic datasets (Eddakrouri et al., 17 Dec 2025).

1. Mathematical Definition and Formula

Brookes’ Dispersion Index Δ is formally expressed as

$\Delta = \frac{M - 1}{N - 1}$

where:

• $N$ denotes the total number of non-empty categories.
• $M$ is the weighted mean rank of the frequency distribution:

$$M = \frac{\sum_{i=1}^{N} f_i r_i}{\sum_{i=1}^{N} f_i} = \frac{\sum f_i r_i}{E_f}$$

• $f_i$ is the count of items in category $i$.
• $r_i$ is the rank of category $i$, assigned inversely (i.e., the category with the largest $f_i$ is assigned $r=1$; the next, $r=2$, etc.).
• $E_f$ is the total number of items ($E_f = \sum f_i$).

Values of Δ are strictly normalized to the $[0,1]$ interval: $\Delta = 0$ when material is distributed equally across all categories (maximum dispersion); $\Delta = 1$ when all items reside in a single category (maximum concentration) (Eddakrouri et al., 17 Dec 2025).

2. Stepwise Calculation Procedure

To compute Brookes’ Δ from empirical data:

1. Tabulate absolute frequencies $\{f_1, f_2, ..., f_N\}$ across all non-empty categories.
2. Rank categories inversely: Sort $f_i$ in descending order and assign ranks $r_i$ starting at 1. For ties, allocate average rank values across tied positions.
3. Compute the weighted mean rank:

$M = \frac{\sum f_i r_i}{E_f}$

1. Insert into Brookes’ formula to obtain Δ:

$\Delta = \frac{M - 1}{N - 1}$

2. Interpret Δ within $[0,1]$; lower values denote dispersion, higher values denote concentration (Eddakrouri et al., 17 Dec 2025).

3. Worked Example: Analysis of Arabic Applied Linguistics

A comprehensive dataset of 1,564 publications spanning 2019–2025 in Arabic Applied Linguistics, classified into eight sub-disciplines, illustrates the application of the Δ index (Eddakrouri et al., 17 Dec 2025).

Sub-discipline	Absolute Count $f_i$	Inverse Rank $r_i$
Computational Linguistics/NLP	767	1
Sociolinguistics	264	2
Language Teaching	197	3
Discourse Analysis	127	4
Second Language Acquisition	77	5
Corpus Linguistics	53	6
Applied Linguistics (General)	45	7
Language Assessment	34	8

Calculation:

• $\sum f_i r_i = 3,686$, $E_f = 1,564$
• $M = 3,686 / 1,564 \approx 2.3568$
• $\Delta = (2.3568 - 1)/(8-1) \approx 0.194$

This outcome signals exceptionally high thematic dispersion: despite Computational Linguistics comprising 49% of the corpus, the upward shift in mean rank is driven by persistent representation across six additional subfields, confirming pronounced field heterogeneity (Eddakrouri et al., 17 Dec 2025).

4. Interpretation of Δ Values

The index encodes the categorical structure of a dataset along the following continuum:

• $\Delta \approx 0$: Maximum dispersion, items evenly distributed (heterogeneous structure).
• $\Delta \approx 1$: Maximum concentration, items overwhelmingly in one category.
• Empirical thresholds:
  • $\Delta < 0.2$: Very high dispersion (broad thematic sweep).
  • $0.2 \leq \Delta < 0.5$: Moderate dispersion (dominance with substantial diversity).
  • $\Delta \approx 0.5$: Balanced state.
  • $\Delta > 0.5$: Increasing concentration (focus within a few categories).

For instance, $\Delta = 0.194$ establishes Arabic Applied Linguistics as exceptionally dispersed, with no hegemonic subfield dominating the research landscape (Eddakrouri et al., 17 Dec 2025).

5. Methodological Constraints and Pitfalls

Accurate application of Brookes’ Δ requires careful attention to several methodological aspects:

• Dataset integrity: Each item must be uniquely classified; data should be comprehensive and curated to ensure only mutually exclusive, non-empty categories.
• Classification granularity: Category number ($N$) must be justified—as overly granular partitions can artificially depress Δ, while excessive coarseness can inflate it. Categories must cover all relevant domain facets without overlap.
• Ranking procedure: Always employ inverse ranking. For frequency ties, use averaged ranks.
• Sample size effects: Small $N$ increases sensitivity to ranking artifacts; sufficient data volume is essential for stable estimation.
• Interpretive limitations: Δ does not accommodate multi-thematic assignments, nor does it reflect second-order relations or overlaps between categories.
• Comparative cautions: Cross-field comparisons require consistent classification schemes; differences in $N$ or the framing of categories impact Δ’s meaning (Eddakrouri et al., 17 Dec 2025).

6. Applications and Utility in Field Characterization

Brookes’ Δ yields replicable, field-independent insight into disciplinary structure:

• Field characterization: Quantifies whether research output is narrowly or broadly distributed, enabling assessments of thematic focus.
• Comparative bibliometrics: Supports cross-domain comparison, contingent on harmonized classification logic.
• Research portfolio analysis: Guides strategic evaluation of diversity/concentration in grant funding, institutional output, or subfield evolution (Eddakrouri et al., 17 Dec 2025).

A plausible implication is that, when applied rigorously, Brookes’ Δ provides a transparent, scalable paradigm for analyzing the distributional heterogeneity of scholarly endeavors, with direct methodological implications for bibliometrics and science policy analysis.