Data Points Cast Ballots to Identify the Most Representative Dataset Average

A newly formulated mathematical voting procedure enables individual numerical observations to directly choose their most representative average by casting ballots based on geometric closeness, resolving persistent analytical dilemmas when statistical modes fail to exist or yield multiple competing values. Developed by Kayode Ayinde, Brian Haile, Taylor Harrison, David Vlieger, and Love J. Ayinde, the methodology reframes central tendency evaluation from traditional top-down statistical aggregation into a participatory framework governed by plurality rule. Rather than calculating a single metric that minimizes cumulative squared dispersion or pinpoints a central rank ordering, the technique requires every data point in a numerical collection to evaluate candidate averages and cast an equal-weight vote for whichever value lies at the smallest absolute distance from itself. Demonstrating the process across eighteen empirical educational datasets of state-level United States Scholastic Assessment Test scores collected over a six-year period from 2017 through 2022, the authors revealed that the resulting Most Representative Average does not exclusively mirror the sample mode, as generalized and positional averages frequently capture the collective preference of underlying observations without producing a single empirical tie.

The mathematical architecture of the voting technique establishes an objective five-step algorithm that evaluates proximity between observations and candidate summary measures using the metric of absolute deviation. Because absolute deviation and squared deviation preserve identical proximity rankings relative to any individual datum, absolute differences directly measure the discrepancy of each subject from candidate central values. In the initial stage of computation, analysts compute a designated collection of p candidate averages from an observed numeric dataset of n observations. The second stage calculates the absolute deviation between each observation and every candidate average, establishing a comprehensive deviation matrix. In the third stage, an indicator function assigns a discrete vote of one to the specific average that produces the minimum deviation for that given observation, while assigning zero to all competing alternatives, precisely mirroring the classic plurality rule in voting theory where each voter endorses exactly one candidate. In the fourth stage, the individual votes allocated to each average are summed across all n observations to generate total vote counts or converted into proportions by dividing by the sample size. Finally, the fifth stage identifies the Most Representative Average as the unique average that maximizes the total vote tally or sample proportion among all evaluated candidates.

To evaluate the competitive performance of candidate metrics, the authors incorporated an extensive suite of classical and generalized averages defined across statistical literature, including measures whose historical behaviors and properties have been evaluated by researchers such as Burk, Havil, Mario, Peck, Frederick, Larry, Siegel, Castellan, Davis, Turner, and Vogel. The mid-range computes the arithmetic sum of the sample minimum and maximum divided by two, offering extreme computational simplicity but remaining acutely sensitive to outlier values while ignoring intermediate data points. The standard arithmetic mean divides the total sum of all observations by sample size, providing established statistical properties but deteriorating significantly under asymmetric skewness or extreme data values. The geometric mean takes the nth root of the continuous product of all observations, and the harmonic mean calculates the reciprocal of the arithmetic mean of reciprocals, both providing utility for rates and proportional data but failing completely whenever any observation is zero or negative. Power-based formulations extend this spectrum through the quadratic mean, also known as the root mean square, the cubic mean, and the quartic mean, each elevating observations to higher exponential powers before computing arithmetic means and extracting corresponding fractional roots, though each remains sensitive to distributional extremes.

The methodological framework also incorporates positional and generalized parameter-driven averages alongside traditional measures to broaden the candidate election pool. The median arranges all observations in non-decreasing order of magnitude, taking the exact central value when sample size is odd or the arithmetic mean of the two middle observations when sample size is even, thereby maintaining noted robustness against extreme values as documented by Bhandari, Dodge, and modern medical analyses by Khorana. The mode identifies the most frequent observation within the data array, remaining usable across qualitative and quantitative data but introducing structural failure modes when no value repeats or when bimodal and multimodal distributions generate non-unique candidates. To bridge these disparate measures, the researchers introduced the Lehmer mean, denoted as LHM with two real-valued parameters p and q, which computes the quotient of the sum of observations raised to power p over the sum of observations raised to power q. When q equals p minus one, the Lehmer mean establishes a continuous parametric spectrum balancing large values when p exceeds one against smaller values when p is less than one, reproducing the harmonic mean at p equals zero, the geometric mean for pairs of values at p equals one-half, the arithmetic mean at p equals one, and the contra-harmonic mean at p equals two, while approaching the sample minimum and maximum as p approaches negative and positive infinity respectively.

Practical validation of the voting technique was conducted using historical educational records sourced from the National Center for Education Statistics repository, covering average Scholastic Assessment Test results across all fifty United States and the District of Columbia from 2017 through 2022. The testing regimen comprised eighteen distinct numerical datasets formed by evaluating three core academic variables across the six individual testing years: reading mean scores, mathematics mean scores, and composite total mean scores. To accommodate varying distributional modalities, the researchers identified empirical modes using the most frequent value function from the modeest package in R, sequentially indexing multiple occurrences as Mode 1, Mode 2, and continuing outward according to their numerical appearance in sorted data. Across the evaluation, at least thirteen distinct candidate averages competed simultaneously for the votes cast by the fifty-one administrative jurisdictions: the mid-range, the arithmetic mean, the geometric mean, the harmonic mean, the quadratic mean, the cubic mean, the quartic mean, the median, empirical modes, and four distinct parameterized Lehmer variants designated as LHM(2,1), LHM(3,2), LHM(4,3), and LHM(5,4).

Box plot summaries characterizing the five-number summary statistics—minimum, lower quartile, median, upper quartile, and maximum—across the six testing years indicated distinct temporal patterns in student performance alongside variable degrees of positive skewness. Overall student performance attained its highest levels during 2021, closely followed by performance metrics observed in 2022 and 2017. Across the multi-year timeline, the distributions of mean SAT scores exhibited rightward skewness, reflecting testing instruments that were demanding yet not excessively severe. Median performance levels displayed noticeable longitudinal fluctuations, undergoing an observable decline beginning after 2016 before experiencing a recovery in 2021, while maximum recorded state mean scores exhibited a progressive downward trend across successive testing years. Critically, visual inspection of the underlying distributions confirmed that these standardized state-level educational scores contained no extreme outliers, demonstrating stable behavioral conditions that supported standard and generalized means without requiring robust modifications such as trimmed or Winsorized adjustments.

Evaluating the modality of the eighteen educational datasets in Table 1 revealed considerable distributional diversity, with observed mode counts ranging across single, double, triple, quadruple, quintuple, sextuple, and octuple values. In empirical testing across these complex configurations, the voting algorithm selected a single, decisive Most Representative Average in every single instance without producing a single tied outcome, confirming that ties remain exceptionally rare in real-valued empirical datasets. While the mode emerged as the most frequent overall winner across the eighteen datasets, closer inspection showed that it was far from universally dominant, followed in selection frequency by the parameterized Lehmer mean LHM(5,4), the mid-range, and the median. In datasets possessing only a single unique mode, such as the 2020 reading mean scores, the 2021 total mean scores, and the 2022 reading mean scores, the single mode failed to win the election, finishing second while the voting technique designated the median, LHM(5,4), and the mid-range as the true Most Representative Averages respectively.

Similar departures from conventional mode selection materialized across datasets characterized by bimodal and multimodal distributions, where one of the candidate modes frequently secured the plurality of votes but failed to do so universally. Specifically, within the 2017 reading mean scores, the 2017 total mean scores, and the 2019 total mean scores, the voting mechanism favored the generalized Lehmer mean LHM(5,4) over every empirical mode present, relegating the most popular modal values to second place in voter tallies. These empirical selections demonstrated that the Most Representative Average does not conform to traditional distributional expectations that prescribe the arithmetic mean for symmetric data and the median for skewed samples. Within positively skewed score distributions from 2017 through 2019, the voting process favored modes and higher-order Lehmer means rather than the robust median, while in approximately symmetric distributions from 2020 and 2021, the algorithm aligned with modes, Lehmer means, and mid-ranges rather than the mathematically efficient arithmetic mean, demonstrating that proximity-based consensus captures underlying data affinities rather than rigid statistical definitions.

Comparisons between the selected Most Representative Averages and the baseline reported average provided by the data source revealed consistent divergences in central estimation across the six-year observation window. Graphical trajectories tracking competing metrics across reading, mathematics, and total scores demonstrated that the published reported average consistently underreported overall student performance relative to the consensus identified by voter plurality. Although the underlying computation method for the reported average was not disclosed by the repository, the Most Representative Average and primary modes exhibited more dynamic, rapid temporal fluctuations across successive years compared to conventional static measures. The authors observed that this fluidity does not represent a flaw but rather reflects genuine shifts in collective proximity preferences across constituent states over time, enabling datasets to adaptively select context-sensitive representations that account for empirical dispersion and clustering.

Addressing the operational characteristics of the method, the authors examined computational complexity alongside future implementation strategies for large-scale data analytics. Calculating absolute deviations between n observations and p candidate averages requires O(np) operations, which can impose substantial processing overhead as datasets expand into massive scales. Because each observation evaluates its distance to candidate averages independently, the algorithm is naturally suited for parallel computing architectures, including MapReduce frameworks, GPU-accelerated computing, and parallel processing loops in R utilizing packages like foreach and doParallel. Computational overhead may be reduced through approximation techniques such as preliminary data clustering or adaptive candidate selection that limits evaluation to meaningful subsets of averages based on initial data properties. Future research directions highlighted by the authors include evaluating the statistical properties of the Most Representative Average through bootstrapping simulations, incorporating trimmed and Winsorized means for noisy data containing severe outliers, and extending the proximity voting framework to high-dimensional spaces, weighted observations, and applied domains such as finance, behavioral science, and health analytics.

Subject of Research: A proximity-based voting technique for determining the most representative average of numerical datasets

Article Title: Voting technique for determining the most representative average of numerical data sets

Article References: Ayinde, K., Haile, B., Harrison, T., Vlieger, D., & Ayinde, L. J. (2025). Voting technique for determining the most representative average of numerical data sets. Discover Data, 3(1), Article 57. https://doi.org/10.1007/s44248-025-00073-z

Image Credits: AI Generated

DOI: 10.1007/s44248-025-00073-z

Keywords: central tendency, voting technique, most representative average, arithmetic mean, median, statistical mode, Lehmer mean, SAT scores, Voting, technique, determining, most