Empirical Investigation of Alternative Measures of Central Tendency

2020 ◽  
Author(s):  
William H Black ◽  
Lari B Masten
Keyword(s):  
Geometric Mean ◽  
Predictive Performance ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Central Tendency ◽  
Business Valuation ◽  
Alternative Measures ◽  
Data Points ◽  
Using Data ◽  
Performance Results

There is ongoing controversy in the business valuation literature regarding the preferability of the arithmetic mean or the harmonic mean when estimating ratios for use in business valuation. This research conducts a simulation using data reported from actual market transactions. Successive random samples were taken from data on valuation multiples and alternative measures of central tendency were calculated, accumulating more than 3.7 million data points. The measures (arithmetic mean, median, harmonic mean, geometric mean) were compared using hold-out sampling to identify which measure provided the closest approximation to actual results, evaluated in terms of least squares differences. Results indicated the harmonic mean delivered superior predictions to the other measures of central tendency, with less overstatement. Further, differences in sample size from 5 to 50 observations were evaluated to assess their impact on predictive performance. Results showed substantial improvements up to sample sizes of 20 or 25, with diminished improvements thereafter.

When Averaging Multiples, the Arithmetic Mean Is Inferior to the Harmonic Mean

2021 ◽  
Vol 40 (2) ◽  
pp. 61-67
Author(s):  
Gilbert E. Matthews
Keyword(s):  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Central Tendency

This article posits that using the arithmetic mean to average multiples is mathematically inferior. A multiple is an inverted ratio with price in the numerator. The harmonic mean is a statistically sound method for averaging inverted ratios. It should be used as a measure of central tendency for multiples, along with the median. Empirically, the harmonic mean and the median of a set of multiples are usually similar. Because the harmonic mean can be overly affected by abnormally low multiples, the valuator must use judgment to exclude outliers.

scholarly journals Unification Theories: Means and Generalized Euler Formulas

Axioms ◽  
2020 ◽  
Vol 9 (4) ◽  
pp. 144
Author(s):  
Radu Iordanescu ◽  
Florin Felix Nichita ◽  
Ovidiu Pasarescu
Keyword(s):  
Geometric Mean ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Current Paper ◽  
Baxter Equation ◽  
Quadratic Mean ◽  
Euler’S Formula ◽  
Associative Structures ◽  
Generalized Mean

The main concepts in this paper are the means and Euler type formulas; the generalized mean which incorporates the harmonic mean, the geometric mean, the arithmetic mean, and the quadratic mean can be further generalized. Results on the Euler’s formula, the (modified) Yang–Baxter equation, coalgebra structures, and non-associative structures are also included in the current paper.

scholarly journals Comparison of different ways of computing grades in continuous assessment into the final grade

2017 ◽  
Vol 8 ◽  
pp. 1
Author(s):  
Juan A. Marin-Garcia ◽  
Julien Maheut ◽  
Julio J. Garcia Sabater
Keyword(s):  
Learning Outcomes ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Alternative Methods ◽  
Continuous Assessment ◽  
Final Grade ◽  
Weighted Arithmetic ◽  
Student’S Performance ◽  
The Subject

<p>We present the results of comparing various ways of calculating students' final grades from continuous assessment grades. Traditionally the weighted arithmetic mean has been used and we compare this method with other alternatives: arithmetic mean, geometric mean, harmonic mean and multiplication of the percentage of overcoming of each activi-ty. Our objective is to verify, if any of the alternative methods, agree with the student’s performance proposed by the teacher of the subject, further discriminating the grade be-tween high and low learning outcomes and reducing the number of approved opportunists.</p><p> </p><p>[Comparación del efecto de diferentes modos de agregar las califica-ciones de evaluación continua en la nota final]</p>

scholarly journals The Efficiency of Aggregation Methods in Ensemble Filter Feature Selection Models

2021 ◽  
Vol 9 (4) ◽  
pp. 39-51
Author(s):  
Noureldien Noureldien ◽  
Saffa Mohmoud
Keyword(s):  
Feature Selection ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Selection Methods ◽  
Aggregation Method ◽  
Ensemble Models ◽  
Aggregation Methods ◽  
Single List ◽  
The Individual ◽  
Performance Results

Ensemble feature selection is recommended as it proves to produce a more stable subset of features and a better classification accuracy when compared to the individual feature selection methods. In this approach, the output of feature selection methods, called base selectors, are combined using some aggregation methods. For filter feature selection methods, a list aggregation method is needed to aggregate output ranked lists into a single list, and since many list aggregation methods have been proposed the decision on which method to use to build the optimum ensemble model is a de facto question.       In this paper, we investigate the efficiency of four aggregation methods, namely; Min, Median, Arithmetic Mean, and Geometric Mean. The performance of aggregation methods is evaluated using five datasets from different scientific fields with a variant number of instances and features. Besides, the classifies used in the evaluation are selected from three different classes, Trees, Rules, and Bayes.       The experimental results show that 11 out of the 15 best performance results are corresponding to ensemble models. And out of the 11 best performance ensemble models, the most efficient aggregation methods are Median (5/11), followed by Arithmetic Mean (3/11) and Min (3/11). Also, results show that as the number of features increased, the efficient aggregation method changes from Min to Median to Arithmetic Mean. This may suggest that for a very high number of features the efficient aggregation method is the Arithmetic Mean. And generally, there is no aggregation method that is the best for all cases.

scholarly journals Statistical measures of the central tendency for H+ activity and pH

2017 ◽  
Vol 68 (4) ◽  
pp. 174-181 ◽  
Author(s):  
Izabela Kuna-Broniowska ◽  
Halina Smal
Keyword(s):  
Probability Distributions ◽  
Geometric Mean ◽  
Arithmetic Mean ◽  
Central Tendency ◽  
Statistical Measures ◽  
Normal Probability ◽  
Explicit Opinion ◽  
Normal Probability Distribution ◽  
Theoretical Considerations

Abstract Despite the numerous papers on the statistical analyses of pH, there is no explicit opinion on the use of arithmetic mean as a measure of the central tendency for pH and H+ activity. The problem arises because the transformation of the arithmetic mean for one does not give the arithmetic mean for the other. The paper presents 1) the theoretical considerations on the distribution of pH and H+ activity and relation between them, properties of these distributions, the choice of distributions which should be consistent with the distribution of pH and the distribution of H+ activity and measures of central tendency for features of such distributions and 2) examples of calculations of measures of central tendency for pH and H+ activity based on the literature data on soil and lake water pH. These data analyses included distributions of pH and H+ activities, properties of distribution, descriptive statistics for pH and for the H+ activity and comparison of arithmetic mean with the geometric mean. From the results, it could be concluded that a uniform approach to the choice of measure for the central tendency of pH and H+ activity requires the determination of the type of measure (mean) for one of them and then consistent transformation of this measure. The choice of measure of the central tendency for the variable should be preceded by determination of its distribution. Normal probability distribution of pH and thus lognormal distribution of H+ activity indicate that the arithmetic mean, and its corresponding geometric mean should be used as proper measures of the central tendency for pH and for H+ activity. Besides, the position statistic that is a median can be used for each of those variables, irrespective of their probability distributions.

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