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.

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.

A Neutral Model With Fluctuating Population Size and Its Effective Size

Genetics ◽  
2002 ◽  
Vol 161 (1) ◽  
pp. 381-388
Author(s):  
Masaru Iizuka ◽  
Hidenori Tachida ◽  
Hirotsugu Matsuda
Keyword(s):  
Asymptotic Behavior ◽  
Population Size ◽  
Diffusion Model ◽  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Effective Size ◽  
Neutral Model ◽  
Average Heterozygosity ◽  
Special Case ◽  
Concrete Model

Abstract We consider a diffusion model with neutral alleles whose population size is fluctuating randomly. For this model, the effects of fluctuation of population size on the effective size are investigated. The effective size defined by the equilibrium average heterozygosity is larger than the harmonic mean of population size but smaller than the arithmetic mean of population size. To see explicitly the effects of fluctuation of population size on the effective size, we investigate a special case where population size fluctuates between two distinct states. In some cases, the effective size is very different from the harmonic mean. For this concrete model, we also obtain the stationary distribution of the average heterozygosity. Asymptotic behavior of the effective size is obtained when the population size is large and/or autocorrelation of the fluctuation is weak or strong.

scholarly journals Impact of different centroid means on the accuracy of orthometric height modelling by geometric geoid method

2020 ◽  
Vol 6 (4) ◽  
pp. 124 ◽  
Author(s):  
Oluyori P. Dare ◽  
Eteje S. Okiemute
Keyword(s):  
Root Mean Square ◽  
Arithmetic Mean ◽  
Orthometric Height ◽  
Harmonic Mean ◽  
Geometric Method ◽  
Mean Square ◽  
Geoid Model ◽  
Microsoft Excel ◽  
Orthometric Heights ◽  
The Impact

<p class="abstract"><strong>Background:</strong> Orthometric height, as well as geoid modelling using the geometric method, requires centroid computation. And this can be obtained using various models, as well as methods. These methods of centroid mean computation have impacts on the accuracy of the geoid model since the basis of the development of the theory of each centroid mean type is different. This paper presents the impact of different centroid means on the accuracy of orthometric height modelling by geometric geoid method.</p><p class="abstract"><strong>Methods:</strong> DGPS observation was carried out to obtain the coordinates and ellipsoidal heights of selected points. The centroid means were computed with the coordinates using three different centroid means models (arithmetic mean, root mean square and harmonic mean). The computed centroid means were entered accordingly into a Microsoft Excel program developed using the Multiquadratic surface to obtain the model orthometric heights at various centroid means. The root means square error (RMSE) index was applied to obtain the accuracy of the model using the known and the model orthometric heights obtained at various centroid means.  </p><p class="abstract"><strong>Results:</strong> The computed accuracy shows that the arithmetic mean method is the best among the three centroid means types.</p><p class="abstract"><strong>Conclusions:</strong> It is concluded that the arithmetic mean method should be adopted for centroid computation, as well as orthometric height modelling using the geometric method.</p>

Comparison of Harmonic Mean versus Arithmetic Mean Clearance Values

1986 ◽  
Vol 75 (4) ◽  
pp. 427-429 ◽  
Author(s):  
L.J. Schaaf ◽  
F.C. Lam ◽  
D.G. Perrier
Keyword(s):  
Arithmetic Mean ◽  
Harmonic Mean

Remarks on the Neglected Mean

1973 ◽  
Vol 66 (3) ◽  
pp. 253-255
Author(s):  
Joseph L. Ercolano
Keyword(s):  
Arithmetic Mean ◽  
Harmonic Mean ◽  
Real Numbers ◽  
Positive Real ◽  
Golden Proportion ◽  
The One

The harmonic mean of two positive, real numbers was known to early Greek mathematicians. In fact, it is alleged that “Pythagoras learned in Mesopotamia of three means—the arithmetic, the geometric, and the subcontrary (later called the harmonic)—and of the ‘golden proportion’ relating two of these: the first of two numbers is to their arithmetic mean as their harmonic mean is to the second of the numbers” (Boyer 1968). Archytas, a disciple of Pythagoras (whose most important contribution to mathematics may very well have been his intervention with Dionysius to save the life of his friend, Plato), wrote on the application of these three means to music, and is possibly the one who is responsible for renaming the suhcontrary mean the harmonic mean (Boyer 1968).

scholarly journals LAS MEDIDAS DE TENDENCIA CENTRAL EN LIBROS DE TEXTO DE EDUCACIÓN PRIMARIA EN MÉXICO

PARADIGMA ◽  
2020 ◽  
pp. 706-729
Author(s):  
Danilo Díaz-Levicoy ◽  
Lizzet Morales-Garcia ◽  
Francisco Rodríguez-Alveal
Keyword(s):  
Content Analysis ◽  
Public Education ◽  
Primary Education ◽  
Publishing House ◽  
Arithmetic Mean ◽  
Central Tendency ◽  
The Public ◽  
Use Of Data ◽  
Personal Context ◽  
Educación Primaria

En este trabajo se describen los principales resultados de analizar las actividades en que intervienen las medidas de tendencia central en libros de texto de Educación Primaria en México. Para ello, se realiza un análisis de contenido en los libros de 4º, 5º y 6º de primaria que entrega la Secretaría de Educación Pública y los de una editorial privada, seis textos en total. Los resultados muestran el predominio de la media aritmética y la moda, la tarea de calcular, el uso de las representaciones de listado de datos y tabla de datos y el contexto personal; en el caso de la forma de trabajo, ésta depende de la editorial. Se observa la necesidad de aumentar la cantidad de las actividades sugeridas en los libros de texto.Palabras clave: Libro de texto. Media aritmética. Moda. Mediana. Actividades.As medidas de tendência central nos livros didáticos da Educação Primária no MéxicoResumoEste trabalho descreve os principais resultados da análise das atividades nas quais medidas de tendência central intervêm nos livros didáticos da educação primária no México. Para isso, realizou-se uma análise de conteúdo nos livros da 4ª, 5ª e 6ª séries do ensino fundamental distribuídos pela Secretaria da Educação Pública, e nos de uma editora privada, num total de seis textos. Os resultados mostram a predominância da média aritmética e da moda, a tarefa de calcular, o uso das representações de lista de dados e tabela de dados e o contexto pessoal; no caso da forma de trabalho, depende da editora. Observa-se a necessidade de aumentar o número de atividades sugeridas nos livros didáticos.Palavras chave: Livro didático. Média aritmética. Moda. Mediana. Atividades.Measures of central tendency in textbooks of Primary Education from MexicoAbstractThis article describes the main results of analyzing the activities in which the central tendency measures in Mexican textbooks of Primary Education are involved. To do this, a content analysis is carried out in books of 4th, 5th and 6th grade of primary education provided by the Public Education Secretariat and also those provided by a private publishing house, making six texts in total. The results show the predominance of the arithmetic mean and the mode, the calculating task, the use of data list representations and data table and the personal context; regarding the way of work, it depends on the publisher. The need to increase the amount of suggested activities in textbooks is observed.Keywords: Textbook. Arithmetic mean. Mode. Median. Activities.

scholarly journals Motivation to Higher Education and its Structure

2019 ◽  
Vol 11 (2) ◽  
pp. 43-57 ◽  
Author(s):  
O.B. Krushelnitskaya ◽  
M.V. Polevaya ◽  
A.N. Tretiyakova
Keyword(s):  
Higher Education ◽  
Well Being ◽  
Arithmetic Mean ◽  
Full Time ◽  
Relative Importance ◽  
Central Tendency ◽  
Subsequent Work ◽  
Motivational Structure ◽  
And Cluster Analysis ◽  
Regions Of Russia

Features of motivation for obtaining higher education, leading motives, motivational structure are investigated. The relevance of the research is determined by the influence of the motivation peculiarities on the success in studies educational activities and subsequent work. The students of the first courses of full-time education of 12 universities of different regions of Russia took part in the research. N = 559. The analysis of such indicators of central tendency as median, mode and arithmetic mean was used to assess the relative importance of motives. Correlation analysis and cluster analysis were used to identify the structure of motivation for obtaining higher education. Motivation for obtaining higher education was defined as a distant, external to the training activity, an anticipatory motivation. The motives for ensuring material well-being in the future, professionalization, self-realization are the most significant. The structure of motivation includes two clusters of motives that are grouped around the motives of material well-being and self-realization, and a separate motive of professionalization. The motives of the first cluster can be marked as motives for socialization. Motives of the second cluster can be marked as motivation for individuation.

scholarly journals Medidas de Tendência Central: Análise da Qualidade das Questões do ENEM de 2016 a 2018

2021 ◽  
Vol 14 (1) ◽  
pp. 119-128
Author(s):  
Talita Emidio Andrade Soares ◽  
Denilson Junio Marques Soares ◽  
Wagner Dos Santos
Keyword(s):  
High School ◽  
Quality Evaluation ◽  
Basic Education ◽  
Item Analysis ◽  
Arithmetic Mean ◽  
Central Tendency ◽  
Weighted Arithmetic ◽  
Evaluation Policies ◽  
Teaching Learning ◽  
Different Levels

ResumoO Exame Nacional do Ensino Médio (ENEM) tem se constituido como uma das principais políticas de avaliação da qualidade da educação básica brasileira. A expansão do exame e as atribuições a ele investidas tornaram imprescindíveis as discussões acerca da sua estrutura e qualidade. Com o intuito de contribuir para esta temática, este artigo tem o objetivo de analisar a qualidade de 11 itens de Matemática das provas do ENEM de 2016 a 2018, que se referem às Medidas de Tendência Central: média aritmética simples e ponderada, mediana e moda. Para isto, adotou-se uma metodologia predominantemente quantitativa e exploratória, desenvolvida a partir da aplicação de técnicas psicométricas a uma amostra representativa de participantes do exame. Os resultados apontaram para itens adequados e bem estruturados, que privilegiam o raciocínio lógico e a interpretação. A análise pedagógica indicou para a comum confusão entre os conceitos de média aritmética simples e ponderada, que representam a grande maioria dos itens analisadas. Espera-se que os resultados obtidos possam esclarecer alguns conceitos da psicometria e auxiliar no processo de ensino e aprendizagem de estatística na educação básica. Palavras-chave: Medidas de Tendência Central. Psicometria. Análise de Itens. ENEM. AbstractThe National High School Examination (ENEM) has become one of the most of the main Brazilian quality evaluation policies in the field of basic education. The expansion and attributions invested in the test make indispensable the discussions about its structure and quality. In order to contribute to this theme, we propose to analyze 11 Maths questions to ENEM from 2016 to 2018 and related to Measures of Central Tendency: arithmetic mean; weighted arithmetic mean, median and mode. It was used a quantitative and exploratory methodology, developed through psychometrics methods applied to a significant sample of students that made the exam. The results pointed to well-structured items, that favor logical reasoning and interpretation and that have an excellent ability discriminate groups with different levels of. Its highlight the confusion between concepts of arithmetic mean and weighted arithmetic mean, that represent the most of the questions analyzed. It is hoped that this research may help clarify some concepts of psychometry and can assist in the teaching-learning process of statistics in basic education.Keywords: Measures of Central Tendency. Psychometry. Item Analysis. ENEM.

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