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An overview of the elementary statistics of correlation, R-squared, cosine, sine, and regression through the origin, with application to votes and seats for Parliament

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  • Colignatus, Thomas

Abstract

The correlation between two vectors is the cosine of the angle between the centered data. While the cosine is a measure of association, the literature has spent little attention to the use of the sine as a measure of distance. A key application of the sine is a new “sine-diagonal inequality / disproportionality” (SDID) measure for votes and their assigned seats for parties for Parliament. This application has nonnegative data and uses regression through the origin (RTO) with non-centered data. Textbooks are advised to discuss this case because the geometry will improve the understanding of both regression and the distinction between descriptive statistics and statistical decision theory. Regression may better be introduced and explained by looking at the angles relevant for a vector and its estimate rather than looking at the Euclidean distance and the sum of squared errors. The paper provides an overview of the issues involved. Also a new relation between the sine and the Euclidean distance is derived.

Suggested Citation

  • Colignatus, Thomas, 2018. "An overview of the elementary statistics of correlation, R-squared, cosine, sine, and regression through the origin, with application to votes and seats for Parliament," MPRA Paper 84722, University Library of Munich, Germany, revised 20 Feb 2018.
    • Handle: RePEc:pra:mprapa:84722
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    References listed on IDEAS

    as
    1. Colignatus, Thomas, 2010. "Single vote multiple seats elections. Didactics of district versus proportional representation, using the examples of the United Kingdom and The Netherlands," MPRA Paper 22671, University Library of Munich, Germany, revised 06 Jul 2007.
    2. Colignatus, Thomas, 2018. "Measures of policy distance and inequality / disproportionality of votes and seats," MPRA Paper 84324, University Library of Munich, Germany, revised 02 Feb 2018.
    3. Gelman, Andrew & Stern, Hal, 2006. "The Difference Between," The American Statistician, American Statistical Association, vol. 60, pages 328-331, November.
    4. Colignatus, Thomas, 2017. "One woman, one vote. Though not in the USA, UK and France," MPRA Paper 82513, University Library of Munich, Germany, revised 08 Nov 2017.
    5. Colignatus, Thomas, 2007. "Correlation and regression in contingency tables. A measure of association or correlation in nominal data (contingency tables), using determinants," MPRA Paper 3394, University Library of Munich, Germany, revised 07 Jun 2007.
    6. Colignatus, Thomas, 2017. "Two conditions for the application of Lorenz curve and Gini coefficient to voting and allocated seats," MPRA Paper 80297, University Library of Munich, Germany, revised 21 Jul 2017.
    7. Colignatus, Thomas, 2018. "Comparing the Aitchison distance and the angular distance for use as inequality or disproportionality measures for votes and seats," MPRA Paper 84334, University Library of Munich, Germany, revised 03 Feb 2018.
    8. Moshe Koppel & Abraham Diskin, 2009. "Measuring disproportionality, volatility and malapportionment: axiomatization and solutions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 33(2), pages 281-286, August.
    9. Colignatus, Thomas, 2007. "A measure of association (correlation) in nominal data (contingency tables), using determinants," MPRA Paper 2662, University Library of Munich, Germany, revised 10 Apr 2007.
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    Cited by:

    1. Colignatus, Thomas, 2020. "Forum Theory & A National Assembly of Science and Learning," MPRA Paper 98568, University Library of Munich, Germany, revised 09 Feb 2020.

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    More about this item

    Keywords

    General Economics; Social Choice; Social Welfare; Election; Parliament; Party System; Representation; Sine Diagonal Inequality / Disproportionality; SDID; Proportion; District; Voting; Seat; Euclid; Distance; Cosine; Sine; Gallagher; Loosemore-Hanby; Sainte-Laguë; Webster; Jefferson; Hamilton; Largest Remainder; Correlation; Diagonal regression; Regression through the origin; Apportionment; Disproportionality; Equity; Inequality;
    All these keywords.

    JEL classification:

    • A10 - General Economics and Teaching - - General Economics - - - General
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations
    • D72 - Microeconomics - - Analysis of Collective Decision-Making - - - Political Processes: Rent-seeking, Lobbying, Elections, Legislatures, and Voting Behavior

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