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Statistics as unbiased estimators: exploring the teaching of standard deviation

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Abstract

This manuscript presents findings from a study about the knowledge for and planned teaching of standard deviation. We investigate how understanding variance as an unbiased (inferential) estimator – not just a descriptive statistic for the variation (spread) in data – is related to teachers’ instruction regarding standard deviation, particularly around the issue of division by n-1. In this regard, the study contributes to our understanding about how knowledge of mathematics beyond the current instructional level, what we refer to as nonlocal mathematics, becomes important for teaching. The findings indicate that acquired knowledge of nonlocal mathematics can play a role in altering teachers’ planned instructional approaches in terms of student activity and cognitive demand in their instruction.

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... Formula untuk perhitungan root mean square error (RMSE) dapat ditunjukkan pada persamaan (1), (Li et al., 2015): Selain perhitungan root mean square error (RMSE), perkiraan nilai parameter standar deviasi juga penting dalam menentukan standar akurasi dari pekerjaan yang telah dilakukan dengan definisi sebagai nilai statistik dalam menentukan persebaran data dalam sampel, serta pendekatan titik data individu ke rata-rata nilai sampel (Hidayat et al., 2019). Standar deviasi dapat dilakukan perhitungan dengan menggunakan persamaan seperti dibawah ini (Wasserman et al., 2017): ...
... This paper is motivated by our teaching statistical inferences about the population standard deviation σ for undergraduate students (cf. [1], [4], [5]). The common rendition is to use the sample-variance s 2 to estimate σ 2 by the fact that the mathematical expectation E s 2 = σ 2 , thereby s to estimate σ, an approach that is so universally adopted that few studies exclusively on s have been found by our extensive literature research (see [4]). ...
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Ministério da Educação. (2006). National curricular guidelines for secondary school: Nature sciences, mathematics, and their technologies. Brasilia: Author.
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  • R Peck
  • R Gould
  • S Miller
  • R M Zbiek
Peck, R., Gould, R., Miller, S., & Zbiek, R. M. (2013). Developing essential understanding of statistics for teaching mathematics in grades 9-12. Reston, VA: NCTM.
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  • T Speed
Speed, T. (2013). Terrence's stuff: N versus n-1. IMS Bulletin, 42(1), 15.
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American Statistical Association. (2015). Statistical education of teachers. Alexandria, VA: Author.
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Common Core State Standards. (2010). Common core state standards (mathematics). Washington, DC: National Governors Association Center for Best Practices & Council of Chief State School Officers.
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Department of Education. (2002). Revised national curriculum statement grades R-9, mathematics. Pretoria: Author.
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National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.
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  • R K Yin
  • R Zazkis
  • A Mamolo
Yin, R. K. (2014). Case study research: Design and methods (5th ed.). Thousand Oaks, CA: Sage. Zazkis, R., & Mamolo, A. (2011). Reconceptualizing knowledge at the mathematical horizon. For the Learning of Mathematics, 31(2), 8-13.
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  • Educação Ministério Da