Statistics as unbiased estimators: exploring the teaching of standard deviation

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.

KEYWORDS:

• Standard deviation
• unbiased estimator
• statistical knowledge for teaching

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Joe Champion http://orcid.org/0000-0003-0788-282X

Notes

1. We say “potentially” different formulas because, for a normal distribution, the formulas for the maximum likelihood estimator of standard deviation from a sample and the population standard deviation are the same.

2. Two participants did not respond to the interview query relevant for the Post-interview, and thus were removed from the post-UEL analysis.

3. In fact, it makes sense to explore other possibilities at times: division by n-1.5 in a normal distribution provides an estimate for standard deviation with even less bias than n-1 (e.g. Bolch, 1968; Brugger, 1969).