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Are we telling the whole truth about the Standard Deviation?

Daniel Wolf-Root

Hutchesons’ Grammar School

wolfrootd@hutchesons.org

Scottish Mathematical Council Conference

16 May, 2026

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What is the whole story of the Standard Deviation?

  • My story on first teaching Standard Deviation in Scotland
  • Descriptive vs Inferential Statistics
  • How many Standard Deviations are there?
  • What do we divide by? Why?
  • Why not absolute values?
  • The equivalence of the two formulas for National 5
  • How is Standard Deviation used in AH Geography and Biology?
  • What story are we telling?
  • What do you think?

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wolfrootd@hutchesons.org

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My story on first teaching Standard Deviation in Scotland

wolfrootd@hutchesons.org

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What is the whole story of the Standard Deviation?

  • My story on first teaching Standard Deviation in Scotland
  • Descriptive vs Inferential Statistics
  • How many Standard Deviations are there?
  • What do we divide by? Why?
  • Why not absolute values?
  • The equivalence of the two formulas for National 5
  • How is Standard Deviation used in AH Geography and Biology?
  • What story are we telling?
  • What do you think?

​

wolfrootd@hutchesons.org

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Descriptive vs. Inferential Statistics

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Descriptive vs. Inferential Statistics

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wolfrootd@hutchesons.org

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What is the whole story of the Standard Deviation?

  • My story on first teaching Standard Deviation in Scotland
  • Descriptive vs Inferential Statistics
  • How many Standard Deviations are there?
  • What do we divide by? Why?
  • Why not absolute values?
  • The equivalence of the two formulas for National 5
  • How is Standard Deviation used in AH Geography and Biology?
  • What story are we telling?
  • What do you think?

​

wolfrootd@hutchesons.org

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What kinds of standard deviation are there?

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What is the whole story of the Standard Deviation?

  • My story on first teaching Standard Deviation in Scotland
  • Descriptive vs Inferential Statistics
  • How many Standard Deviations are there?
  • What do we divide by? Why?
  • Why not absolute values?
  • The equivalence of the two formulas for National 5
  • How is Standard Deviation used in AH Geography and Biology?
  • What story are we telling?
  • What do you think?

​

wolfrootd@hutchesons.org

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What do we divide by? Why?

  • N?
  • n?
  • (n-1)?

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  • It depends!
  • What is it you want from your standard deviation?

a measure of dispersion within your population

a measure of dispersion within your sample

an unbiased estimate of the population’s standard deviation from the sample

wolfrootd@hutchesons.org

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A simple example

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{3,4}

{3,5}

{4,5}

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What is the mean of the three possible sample variances?

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What is the mean of the three possible sample variances?

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What is the mean of the three possible sample variances?

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What is the mean of the three possible sample variances?

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What is the mean of the three possible sample variances?

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A biased estimate!

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Foster, C. 2023 “Questions Pupils Ask: Why do we divide by n-1?”, Mathematics in School 52(3), pp.20-22

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The unbiased estimate of the population standard deviation, from a sample

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Upton, G. and Cook, I. 1996 Understanding Statistics. Oxford.

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What did we show?

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What is the whole story of the Standard Deviation?

  • My story on first teaching Standard Deviation in Scotland
  • Descriptive vs Inferential Statistics
  • How many Standard Deviations are there?
  • What do we divide by? Why?
  • Why not absolute values?
  • The equivalence of the two formulas for National 5
  • How is Standard Deviation used in AH Geography and Biology?
  • What story are we telling?
  • What do you think?

​

wolfrootd@hutchesons.org

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Why not absolute values?

  • Big deviations
  • Small deviations
  • Differentiability
  • The Variance in Linear Regression
  • Moments

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Why not absolute values?

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Image from Open Intro Statistics (p 306)

Open Intro Statistics (2019) Diez, D. et al (p306) available at openintro.org/book/os

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Differentiability

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The Variance in Linear Regression

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Moments

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Hoel, P.G. 1948 Introduction to Mathematical Statistics, Wiley.

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Why not absolute values?

  • Big deviations
  • Small deviations
  • Differentiability
  • The Variance in Linear Regression
  • Moments

wolfrootd@hutchesons.org

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What is the whole story of the Standard Deviation?

  • My story on first teaching Standard Deviation in Scotland
  • Descriptive vs Inferential Statistics
  • How many Standard Deviations are there?
  • What do we divide by? Why?
  • Why not absolute values?
  • The equivalence of the two formulas for National 5
  • How is Standard Deviation used in AH Geography and Biology?
  • What story are we telling?
  • What do you think?

​

wolfrootd@hutchesons.org

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The two formulas in National 5

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The equivalence of the two formulas

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The equivalence of the two formulas

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wolfrootd@hutchesons.org

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What is the whole story of the Standard Deviation?

  • My story on first teaching Standard Deviation in Scotland
  • Descriptive vs Inferential Statistics
  • How many Standard Deviations are there?
  • What do we divide by? Why?
  • Why not absolute values?
  • The equivalence of the two formulas for National 5
  • How is Standard Deviation used in AH Geography and Biology?
  • What story are we telling?
  • What do you think?

​

wolfrootd@hutchesons.org

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The Standard Deviation in Geography and Biology

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AH Geography Course Specification:

SQA (2019) Advanced Higher Geography Course Specification. www.sqa.org.uk/sqa/48465.html

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The Standard Deviation in Geography and Biology

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What is the whole story of the Standard Deviation?

  • My story on first teaching Standard Deviation in Scotland
  • Descriptive vs Inferential Statistics
  • How many Standard Deviations are there?
  • What do we divide by? Why?
  • Why not absolute values?
  • The equivalence of the two formulas for National 5
  • How is Standard Deviation used in AH Geography and Biology?
  • What story are we telling?
  • What do you think?

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wolfrootd@hutchesons.org

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Standard Deviation: What story are we telling?

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The course specification National 5 Mathematics

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SQA (2023) National 5 Mathematics Course Specification. www.sqa.org.uk/sqa/47419.html

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Which standard deviation is covered?

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Textbook

n-1

n

Motivation or Reason for distinction

Heinemann (R3)

Y

Y

X

Zeta (N5+)

Y

X

X

Leckie and Leckie (N5)

Y

Y

Y

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Y

Y

Y

Hodder Gibson (N5)

Y

X

X

Teejay (N5)

Y

Y

X

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Teejay needs its own page

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Applications of Mathematics

  • Higher Course spec uses “sample measures of location and dispersion” including the standard deviation.
  • N5 course mentions standard deviation to “anaylse and compare data sets” (pages 7, 32). Formula sheet like N5, uses n-1 only.
  • Zeta National 5 Applications of Mathematics book uses n-1 and refers to the standard deviation of a “data set”. There is no distinction or explicit mention of a population or a sample. (pages 177-178). (National 5 Applications of Mathematics Course Specification May 2023).
  • I couldn’t find an explicit mention of sample or population or estimate in either course spec or in the textbook. It appears that all standard deviations in these courses are assumed to come from samples and be used as unbiased estimates of the population standard deviations.

wolfrootd@hutchesons.org

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What is the whole story of the Standard Deviation?

  • My story on first teaching Standard Deviation in Scotland
  • Descriptive vs Inferential Statistics
  • How many Standard Deviations are there?
  • What do we divide by? Why?
  • Why not absolute values?
  • The equivalence of the two formulas for National 5
  • How is Standard Deviation used in AH Geography and Biology?
  • What story are we telling?
  • What do you think?

​

wolfrootd@hutchesons.org

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Thank you!

  • Thank you to Dr. Clara Cohen for technology support
  • Thank you Mrs. Lucy Anderson who suggested there would be interest in this topic and for helpful feedback
  • To you for being here

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wolfrootd@hutchesons.org

wolfrootd@hutchesons.org

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References

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