Kenneth ChengMathematical modeling
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Correlation of paired data

Read the idea, work independently, then explain what changed.

高三選擇性必修 第三册(A版).pdf · 8.1 · PDF 98 / printed page 93

TOPIC 01

Correlation of paired data

Interpret scatter patterns and compute a correlation from centered paired summaries.

What you will be able to explain

  • Interpret scatter patterns and compute a correlation from centered paired summaries.
  • Justify the method and check the conditions in a new situation.

Defining relation

Interpret scatter patterns and compute a correlation from centered paired summaries.

r=Sxy/SxxSyyr=S_{xy}/\sqrt{S_{xx}S_{yy}}

Conditions

Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Can a clear U-shaped relation have zero Pearson correlation?

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Synthetic data: slope=2, intercept=1, r=1. Correlation alone does not establish causation.

Synthetic data: slope=2, intercept=1, r=1. Correlation alone does not establish causation.

Explain: Calculate two valid cases and explain the change using the defining relation.

Transfer: Compare a symmetric U-shaped scatter with a straight-line scatter.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

Find r from centered summaries.

Sxx=4,Syy=4,Sxy=2S_{xx}=4,\quad S_{yy}=4,\quad S_{xy}=2
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
r=Sxy/SxxSyyr=S_{xy}/\sqrt{S_{xx}S_{yy}}
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    r=2/(4)=1/2r=2/(4)=1/2

  3. The positive square root gives the denominator scale.

The requested value is 0.5.

Checks and common pitfalls: The positive square root gives the denominator scale.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Explain whether adding t to all y-values changes r.

t=3t=3
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
(yi+t)−(yˉ+t)=yi−yˉ(y_i+t)-(\bar y+t)=y_i-\bar y
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    Sxy,Syy unchangedS_{xy},S_{yy}\text{ unchanged}

  3. Correlation is invariant under translation because centered deviations are unchanged.

The requested relation or conclusion is shown below.

r unchangedr\text{ unchanged}

Checks and common pitfalls: Correlation is invariant under translation because centered deviations are unchanged.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Can Pearson r be computed when all x-values equal t? Explain.

xi=4x_i=4
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
Sxx=0S_{xx}=0
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    r=Sxy/0⋅Syyr=S_{xy}/\sqrt{0\cdot S_{yy}}

  3. A constant variable has no variation against which to measure linear correlation.

The requested relation or conclusion is shown below.

r undefinedr\text{ undefined}

Checks and common pitfalls: A constant variable has no variation against which to measure linear correlation.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Find r from centered summaries.

Sxx=4,Syy=25,Sxy=5S_{xx}=4,\quad S_{yy}=25,\quad S_{xy}=5
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
r=Sxy/SxxSyyr=S_{xy}/\sqrt{S_{xx}S_{yy}}
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    r=5/(10)=1/2r=5/(10)=1/2

  3. The positive square root gives the denominator scale.

The requested value is 0.5.

Checks and common pitfalls: The positive square root gives the denominator scale.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Find r when y=tx+2 and x varies.

t=6t=6
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
Sxy=tSxx;Syy=t2SxxS_{xy}=tS_{xx};\quad S_{yy}=t^2S_{xx}
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    r=t/∣t∣=1r=t/|t|=1

  3. The positive slope gives perfect positive linear correlation.

The requested value is 1.

Checks and common pitfalls: The positive slope gives perfect positive linear correlation.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

Find r when y=−tx+2 and x varies.

t=7t=7
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
Sxy=−tSxxS_{xy}=-tS_{xx}
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    r=−1r=-1

  3. Correlation records direction, not just closeness to a line.

The requested value is -1.

Checks and common pitfalls: Correlation records direction, not just closeness to a line.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

For pairs (−1,t),(0,0),(1,t), find Sxy.

t=8t=8
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
xˉ=0;Sxy=∑xi(yi−yˉ)\bar x=0;\quad S_{xy}=\sum x_i(y_i-\bar y)
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    Sxy=−1(8−16/3)+1(8−16/3)=0S_{xy}=-1(8-16/3)+1(8-16/3)=0

  3. Symmetry cancels the centered products even though the pattern is nonlinear.

The requested value is 0.

Checks and common pitfalls: Symmetry cancels the centered products even though the pattern is nonlinear.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

For pairs (−1,t),(0,0),(1,t), find Sxy.

t=9t=9
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
xˉ=0;Sxy=∑xi(yi−yˉ)\bar x=0;\quad S_{xy}=\sum x_i(y_i-\bar y)
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    Sxy=−1(9−18/3)+1(9−18/3)=0S_{xy}=-1(9-18/3)+1(9-18/3)=0

  3. Symmetry cancels the centered products even though the pattern is nonlinear.

The requested value is 0.

Checks and common pitfalls: Symmetry cancels the centered products even though the pattern is nonlinear.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Explain whether adding t to all y-values changes r.

t=10t=10
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
(yi+t)−(yˉ+t)=yi−yˉ(y_i+t)-(\bar y+t)=y_i-\bar y
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    Sxy,Syy unchangedS_{xy},S_{yy}\text{ unchanged}

  3. Correlation is invariant under translation because centered deviations are unchanged.

The requested relation or conclusion is shown below.

r unchangedr\text{ unchanged}

Checks and common pitfalls: Correlation is invariant under translation because centered deviations are unchanged.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

An observational survey reports the displayed positive correlation between study time and score. Does it prove a causal effect?

r=0.4r=0.4
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Consider confounding and study design.
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    Prior preparation may influence both variables.\text{Prior preparation may influence both variables.}

  3. The size of an observational correlation does not replace a causal design or justified control of confounding.

The requested relation or conclusion is shown below.

association does not establish causation\text{association does not establish causation}

Checks and common pitfalls: The size of an observational correlation does not replace a causal design or justified control of confounding.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

Can Pearson r be computed when all x-values equal t? Explain.

xi=12x_i=12
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
Sxx=0S_{xx}=0
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    r=Sxy/0⋅Syyr=S_{xy}/\sqrt{0\cdot S_{yy}}

  3. A constant variable has no variation against which to measure linear correlation.

The requested relation or conclusion is shown below.

r undefinedr\text{ undefined}

Checks and common pitfalls: A constant variable has no variation against which to measure linear correlation.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

An observational survey reports the displayed positive correlation between study time and score. Does it prove a causal effect?

r=0.6r=0.6
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Consider confounding and study design.
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    Prior preparation may influence both variables.\text{Prior preparation may influence both variables.}

  3. The size of an observational correlation does not replace a causal design or justified control of confounding.

The requested relation or conclusion is shown below.

association does not establish causation\text{association does not establish causation}

Checks and common pitfalls: The size of an observational correlation does not replace a causal design or justified control of confounding.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

Can Pearson r be computed when all x-values equal t? Explain.

xi=14x_i=14
  • Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the paired-data summaries, preserving which observations belong together.
Hint 2
Use this intermediate relation.
Sxx=0S_{xx}=0
Worked solution

  1. Use the paired-data summaries, preserving which observations belong together.

  2. Apply the stated relation and retain its conditions.

    r=Sxy/0⋅Syyr=S_{xy}/\sqrt{0\cdot S_{yy}}

  3. A constant variable has no variation against which to measure linear correlation.

The requested relation or conclusion is shown below.

r undefinedr\text{ undefined}

Checks and common pitfalls: A constant variable has no variation against which to measure linear correlation.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • Interpret scatter patterns and compute a correlation from centered paired summaries.
    • Which condition is essential in correlation of paired data?
    • Can a clear U-shaped relation have zero Pearson correlation?

    Board plan

    • Defining relation: Interpret scatter patterns and compute a correlation from centered paired summaries.
      r=Sxy/SxxSyyr=S_{xy}/\sqrt{S_{xx}S_{yy}}
    • Conditions: Both centered sums of squares must be positive; correlation measures linear association and does not establish causation.

    Anticipated thinking

    • Zero correlation does not rule out a nonlinear relationship.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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    Curriculum and source notes ↗