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Teacher Guide: Correlation Coefficient

Learn how to measure the strength and direction of a linear relationship between two variables using the correlation coefficient r.

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Correlation. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define the correlation coefficient and explain what it measures
  • Calculate the correlation coefficient rr for a dataset
  • Interpret the sign and magnitude of rr in context
  • Distinguish between correlation and causation
  • Recognize limitations of rr for non-linear relationships
Prerequisites
  • • Understanding of scatter plots and plotting ordered pairs
  • • Ability to calculate mean (average)
  • • Familiarity with squared values and square roots
  • • Basic understanding of linear relationships
Discussion Starters
  • 1. If two variables have r=0.95r = 0.95, can we conclude that one causes the other? Why or why not?
  • 2. Why might it be misleading to report only the correlation coefficient without showing a scatter plot?
  • 3. A study finds r=0.85r = 0.85 between shoe size and reading ability in elementary students. What explains this correlation?
  • 4. How would you explain to someone that a perfect correlation of r=1r = 1 is rare in real-world data?
Common Misconceptions

A correlation of 0.5 means the variables are 50% related

Remediation: Explain that rr is not a percentage. Instead, r2r^2 (coefficient of determination) gives the proportion of variance explained. For r=0.5r = 0.5, only 0.250.25 or 25% of variance is explained.

Negative correlation means no relationship

Remediation: Use concrete examples: As altitude increases, temperature decreases (r<0r < 0). This is still a strong, predictable relationship - just inverse.

Differentiation Ideas

For Struggling Students:

  • • Focus on interpreting given rr values before calculating
  • • Use technology (calculator/spreadsheet) for computation
  • • Provide scatter plots with clearly positive, negative, or no correlation

For On-Level Students:

  • • Calculate rr by hand for small datasets (5-6 points)
  • • Interpret rr in various real-world contexts
  • • Analyze the effect of outliers on correlation

For Advanced Students:

  • • Explore the coefficient of determination r2r^2
  • • Compare Pearson and Spearman correlation for ranked data
  • • Investigate partial correlation controlling for third variables
Standards Alignment
  • HSS.ID.C.8 (CCSS.MATH.CONTENT.HSS.ID.C.8)

    Compute (using technology) and interpret the correlation coefficient of a linear fit

  • HSS.ID.C.9 (CCSS.MATH.CONTENT.HSS.ID.C.9)

    Distinguish between correlation and causation

Lesson Resources
  • visualInteractive Scatter Plot

    Students add points and see rr update in real-time

  • activityCorrelation Guessing Game

    Estimate rr from scatter plots before revealing the value

  • worksheetCorrelation vs Causation Scenarios

    Identify whether scenarios show correlation, causation, or neither

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

The correlation coefficient (denoted rr) measures the strength and direction of a linear relationship between two variables.
r=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2⋅∑(yi−yˉ)2r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \cdot \sum(y_i - \bar{y})^2}}
**Key properties of rr:**
  • rr is always between −1-1 and 11: −1≤r≤1-1 \le r \le 1
  • r=1r = 1: Perfect positive correlation
  • r=−1r = -1: Perfect negative correlation
  • r=0r = 0: No linear correlation
**Interpreting ∣r∣|r|:**
Value of $rStrength
0.00−0.190.00 - 0.19Very weak
0.20−0.390.20 - 0.39Weak
0.40−0.590.40 - 0.59Moderate
0.60−0.790.60 - 0.79Strong
0.80−1.000.80 - 1.00Very strong

Worked Examples

Find the correlation coefficient for the data: (1,2),(2,4),(3,5),(4,4),(5,5)(1, 2), (2, 4), (3, 5), (4, 4), (5, 5)
1

Find the means xˉ\bar{x} and yˉ\bar{y}

xˉ=1+2+3+4+55=3\bar{x} = \frac{1+2+3+4+5}{5} = 3, yˉ=2+4+5+4+55=4\bar{y} = \frac{2+4+5+4+5}{5} = 4 → xˉ=3\bar{x} = 3, yˉ=4\bar{y} = 4
2

Calculate deviations (xi−xˉ)(x_i - \bar{x}) and (yi−yˉ)(y_i - \bar{y})

Point (1,2)(1,2): (−2,−2)(-2, -2) Point (2,4)(2,4): (−1,0)(-1, 0) Point (3,5)(3,5): (0,1)(0, 1) Point (4,4)(4,4): (1,0)(1, 0) Point (5,5)(5,5): (2,1)(2, 1) → Deviations calculated
3

Calculate products (xi−xˉ)(yi−yˉ)(x_i - \bar{x})(y_i - \bar{y})

(−2)(−2)=4(-2)(-2) = 4 (−1)(0)=0(-1)(0) = 0 (0)(1)=0(0)(1) = 0 (1)(0)=0(1)(0) = 0 (2)(1)=2(2)(1) = 2 Sum =4+0+0+0+2=6= 4 + 0 + 0 + 0 + 2 = 6 → ∑(xi−xˉ)(yi−yˉ)=6\sum(x_i - \bar{x})(y_i - \bar{y}) = 6
4

Calculate ∑(xi−xˉ)2\sum(x_i - \bar{x})^2

(−2)2+(−1)2+02+12+22=4+1+0+1+4=10(-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 = 4 + 1 + 0 + 1 + 4 = 10 → ∑(xi−xˉ)2=10\sum(x_i - \bar{x})^2 = 10
5

Calculate ∑(yi−yˉ)2\sum(y_i - \bar{y})^2

(−2)2+02+12+02+12=4+0+1+0+1=6(-2)^2 + 0^2 + 1^2 + 0^2 + 1^2 = 4 + 0 + 1 + 0 + 1 = 6 → ∑(yi−yˉ)2=6\sum(y_i - \bar{y})^2 = 6
6

Apply the formula

r=610⋅6=660=67.746≈0.775r = \frac{6}{\sqrt{10 \cdot 6}} = \frac{6}{\sqrt{60}} = \frac{6}{7.746} \approx 0.775 → r≈0.77r \approx 0.77

Common Mistakes

Assuming correlation implies causation

Why it's wrong: A strong correlation between AA and BB doesn't mean AA causes BB. There could be a third variable affecting both, or the relationship could be coincidental.

Correct: Always say "associated with" or "correlated with" rather than "causes." Look for experimental evidence to establish causation.

Ignoring outliers when interpreting rr

Why it's wrong: A single outlier can dramatically change the correlation coefficient. rr is sensitive to extreme values.

Correct: Always plot the data first. Look for outliers and consider calculating rr with and without them.

Using rr for non-linear relationships

Why it's wrong: rr only measures linear relationships. A perfect curved pattern could have r=0r = 0.

Correct: Examine the scatter plot. If the pattern is curved, rr is not the right measure. Consider transformations or non-linear models.

Why It Matters

The correlation coefficient is fundamental in data analysis:
  • Science: Researchers measure how strongly variables like temperature and ice cream sales relate
  • Finance: Analysts track how stock prices move together (portfolio diversification)
  • Medicine: Doctors study relationships between risk factors and health outcomes
  • Sports: Coaches analyze which statistics best predict team success
Understanding rr helps you:
  • Quantify relationships that "look" linear in scatter plots
  • Make predictions using regression analysis
  • Evaluate the reliability of statistical models
  • Communicate findings precisely ("strong positive correlation" vs "weak negative correlation")

Real World Applications

Stock Market Diversification

Investment analysts use correlation to build diversified portfolios. Stocks with low or negative correlation balance risk.

Example:

If Stock A and Stock B have r=−0.3r = -0.3, when A drops, B tends to rise slightly, reducing portfolio volatility.

1Try It Yourself

Two tech stocks have r=0.92r = 0.92. Two stocks from different sectors have r=0.15r = 0.15.

Which pair provides better diversification?

Step 1: Write the mathematical expression

Compare: ∣0.92∣|0.92| vs ∣0.15∣|0.15|

Medical Research

Researchers use correlation to identify potential risk factors for diseases.

Example:

A study finds r=0.68r = 0.68 between blood pressure and risk of heart disease, indicating a strong positive relationship.

2Try It Yourself

Three health factors show correlations with diabetes risk: Exercise (r=−0.45r = -0.45), Sugar intake (r=0.52r = 0.52), Sleep quality (r=−0.31r = -0.31).

Which factor has the strongest relationship with diabetes risk?

Step 1: Write the mathematical expression

Compare absolute values: ∣−0.45∣|-0.45|, ∣0.52∣|0.52|, ∣−0.31∣|-0.31|

Key Takeaways

  • 1The correlation coefficient rr measures the strength and direction of a linear relationship
  • 2rr ranges from −1-1 (perfect negative) to +1+1 (perfect positive), with 00 indicating no linear relationship
  • 3The absolute value ∣r∣|r| indicates strength: closer to 11 is stronger, closer to 00 is weaker
  • 4Always visualize data with a scatter plot before interpreting rr
  • 5Correlation does not imply causation - association is not the same as cause-and-effect

Frequently Asked Questions

What is the difference between correlation and causation?

Correlation shows that two variables move together, but causation means one variable directly causes changes in another. Ice cream sales and drowning rates are correlated (both increase in summer), but ice cream doesn't cause drowning - hot weather affects both.

Can rr be greater than 1 or less than -1?

No. By mathematical definition, rr is always between −1-1 and 11. If you calculate a value outside this range, there's an error in your calculations.

What does r=0r = 0 mean?

r=0r = 0 means there is no linear relationship between the variables. However, there could still be a non-linear relationship (like a parabola or sine curve).

Glossary

Correlation coefficient (rr)
A numerical measure from −1-1 to +1+1 that quantifies the strength and direction of a linear relationship between two variables
Positive correlation
When r>0r > 0: as one variable increases, the other tends to increase
Negative correlation
When r<0r < 0: as one variable increases, the other tends to decrease
Pearson correlation
The most common type of correlation coefficient, measuring linear relationships (also called Pearson's rr)
Scatter plot
A graph showing points for paired data, used to visualize relationships between variables

Formula Card

Correlation Coefficient Formula

r=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2⋅∑(yi−yˉ)2r = \frac{\sum(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i - \bar{x})^2 \cdot \sum(y_i - \bar{y})^2}}

Alternative form: $r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$

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