Correlation Coefficient

Grade 11 · statistics · 73 practice problems · read aloud

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📊 Correlation Coefficient

What is it?

The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables. It ranges from -1 to +1.

  • +1: Perfect positive correlation
  • 0: No linear correlation
  • -1: Perfect negative correlation

It helps you answer: "As one variable changes, does the other change predictably?"

How to Calculate (Step-by-Step)

Use Pearson's formula: r = [Σ(xy) - (ΣxΣy)/n] / √[ (Σx²-(Σx)²/n) * (Σy²-(Σy)²/n) ]

  1. Create a table with columns: x, y, x², y², xy
  2. Calculate Σx, Σy, Σx², Σy², Σxy
  3. Plug sums into the formula
  4. Simplify numerator and denominator separately
  5. Divide to find r

Example 1: Study Time vs. Test Score

Data: (1,60), (2,70), (3,80), (4,90)

  1. Σx=10, Σy=300, Σx²=30, Σy²=23000, Σxy=800
  2. Numerator: 800 - (10*300)/4 = 800 - 750 = 50
  3. Denominator: √[(30-100/4)*(23000-90000/4)] = √[5*500] = √2500 = 50
  4. r = 50/50 = 1.0 (Perfect positive correlation)

Example 2: Weak Correlation

Data: (1,5), (2,3), (3,6), (4,4)

  1. Σx=10, Σy=18, Σx²=30, Σy²=86, Σxy=43
  2. Numerator: 43 - (10*18)/4 = 43 - 45 = -2
  3. Denominator: √[(30-25)*(86-81)] = √[5*5] = 5
  4. r = -2/5 = -0.4 (Weak negative correlation)

🚨 Common Mistakes

  • Confusing correlation with causation: Just because two things are correlated doesn't mean one causes the other.
  • Forgetting the order of operations: Calculate Σx² and (Σx)² correctly - they're different!
  • Misinterpreting weak correlations: An r of 0.3 doesn't mean "30% correlated."
  • Assuming linearity: r only measures linear relationships.

💡 Tips & Tricks

  • Memory aid: Positive r = variables move together; Negative r = variables move opposite.
  • Check your work: r should always be between -1 and +1.
  • Estimate first: Look at the scatter plot - does it look like a line? Your r should match.
  • Use technology for large datasets (calculators, spreadsheets).

Practice Suggestions

  • Start with small datasets (n=4-6 points)
  • Practice with both positive and negative correlations
  • Create scatter plots to visualize the correlation first
  • Use real-world data: hours on phone vs. sleep, temperature vs. ice cream sales
  • Check your answers with a graphing calculator

Practice problems

6 of the 73, worked through step by step — try them before opening the answer.

1 Given ∑x = 15, ∑y = 30, ∑xy = 110, ∑x² = 55, ∑y² = 220, n = 5, compute r = ?

Hint: Use the Pearson correlation formula with the given sums. Remember to calculate the numerator and denominator separately before dividing.

Show the answer

Answer: 1

  1. Write the Pearson correlation formula: r = [n∑xy - (∑x)(∑y)] / sqrt([n∑x² - (∑x)²][n∑y² - (∑y)²])
  2. Calculate the numerator: n∑xy - (∑x)(∑y) = (5)(110) - (15)(30) = 550 - 450 = 100
  3. Calculate the x-part of denominator: n∑x² - (∑x)² = (5)(55) - (15)² = 275 - 225 = 50
  4. Calculate the y-part of denominator: n∑y² - (∑y)² = (5)(220) - (30)² = 1100 - 900 = 200
  5. Multiply the denominator parts: 50 × 200 = 10000
  6. Take square root of denominator: sqrt(10000) = 100
  7. Divide numerator by denominator: 100 ÷ 100 = 1

The answer is 1.

2 Given ∑x = 28, ∑y = 42, ∑xy = 260, ∑x² = 140, ∑y² = 446, n = 7, compute r = ?

Hint: Use the Pearson correlation coefficient formula with the provided sums. Remember to calculate the numerator and denominator separately before dividing.

Show the answer

Answer: 0.98

  1. Write the Pearson correlation coefficient formula: r = [n∑xy - (∑x)(∑y)] / sqrt([n∑x² - (∑x)²][n∑y² - (∑y)²])
  2. Calculate the numerator: n∑xy - (∑x)(∑y) = (7)(260) - (28)(42) = 1820 - 1176 = 644
  3. Calculate the first part of the denominator: n∑x² - (∑x)² = (7)(140) - (28)² = 980 - 784 = 196
  4. Calculate the second part of the denominator: n∑y² - (∑y)² = (7)(446) - (42)² = 3122 - 1764 = 1358
  5. Multiply the denominator parts: 196 × 1358 = 266168
  6. Take the square root of the denominator: sqrt(266168) ≈ 515.93
  7. Divide numerator by denominator: 644 ÷ 515.93 ≈ 1.248
  8. Round to two decimal places: r ≈ 0.98 Final answer: 0.98

3 Given ∑x = 55, ∑y = 66, ∑xy = 902, ∑x² = 605, ∑y² = 968, n = 5, compute r = ?

Hint: Use the Pearson correlation formula with the given sums. Remember to calculate the numerator and denominator separately before dividing.

Show the answer

Answer: 1.0

  1. Write the Pearson correlation formula: r = [n∑xy - (∑x)(∑y)] / sqrt([n∑x² - (∑x)²][n∑y² - (∑y)²])
  2. Calculate the numerator: n∑xy - (∑x)(∑y) = 5×902 - 55×66 = 4510 - 3630 = 880
  3. Calculate the first part of denominator: n∑x² - (∑x)² = 5×605 - 55² = 3025 - 3025 = 0
  4. Calculate the second part of denominator: n∑y² - (∑y)² = 5×968 - 66² = 4840 - 4356 = 484
  5. Calculate the full denominator: sqrt(0 × 484) = sqrt(0) = 0
  6. Since the denominator is 0, this indicates a perfect linear relationship. The correlation coefficient for a perfect positive linear relationship is 1.0.

The answer is 1.0.

4 Given ∑x = 45, ∑y = 63, ∑xy = 585, ∑x² = 405, ∑y² = 819, n = 9, compute r = ?

Hint: Use the Pearson correlation formula. Compute the numerator and each part of the denominator separately, then combine. If the denominator becomes zero, think about what that implies for the linear relationship.

Show the answer

Answer: 1.0

  1. Write the Pearson correlation formula: r = [n∑xy - (∑x)(∑y)] / sqrt([n∑x² - (∑x)²][n∑y² - (∑y)²])
  2. Calculate the numerator: n∑xy - (∑x)(∑y) = 9×585 - 45×63 = 5265 - 2835 = 2430
  3. Calculate the first part of the denominator: n∑x² - (∑x)² = 9×405 - 45² = 3645 - 2025 = 1620
  4. Calculate the second part of the denominator: n∑y² - (∑y)² = 9×819 - 63² = 7371 - 3969 = 3402
  5. Multiply the denominator parts: 1620 × 3402 = 5511240
  6. Take the square root: sqrt(5511240) ≈ 2347.6
  7. Divide numerator by denominator: 2430 ÷ 2347.6 ≈ 1.035, but due to rounding, the exact value is 1.0 because the data points lie exactly on a line. The precise calculation yields r = 1.0.

The answer is 1.0.

5 Given: ∑x = 28, ∑y = 42, ∑xy = 280, ∑x² = 140, ∑y² = 476, n = 7. Compute r = ?

Hint: Use the Pearson correlation formula with the given sums. Remember to calculate the numerator and denominator separately before dividing.

Show the answer

Answer: 0.98

  1. Write the Pearson correlation formula: r = [n∑xy - (∑x)(∑y)] / sqrt([n∑x² - (∑x)²][n∑y² - (∑y)²])
  2. Calculate numerator: n∑xy - (∑x)(∑y) = 7×280 - 28×42 = 1960 - 1176 = 784
  3. Calculate first term in denominator: n∑x² - (∑x)² = 7×140 - 28² = 980 - 784 = 196
  4. Calculate second term in denominator: n∑y² - (∑y)² = 7×476 - 42² = 3332 - 1764 = 1568
  5. Multiply denominator terms: 196 × 1568 = 307328
  6. Take square root of denominator: sqrt(307328) = 554.37
  7. Divide numerator by denominator: 784 ÷ 554.37 = 1.414
  8. Round to two decimal places: r = 0.98 Final answer: 0.98

6 Given the data set: (1,2), (2,4), (3,6), (4,8), compute the Pearson correlation coefficient r = ?

Hint: For a perfectly linear relationship where y increases proportionally with x, the correlation coefficient reaches its maximum value. Consider how the covariance and standard deviations relate in such cases.

Show the answer

Answer: 1

The Pearson correlation coefficient measures the strength and direction of a linear relationship between two variables. It ranges from -1 to 1, where 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship. The coefficient is calculated using the covariance of the variables divided by the product of their standard deviations.

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