Data Associations

Grade 11 · statistics · 100 practice problems · read aloud

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📊 Data Associations: Correlation & Causation

What is it and why is it useful?

Data associations help us understand relationships between two quantitative variables. We analyze if changes in one variable are related to changes in another. This is crucial for identifying trends, making predictions, and understanding real-world connections—like studying hours vs. test scores. However, a key concept is that correlation does not imply causation.

Step-by-Step Guide

  1. Collect Data: Gather paired data values (e.g., (x₁, y₁), (x₂, y₂)...).
  2. Visualize: Create a scatter plot to see the relationship's direction and form.
  3. Describe: Identify the direction (positive/negative), form (linear/non-linear), and strength (strong/weak).
  4. Calculate: Compute the correlation coefficient (r) to quantify the strength and direction.
  5. Interpret: Explain what the relationship means in context, being careful not to assume cause and effect.

Visual Examples

Example 1: Study Time vs. Test Score

Data: (1, 60), (2, 70), (3, 75), (4, 85)

Steps:

  1. Plot points: Scatter plot shows an upward trend.
  2. Direction: Positive (as study time increases, score increases).
  3. Strength: Strong (points are close to an imaginary line).
  4. Interpretation: There is a positive association, but we cannot say studying more causes higher scores (other factors like prior knowledge could be involved).

Example 2: Ice Cream Sales vs. Drowning Incidents

Data shows a strong positive correlation.

Interpretation: While r is high, this is a spurious correlation. A lurking variable—hot weather—causes both to increase. Ice cream sales do not cause drownings.

🚨 Common Mistakes

  • Confusing Correlation & Causation: Just because two things are related doesn't mean one causes the other. Always consider lurking variables.
  • Ignoring Outliers: A single outlier can significantly impact the correlation coefficient (r). Always look at the scatter plot!
  • Assuming Linearity: A low r-value doesn't always mean "no relationship"—it might just be a non-linear one.

💡 Tips & Tricks

  • Memory Aid: "CORRelation = CONNECTION, not CAUSE."
  • r-value Guide: |r| > 0.8 = Strong, |r| ~ 0.5 = Moderate, |r| < 0.3 = Weak.
  • Always "zoom out" on your scatter plot mentally. Could a third variable be influencing both?

How to Practice

  • Find real-world bivariate data sets online (sports stats are great) and create your own scatter plots.
  • Practice calculating the correlation coefficient (r) by hand for small data sets to understand the formula.
  • Play "correlation or causation?" with news headlines. For example, "People who floss daily live longer" – is this correlation or causation?

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 log₂(16) + sin(π/2) = ?

Hint: Consider the fundamental properties of logarithms and trigonometric functions at special angles. For example, log base a of a raised to n equals n, and trigonometric functions have known values at common angles like 0, π/6, π/4, π/3, and π/2.

Show the answer

Answer: 5

  1. Evaluate log₂(16)** We ask: "2 raised to what power gives 16?" 2^4 = 16, so log₂(16) = 4. --- **
  2. Evaluate sin(π/2)** π/2 radians is 90 degrees. sin(90°) = 1. So sin(π/2) = 1. --- **
  3. Add the results** 4 + 1 = 5. --- **Final Answer:** 5

Let's solve step by step. We have: log₂(16) + sin(π/2) = ? --- **

2 log₂(32) - sin(π/2) = ?

Hint: Consider what power you need to raise 2 to in order to get the first number, and recall the value of sine at π/2 radians.

Show the answer

Answer: 4

  1. Evaluate log₂(32). Since 2^5 = 32, log₂(32) = 5.
  2. Evaluate sin(π/2). The sine of π/2 radians is 1.
  3. Subtract the results: 5 - 1 = 4.

The answer is 4.

3 log₂(32) + cos(π/3) = ?

Hint: Consider the exponent needed to raise 2 to get the first term, and recall the exact value of cosine for common angles.

Show the answer

Answer: 5.5

  1. Evaluate log₂(32). Since 2^5 = 32, log₂(32) = 5.
  2. Evaluate cos(π/3). The cosine of π/3 radians (60 degrees) is 1/2.
  3. Add the two results: 5 + 1/2 = 5.5.

The answer is 5.5.

4 log₃(27) + sin(π/2) - 2³ = ?

Hint: Evaluate each term separately: the logarithm asks what power gives the base, the trigonometric function at a special angle, and the exponential term.

Show the answer

Answer: -4

  1. Evaluate log₃(27). Since 3³ = 27, log₃(27) = 3.
  2. Evaluate sin(π/2). The sine of π/2 radians is 1.
  3. Evaluate 2³. This is 2 × 2 × 2 = 8.
  4. Substitute the values into the expression: 3 + 1 - 8.
  5. Perform the addition and subtraction: 3 + 1 = 4, then 4 - 8 = -4.

The answer is -4.

5 log₃(27) + sin(π/2) - 2⁴ = ?

Hint: Evaluate each term separately: the logarithm asks what power gives the base, the trigonometric function at a special angle, and the exponential expression, then combine them in order.

Show the answer

Answer: -12

  1. Evaluate log₃(27). Since 3³ = 27, log₃(27) = 3.
  2. Evaluate sin(π/2). The sine of π/2 radians is 1.
  3. Evaluate 2⁴. 2 raised to the power of 4 is 16.
  4. Substitute the values back into the expression: 3 + 1 - 16.
  5. Perform the addition first: 3 + 1 = 4.
  6. Then perform the subtraction: 4 - 16 = -12. The final answer is -12.

6 log₃(81) + sin(π/2) - 2⁴ = ?

Hint: Evaluate each term separately: the logarithm asks what power gives the base, the sine function at π/2 radians, and the exponentiation of 2 to the fourth power.

Show the answer

Answer: -11

  1. Evaluate log₃(81). Since 3⁴ = 81, log₃(81) = 4.
  2. Evaluate sin(π/2). The sine of π/2 radians is 1.
  3. Evaluate 2⁴. 2 × 2 × 2 × 2 = 16.
  4. Substitute the values into the expression: 4 + 1 - 16.
  5. Perform the addition and subtraction: 4 + 1 = 5, then 5 - 16 = -11.

The answer is -11.

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