Scatterplots and Correlation

Scatterplots

Structure:

• x-axis: explanatory (independent) variable
• y-axis: response (dependent) variable
• Each point: one observation with both x and y values
• Use: explore relationship between two quantitative variables

Reading a scatterplot:

1. Trend: do points generally move up or down?
2. Strength: how tightly clustered around trend?
3. Form: is pattern linear, curved, or no pattern?
4. Outliers: any isolated points?

Describing Relationships: Direction, Form, Strength

Direction:

• Positive association: as x increases, y tends to increase
• Negative association: as x increases, y tends to decrease
• No association: no discernible trend

Form:

• Linear: points cluster around straight line
• Curved: points follow curved pattern
• No pattern: points scattered randomly

Strength:

• Strong: points tightly clustered
• Moderate: visible trend with some scatter
• Weak: scatter with little visible trend

The Correlation Coefficient (\(r\))

Formula: $r = \frac{1}{n-1} \sum_{i=1}^{n} \left( \frac{x_i - \bar{x}}{s_x} \right) \left( \frac{y_i - \bar{y}}{s_y} \right)$

Properties:

• Range: −1 ≤ r ≤ 1
• r = 1: perfect positive linear relationship
• r = −1: perfect negative linear relationship
• r = 0: no linear relationship
• r > 0: positive association
• r < 0: negative association
• Unitless: r doesn't depend on units (inches vs. cm give same r)
• Symmetric: r(x, y) = r(y, x)

Interpretation of |r|:

• |r| ≥ 0.9: very strong
• 0.7 ≤ |r| < 0.9: strong
• 0.5 ≤ |r| < 0.7: moderate
• 0.3 ≤ |r| < 0.5: weak
• |r| < 0.3: very weak/negligible

Limitations of \(r\)

1. Measures linear association only

   • Two variables may have strong curved relationship but r ≈ 0
   • Always plot scatterplot; don't rely on r alone

2. Sensitive to outliers

   • One extreme point can dramatically change r
   • Example: With outlier, r might change from 0.2 to 0.8

3. r only quantifies strength, not causation

   • Strong r doesn't prove x causes y
   • Confounding variables often explain relationship

4. Restricted range reduces r

   • If data shows only part of potential relationship, r is weaker

Correlation ≠ Causation

Example: Ice cream sales and drowning deaths

• Both increase in summer
• Strong positive correlation (r ≈ 0.9)
• But neither causes the other; temperature is confounding variable

Possible explanations for correlation:

• Causation: x causes y (rare without experiment)
• Reverse causation: y causes x
• Confounding variable: third variable causes both x and y
• Coincidence: random correlation in unrelated variables

Rule: Correlation suggests association; prove causation with randomized experiment, not observational data.

Worked Example

Data: 6 students, hours studied (x) vs. exam score (y)

• \(\bar{x} = 4.67, s_x ≈ 1.97\)
• \(\bar{y} = 83, s_y ≈ 10.05\)
• \(r ≈ 0.97\) (very strong positive linear relationship)

Interpretation: Strong positive correlation suggests more study hours associated with higher scores.

Common Mistakes

1. Assuming r = 0.4 means no relationship: relationship exists; it's just weak
2. Claiming causation from strong r: r alone doesn't prove causation
3. Ignoring scatterplot: r = 0.5 could be weak linear + curved pattern (plot it!)
4. Confusing r and slope: different concepts; strong r doesn't mean steep slope

AP Exam Tip

When asked about relationship:

1. Describe scatterplot: direction, form, strength, outliers
2. Calculate r: state value (e.g., r ≈ 0.82)
3. Interpret r: "Strong positive linear correlation"
4. Caveat: "Correlation does not imply causation. A confounding variable such as _____ may explain the relationship."

Example: "There is a strong positive correlation (r ≈ 0.85) between hours studied and exam score. Students who study more tend to score higher. However, this does not prove studying causes higher scores; student motivation might influence both variables."