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Create scatterplots and calculate the correlation coefficient r to describe linear relationships.
Learn step-by-step with practice exercises built right in.
Structure:
Reading a scatterplot:
Direction:
A scatterplot shows study hours (x-axis) vs. exam scores (y-axis) for 25 students. The points show a clear upward trend from lower-left to upper-right, tightly clustered around a line. Describe the relationship.
This scatterplot shows a strong positive linear correlation:
The correlation coefficient would be close to (e.g., ), indicating a strong positive association. Students who study more score higher; the pattern is predictable.
Avoid these 3 frequent errors
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Strength:
Formula:
Properties:
Interpretation of |r|:
Measures linear association only
Sensitive to outliers
r only quantifies strength, not causation
Restricted range reduces r
Example: Ice cream sales and drowning deaths
Possible explanations for correlation:
Rule: Correlation suggests association; prove causation with randomized experiment, not observational data.
Data: 6 students, hours studied (x) vs. exam score (y)
| Hours | Score |
|---|---|
| 2 | 65 |
| 3 | 78 |
| 4 | 82 |
| 5 | 88 |
| 6 | 90 |
| 8 | 95 |
Interpretation: Strong positive correlation suggests more study hours associated with higher scores.
When asked about relationship:
Example: "There is a strong positive correlation (r โ 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."
Two variables have correlation . Interpret what this means about their relationship.
Interpretation of :
Correlation direction: Negative () โ as one variable increases, the other tends to decrease.
Correlation strength: Moderate to strong (absolute value 0.65 is closer to โ1 than to 0).
Visual pattern: Scatterplot would show points trending downward (from upper-left to lower-right) with moderate scatter โ not perfectly linear, but a clear downward tendency.
Example: Temperature (x) vs. heating costs (y) might show . As temperature rises, heating costs fall. The relationship is clear but not perfect (other factors like insulation affect costs).
Important: describes association, not causation. The two variables move together, but one doesn't necessarily cause the other.
Two scatterplots are shown: Plot A has with points close to a line, but there are three extreme points far from the line at the upper-right corner. Plot B has with all points evenly scattered. Explain the correlation coefficient difference and discuss which might be a better summary.
What the correlations show:
Both indicate strong positive linear relationships (both r > 0.8). But they tell different stories.
Plot A (r = 0.92):
Plot B (r = 0.85):
Which is a better summary?
Neither r alone is sufficient. Always examine the scatterplot visually:
Lesson: Correlation is vulnerable to outliers. Always make and examine scatterplots; don't rely on r alone. Correlation โ causation anyway.