Complete lesson: scatterplots, linear/nonlinear models, correlation, line of best fit. Rules, strategies, solved examples, and 10 practice questions.
📖 What Is Two-Variable Data?
Two-variable data examines the relationship between two variables. We use scatterplots and lines of best fit to analyze these relationships.
Scatterplot
A graph showing the relationship between two variables. Each point represents one data pair.
Positive Correlation
As x increases, y increases.
📈 Height → Weight
Negative Correlation
As x increases, y decreases.
📉 Study Time → Errors
No Correlation
No clear relationship between variables.
📊 Shoe Size → IQ
1. Scatterplots
📌 RULE: Reading Scatterplots
Step 1: Look at the axes — what do x and y represent? Step 2: Look at the overall pattern — is there a trend? Step 3: Identify the direction (positive/negative) and strength (strong/weak). Step 4: Look for outliers (points far from the pattern).
2. Correlation
📌 RULE: Types of Correlation
Positive Correlation: Points go up from left to right. \(x \uparrow \rightarrow y \uparrow\)
Negative Correlation: Points go down from left to right. \(x \uparrow \rightarrow y \downarrow\)
No Correlation: Points are scattered randomly.
Strong vs. Weak: The closer the points are to a straight line, the stronger the correlation.
💡 Strategy — Correlation
1. Look for the overall direction (up or down).
2. Check how tightly the points cluster around a line.
3. Correlation does NOT imply causation!
3. Line of Best Fit
📌 RULE: Line of Best Fit
The line of best fit (or trend line) is a straight line that best represents the data on a scatterplot.
\[
y = mx + b
\]
Use the line to make predictions. Substitute the x-value into the equation to find the predicted y-value.
📌 RULE: Residuals
A residual is the difference between the actual y-value and the predicted y-value from the line of best fit.
Positive residual: Actual is above the line. Negative residual: Actual is below the line.
📌 Key Concept
The line of best fit minimizes the sum of the squared residuals. You don't need to calculate the line yourself, but you need to interpret it.
📝 Solved Examples
Study these examples carefully. Each shows the step-by-step solution process.
📝 SOLVED EXAMPLE 1 — Reading a Scatterplot
A scatterplot shows the relationship between hours studied and test scores. As study hours increase, test scores also increase. What type of correlation is this?
Step 1: As x (study hours) increases, y (test scores) increases
Step 2: This is a positive correlation
✅ Positive correlation
💡 Tip: More study time → higher scores → positive relationship.
📝 SOLVED EXAMPLE 2 — Line of Best Fit
A line of best fit is \(y = 2x + 10\). What is the predicted y-value when \(x = 5\)?
💡 Tip: Positive residual = actual is above the line.
📝 SOLVED EXAMPLE 4 — Interpreting Slope
A line of best fit is \(y = 1.5x + 20\) where x = hours studied and y = test score. What does the slope 1.5 mean?
Step 1: Slope = 1.5 means for every 1 hour increase in study time...
Step 2: The test score increases by 1.5 points
✅ Each additional hour of study increases the predicted test score by 1.5 points
💡 Tip: Slope = rate of change. For each unit increase in x, y changes by the slope.
📝 SOLVED EXAMPLE 5 — Interpreting y-Intercept
A line of best fit is \(y = 1.5x + 20\). What does the y-intercept 20 mean?
Step 1: y-intercept = 20 is the predicted value when \(x = 0\)
Step 2: When \(x = 0\) hours studied, the predicted test score is 20
✅ The predicted test score with 0 hours of study is 20
💡 Tip: y-intercept is the starting value when x = 0.
🧪 Practice Questions
Solve each problem using the rules above. Click "Show Answer" to see the full solution.
Question 1
A scatterplot shows points going up from left to right. What type of correlation is this?
A) Negative correlation
B) Positive correlation
C) No correlation
D) Strong correlation
✓ Answer: B
Points going up from left to right indicate a positive correlation.
📝 Solution: Positive correlation
Question 2
A line of best fit is \(y = 3x + 5\). What is the predicted y-value when \(x = 4\)?
A) 12
B) 15
C) 17
D) 20
✓ Answer: C
\(y = 3(4) + 5 = 12 + 5 = 17\)
📝 Solution: \(3(4) + 5 = 17\)
Question 3
A line of best fit predicts \(y = 25\) for a data point. The actual value is 28. What is the residual?
A) -3
B) 0
C) 2
D) 3
✓ Answer: D
Residual = Actual - Predicted = 28 - 25 = 3
📝 Solution: \(28 - 25 = 3\)
Question 4
A line of best fit is \(y = 2.5x + 10\), where x = hours worked and y = money earned. What does the slope 2.5 mean?
A) The starting pay is $10
B) Each hour worked earns $10
C) Each hour worked earns $2.50
D) You need to work 2.5 hours
✓ Answer: C
Slope = 2.5 means for each additional hour worked, money earned increases by $2.50.
📝 Solution: Slope = rate of change = $2.50 per hour.
Question 5
A scatterplot shows points going down from left to right. What type of correlation is this?
A) Negative correlation
B) Positive correlation
C) No correlation
D) Strong correlation
✓ Answer: A
Points going down from left to right indicate a negative correlation.
📝 Solution: Negative correlation
Question 6
A line of best fit is \(y = 0.8x + 15\). What is the predicted y-value when \(x = 10\)?
A) 15
B) 18
C) 20
D) 23
✓ Answer: D
\(y = 0.8(10) + 15 = 8 + 15 = 23\)
📝 Solution: \(0.8(10) + 15 = 23\)
Question 7
A line of best fit is \(y = 4x - 2\), where x = number of items and y = total cost. What is the predicted total cost for 5 items?
A) 18
B) 18
C) 20
D) 22
✓ Answer: B
\(y = 4(5) - 2 = 20 - 2 = 18\)
📝 Solution: \(4(5) - 2 = 18\)
Question 8
A scatterplot shows points that are randomly scattered with no clear pattern. What type of correlation is this?
A) No correlation
B) Positive correlation
C) Negative correlation
D) Strong correlation
✓ Answer: A
Randomly scattered points with no clear pattern indicate no correlation.
📝 Solution: No correlation
Question 9
A line of best fit predicts \(y = 30\) for a data point. The actual value is 27. What is the residual?
A) -3
B) 0
C) 3
D) 30
✓ Answer: A
Residual = Actual - Predicted = 27 - 30 = -3 (point is below the line)
📝 Solution: \(27 - 30 = -3\)
Question 10
A line of best fit is \(y = 1.2x + 8\), where x = years of experience and y = salary in thousands. What does the y-intercept 8 mean?
A) Each year of experience adds $8,000
B) The salary increases by 8% each year
C) The predicted salary with 0 years experience is $8,000
D) The salary is always $8,000
✓ Answer: C
y-intercept = 8 means when x = 0 (0 years experience), the predicted salary is 8 (thousand) = $8,000.
📝 Solution: y-intercept = predicted value when x = 0.
🎉 WELL DONE!
You've completed the Two-Variable Data lesson. You now know how to read scatterplots, identify correlation, use lines of best fit, and interpret residuals.