Bivariate Data

1. What is Bivariate Data?

Bivariate means "two variables." So, you're looking at data where two things are measured for each case—like height and weight of people, or hours studied and exam scores.

2. Identify EV and RV

• Explanatory Variable (EV): The variable you think influences the other (independent).

• Response Variable (RV): The variable that responds to the change in the EV (dependent).

🧠 Example:

• Hours studied (EV) vs Test score (RV)
  You think the number of hours you study affects your test score.

3. Construct a Scatter Plot

Plot each pair of values on a graph:

• x-axis: Explanatory Variable (EV)

• y-axis: Response Variable (RV)

Use graphing software or a CAS calculator to do this.

4. Interpret a Scatter Plot

• Direction:

  • Positive: As x increases, y increases.

  • Negative: As x increases, y decreases.

• Form:

  • Linear (straight-line pattern)

  • Non-linear (curved or no clear pattern)

• Strength:

  • Strong: points tightly clustered around a line

  • Weak: points widely scattered

5. Pearson’s Correlation Coefficient (r)

This number tells you:

• Direction: Positive or negative

• Strength of a linear relationship

Values range from:

• r = 1: Perfect positive linear relationship

• r = -1: Perfect negative linear relationship

• r = 0: No linear relationship

📈 Strong if |r| is close to 1
📉 Weak if |r| is close to 0

6. Using a CAS Calculator to Find r and r²

• Input your data as two lists (x-values and y-values).

• Use the linear regression tool to calculate:

  • r (correlation coefficient)

  • r² (coefficient of determination): % of variation in y explained by x.

🧠 Interpret r²
If r² = 0.85, then 85% of the variation in the response variable is explained by the explanatory variable.

7. Least Squares Regression Line (LSRL)

This is the best-fit line that minimizes the squared vertical distances from data points to the line.

Equation:

y=a+bxy = a + bxy=a+bx

Where:

• a = y-intercept (value of y when x = 0)

• b = slope (how much y increases per 1 unit increase in x)

Use your CAS to calculate this.

8. Make Predictions Using the LSRL

Plug an x-value into your regression equation to predict y.

🧠 Example:
If the equation is y=50+10xy = 50 + 10xy=50+10x, then:

• If x = 3 → y=50+10(3)=80y = 50 + 10(3) = 80y=50+10(3)=80

9. Interpret Slope and Intercept

• Slope (b): How much the response variable changes per 1 unit increase in the explanatory variable.

• Intercept (a): Predicted value of y when x = 0. (Sometimes meaningful, sometimes not—depends on the context.)