Bivariate Statistics
Analyse relationships between two variables using scatter plots, measure correlation strength, and use the regression line to make predictions.
Scatter Plots
Bivariate data involves two variables measured on the same subject. A scatter plot displays each pair of values as a point on a coordinate plane. The explanatory (independent) variable goes on the x-axis; the response (dependent) variable goes on the y-axis.
Explanatory Variable (x-axis)
The variable thought to explain or predict changes in the other. Example: hours of study.
Response Variable (y-axis)
The variable whose changes we are trying to explain or predict. Example: exam score.
Scatter plot with a strong positive linear relationship
Correlation
Correlation describes the strength and direction of the linear relationship between two variables. It is measured by the Pearson correlation coefficient r, where −1 ≤ r ≤ +1.
Pearson Correlation Coefficient
−1 ≤ r ≤ +1
Strong negative
No linear relationship
Strong positive
Positive Correlation (r > 0)
As x increases, y tends to increase. Example: study hours and test score.
Negative Correlation (r < 0)
As x increases, y tends to decrease. Example: TV hours and test score.
Strength Guidelines (Australian Curriculum)
The Regression Line (Line of Best Fit)
When a linear relationship exists, the least-squares regression line (line of best fit) models the data. Its equation is:
y = a + bx
b = gradient (change in y per unit increase in x) | a = y-intercept (predicted y when x = 0)
Worked Example: Interpreting the regression line
A regression line for study hours (x) and exam score (y) is: y = 42 + 5x. Interpret the equation and predict the score for 7 hours of study.
y-intercept (42): A student who studies 0 hours is predicted to score 42.
Gradient (5): Each additional hour of study increases the predicted score by 5 marks.
Prediction (x = 7): y = 42 + 5(7) = 42 + 35 = 77 marks
Worked Example: Interpolation vs. extrapolation
Interpolation: Predicting within the observed data range. Generally considered reliable. Example: predicting score for 5 hours when data spans 1–10 hours.
Extrapolation: Predicting outside the data range. Considered unreliable as the pattern may not continue. Example: predicting for 20 hours when data only goes to 10 hours.
Always state whether you are interpolating or extrapolating and comment on reliability.
Worked Example: Correlation does not imply causation
Ice-cream sales and drowning rates are strongly positively correlated — but eating ice-cream does not cause drowning. Both are driven by hot weather, a confounding variable. Always consider alternative explanations before claiming a causal relationship.
Key Vocabulary
Bivariate Data
Data involving two variables measured on each subject, analysed together to identify relationships.
Correlation Coefficient (r)
A value between −1 and +1 measuring the strength and direction of a linear relationship. Values closer to ±1 indicate stronger correlation.
Interpolation
Predicting a value within the observed data range using the regression line. Generally reliable.
Confounding Variable
A third variable that influences both the explanatory and response variables, creating the appearance of a relationship without direct causation.
Knowledge Check
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Key Concepts Summary
- ●A scatter plot displays bivariate data: explanatory variable on x-axis, response variable on y-axis.
- ●The correlation coefficient r (−1 ≤ r ≤ 1) measures the strength and direction of a linear relationship.
- ●The regression line y = a + bx is the line of best fit; b is the gradient and a is the y-intercept.
- ●Predicting within the data range is interpolation (reliable); outside is extrapolation (unreliable).
- ●Correlation does not imply causation. A confounding variable may explain an apparent relationship.