What you'll learn
This revision guide covers the complete AP Statistics curriculum on two-variable (bivariate) data analysis. You'll master how to visualize relationships using scatterplots, quantify association with correlation, build and interpret linear regression models, and assess model fit using residuals. These skills are essential for both the multiple-choice and free-response sections of the AP exam.
Key terms and definitions
Scatterplot — a graphical display of the relationship between two quantitative variables, with one variable on each axis and each observation plotted as a point
Correlation coefficient (r) — a numerical measure of the strength and direction of a linear relationship between two quantitative variables, always between -1 and +1
Least-squares regression line — the line that minimizes the sum of squared residuals; the line of best fit for predicting the response variable from the explanatory variable
Residual — the difference between an observed value and the predicted value from the regression line (observed - predicted)
Explanatory variable — the independent variable (x) used to predict or explain changes in the response variable
Response variable — the dependent variable (y) that we are trying to predict or explain
Coefficient of determination (r²) — the proportion of variation in the response variable that is explained by the least-squares regression on the explanatory variable
Influential point — an observation that, if removed, substantially changes the regression line, slope, or correlation
Core concepts
Displaying relationships with scatterplots
A scatterplot displays the relationship between two quantitative variables by plotting ordered pairs (x, y) on a coordinate plane. The explanatory variable always appears on the horizontal axis, and the response variable on the vertical axis.
When describing a scatterplot, always comment on four features using the acronym DFOS:
- Direction: positive (upward trend), negative (downward trend), or no association
- Form: linear, curved (non-linear), or no pattern
- Outliers: any points that deviate from the overall pattern
- Strength: strong, moderate, or weak (how closely points follow the form)
Example context: A scatterplot showing hours studied versus exam marks would typically show a positive, moderately strong, linear relationship.
Correlation and its properties
The correlation coefficient (r) measures the strength and direction of a linear relationship. Key properties include:
- r is always between -1 and +1
- r = +1 indicates perfect positive linear relationship
- r = -1 indicates perfect negative linear relationship
- r = 0 indicates no linear relationship (but a non-linear relationship may exist)
- r is unitless and not affected by changes in scale or units
- r is strongly affected by outliers
- Correlation does not imply causation
Formula (for reference, though typically calculated using technology):
r = (1/(n-1)) × Σ[(xi - x̄)/sx][(yi - ȳ)/sy]
Where sx and sy are the standard deviations of x and y respectively.
Important: Correlation only measures linear relationships. Two variables with a strong curved relationship may have r ≈ 0.
Least-squares regression
The least-squares regression line has equation:
ŷ = a + bx
Where:
- ŷ (y-hat) is the predicted value of y
- b is the slope = r(sy/sx)
- a is the y-intercept = ȳ - b×x̄
- The line always passes through the point (x̄, ȳ)
Interpreting the slope: For each one-unit increase in x, we predict y to increase/decrease by b units (on average).
Interpreting the y-intercept: When x = 0, we predict y to be a. Only interpret this if x = 0 is meaningful in context and within the range of the data.
Extrapolation warning: Using the regression line to predict y for x-values outside the range of the observed data is unreliable and should be avoided.
Residuals and residual plots
A residual is calculated as:
residual = observed y - predicted y = y - ŷ
Residuals measure the prediction error. A positive residual means the regression line underestimated the actual value; a negative residual means it overestimated.
Residual plots (residuals on y-axis, x or ŷ on x-axis) help assess whether a linear model is appropriate:
- Good pattern: random scatter around the horizontal line at residual = 0, with no obvious pattern
- Poor patterns: curved pattern (suggests non-linear relationship), fan shape (non-constant variance), or outliers
The standard deviation of residuals (s) measures the typical prediction error and has the same units as the response variable. Smaller s indicates better predictions.
Coefficient of determination (r²)
The coefficient of determination (r²) is simply the square of the correlation coefficient. It represents the proportion (or percentage, when multiplied by 100) of the variation in y that is explained by the linear relationship with x.
Interpreting r²: "Approximately [r² × 100]% of the variation in [response variable] can be explained by the linear relationship with [explanatory variable]."
For example, if r = 0.8, then r² = 0.64, meaning 64% of variation in y is explained by the linear model. The remaining 36% is unexplained (due to other variables or randomness).
Outliers and influential points
Outliers in bivariate data can be:
- Points far from the regression line (large residuals)
- Points with extreme x-values, y-values, or both
Influential points are observations that substantially change the regression line if removed. Points with extreme x-values far from x̄ have high leverage and are most likely to be influential.
When identifying influential points:
- Note whether the point follows the general pattern or deviates
- Calculate regression with and without the point
- Compare slopes and correlations
- A point is influential if its removal substantially changes these statistics
Transformations to achieve linearity
When a scatterplot shows a curved relationship, transformation of one or both variables may linearize the relationship. Common transformations include:
- Logarithmic: log(y) or log(x) for exponential growth/decay patterns
- Power: √y, 1/x for curved relationships
- Exponential: ex for certain growth patterns
After transformation, analyze the transformed data using standard linear regression techniques. Always interpret results in the context of the transformed variables.
Worked examples
Example 1: Interpreting correlation and regression
A study of 15 sixth-form students recorded hours spent on social media per week (x) and A-level point score (y). The data yielded: x̄ = 12 hours, sx = 4 hours, ȳ = 280 points, sy = 50 points, r = -0.72.
(a) Calculate and interpret the coefficient of determination. [3 marks]
(b) Determine the equation of the least-squares regression line. [3 marks]
(c) Predict the A-level point score for a student using social media 8 hours per week. [2 marks]
Solution:
(a) r² = (-0.72)² = 0.5184 ≈ 0.52 ✓
Approximately 52% ✓ of the variation in A-level point scores is explained by the linear relationship with hours on social media. ✓
(b) Slope: b = r(sy/sx) = -0.72 × (50/4) = -0.72 × 12.5 = -9 ✓
y-intercept: a = ȳ - b×x̄ = 280 - (-9)(12) = 280 + 108 = 388 ✓
Equation: ŷ = 388 - 9x ✓
(c) ŷ = 388 - 9(8) = 388 - 72 = 316 ✓
We predict an A-level point score of 316 points. ✓
Example 2: Analyzing residuals
For a regression of exam mark (%) on hours studied, the regression equation is ŷ = 42 + 6.5x. A student who studied 5 hours scored 78%.
(a) Calculate the residual for this student. [2 marks]
(b) Interpret this residual in context. [1 mark]
Solution:
(a) Predicted mark: ŷ = 42 + 6.5(5) = 42 + 32.5 = 74.5% ✓
Residual = 78 - 74.5 = 3.5% ✓
(b) The student scored 3.5 percentage points higher than predicted by the regression line. ✓
Example 3: Describing relationships
A researcher investigates the relationship between daily temperature (°C) and ice cream sales (£) at a Caribbean beach vendor over 20 days. The scatterplot shows points trending upward in a roughly linear pattern, with one day (a local festival) showing unusually high sales.
Describe the relationship between temperature and ice cream sales. [4 marks]
Solution:
Direction: There is a positive association ✓ between temperature and ice cream sales.
Form: The relationship appears to be approximately linear. ✓
Strength: The relationship is moderately strong, ✓ with most points close to a linear trend.
Outliers: There is one outlier showing higher sales than expected for its temperature, ✓ possibly due to the festival day mentioned.
Common mistakes and how to avoid them
Confusing correlation with causation: Remember that r measures association only. A strong correlation between ice cream sales and drowning deaths doesn't mean one causes the other (both are linked to hot weather). Always state "association" or "relationship," not "cause."
Misinterpreting the slope: The slope tells you the predicted change in y per one-unit change in x, not the total change or the rate of change in percentages. Always include proper units and the phrase "on average" or "we predict."
Extrapolating beyond the data range: Never use the regression equation to predict y for x-values outside the observed range. The relationship may not hold beyond the data collected, and predictions become unreliable.
Ignoring residual plot patterns: A curved residual plot indicates that a linear model is inappropriate, even if r is fairly strong. Always check residual plots before concluding that a linear model fits well.
Forgetting that r² is a proportion: When interpreting r², express it as a decimal or percentage of variation explained. Don't say "r² = 0.64 means the relationship is 64% strong" — this is meaningless. Say "64% of variation in y is explained by the linear relationship with x."
Switching x and y variables: The explanatory variable must be on the x-axis and used to predict the response variable on the y-axis. Switching them gives a different regression line (unless r = ±1).
Exam technique for "Exploring Two-Variable Data"
Command words matter: "Describe" requires DFOS (direction, form, outliers, strength) for scatterplots. "Calculate" requires showing work and the numerical answer. "Interpret" requires context-specific explanation, not just restating the number.
Context is essential: Never give generic interpretations. Always reference the actual variables. Write "for each additional hour studied, we predict exam marks to increase by 6.5 percentage points" rather than "y increases by 6.5 when x increases by 1."
Show your working: Even when using a calculator for regression statistics, write down the formula you're using and substitute values. Partial credit is available for correct method even if your final answer is wrong.
Check residual plots on calculator questions: The AP exam often provides computer output or expects you to analyze residual plots. Practice interpreting these on your calculator and describing what patterns indicate about model appropriateness.
Quick revision summary
Two-variable data analysis involves examining relationships between quantitative variables. Use scatterplots to visualize (describe using DFOS), correlation r to quantify linear association (-1 ≤ r ≤ +1), and least-squares regression (ŷ = a + bx) to model and predict. The coefficient of determination r² measures proportion of variation explained. Residuals (y - ŷ) assess prediction accuracy; random residual plots indicate good model fit. Always interpret in context, avoid extrapolation, and remember correlation ≠ causation. Check for outliers and influential points that affect the regression line.