Two-Variable Data — Models and Scatterplots

Two-Variable Data: Models and Scatterplots

SAT Math — Problem-Solving and Data Analysis

Learning Objectives

By the end of this lesson, you should be able to:

• Interpret two-variable data presented in scatterplots and tables.

• Identify positive, negative, and no clear association.

• Distinguish linear and nonlinear relationships.

• Recognize strong and weak associations and outliers.

• Interpret the slope and y-intercept of a line of best fit.

• Calculate predicted values and residuals.

• Recognize linear, quadratic, and exponential models.

• Evaluate interpolation, extrapolation, and the limitations of predictions.

1. Understanding Two-Variable Data

Two-variable data consist of pairs of numerical values collected from the same observations.

Each observation is represented by an ordered pair (x, y).

For example, suppose a researcher records the number of hours students study and their test scores.

Every row can be plotted as one point on a coordinate plane.

A graph showing these points is called a scatterplot.

The horizontal axis usually shows the explanatory variable (x), while the vertical axis shows the response variable (y).

In this example, study time is the explanatory variable and test score is the response variable.

SAT Tip: Always read the axis labels and units before analyzing a scatterplot.

2. Types of Association

An association describes the overall relationship between two variables.

Positive Association

A positive association occurs when larger values of x generally correspond to larger values of y.

Example: As the number of study hours increases, test scores may also increase.

The points in a scatterplot generally move upward from left to right.

Negative Association

A negative association occurs when larger values of x generally correspond to smaller values of y.

Example: As a machine gets older, its resale value may decrease.

The points generally move downward from left to right.

No Clear Association

No clear association occurs when the data points do not follow a recognizable trend.

Example: The number of letters in a student's first name may have no clear relationship with their mathematics score.

Linear Association

A linear association follows an approximately straight-line pattern.

Nonlinear Association

A nonlinear association follows a curved pattern, such as a U-shaped curve or exponential growth pattern.

Important: A nonlinear association can still be strong. A strong association does not necessarily mean the points follow a straight line.

3. Strength of Association

The strength of an association describes how closely data points follow an overall pattern.

Strong association: Most points are close to a clear line or curve.

Weak association: Points are scattered more widely around the general pattern.

No clear association: No consistent pattern is visible.

A scatterplot showing points close to a downward-sloping straight line has a strong negative linear association.

4. Outliers

An outlier is an observation that lies unusually far from the main data pattern.

Consider:

(1, 4), (2, 6), (3, 8), (4, 10), (5, 24)

The first four points follow:

y = 2x + 2

When x = 5, this rule predicts:

y = 2(5) + 2 = 12

However, the actual observation is 24.

Therefore, (5, 24) is an outlier.

An outlier may influence the position and slope of a fitted line, particularly if it is far from the other observations.

SAT Tip: Do not assume the point with the greatest x-value or y-value is automatically an outlier. Compare it with the overall trend.

5. Lines of Best Fit

A line of best fit is a straight line that models the general trend of data showing an approximately linear relationship.

The line does not need to pass through every point.

It is commonly represented by:

ŷ = mx + b

Where:

• ŷ = predicted value of y

• m = slope or rate of change

• x = input or explanatory variable

• b = y-intercept

Calculating the Slope

The slope of a line through two points is:

m = (y₂ − y₁) / (x₂ − x₁)

Example:

A fitted line passes through (2, 15) and (8, 33).

Step 1: Substitute the coordinates.

m = (33 − 15) / (8 − 2)

Step 2: Simplify.

m = 18 / 6

m = 3

The slope is 3.

This means the model predicts an increase of 3 in y for every increase of 1 in x.

Finding the Y-Intercept

Use:

ŷ = mx + b

We already know m = 3.

Substitute the point (2, 15):

15 = 3(2) + b

15 = 6 + b

b = 9

The equation is:

ŷ = 3x + 9

Making Predictions

Predict the value of y when x = 10.

ŷ = 3(10) + 9

ŷ = 30 + 9

Predicted value = 39

Remember that a model produces an estimate, not necessarily the exact observed value.

6. Interpreting Slope and Intercept in Context

Suppose the total cost C, in dollars, of using a service for h hours is modeled by:

C(h) = 8h + 35

The coefficient 8 represents the additional cost for each hour.

The constant 35 represents the cost when h = 0.

Therefore:

• Slope = $8 per hour

• Y-intercept = $35 initial cost

If the slope is negative, the predicted response decreases when the input increases.

SAT Tip: When interpreting slope, always include the relevant units, such as dollars per hour or centimeters per year.

7. Observed Values, Predicted Values, and Residuals

A predicted value is calculated from a model.

An observed value is the actual recorded measurement.

A residual measures the difference between them.

Residual = Observed value − Predicted value

Example:

The fitted equation is:

ŷ = 2x + 9

At x = 5, the observed y-value is 22.

Step 1: Calculate the predicted value.

ŷ = 2(5) + 9

ŷ = 19

Step 2: Calculate the residual.

Residual = 22 − 19

Residual = +3

A positive residual means the observed point is above the model's prediction.

A negative residual means the observed point is below the prediction.

A zero residual means the observed value equals the prediction.

8. Recognizing Mathematical Models

Two-variable data can follow linear, quadratic, exponential, or other patterns.

Linear Models

A linear model has constant first differences when the x-values are equally spaced.

Example:

First differences:

9 − 5 = 4

13 − 9 = 4

17 − 13 = 4

Therefore, the data follow a linear model.

y = 4x + 5

Quadratic Models

A quadratic model has constant second differences when x-values are equally spaced.

General equation:

y = ax² + bx + c

Example:

First differences: 3, 5, 7

Second differences: 2, 2

The constant second difference indicates a quadratic pattern.

One matching equation is:

y = x² + 2x + 3

Exponential Models

An exponential model has a constant multiplicative ratio for equal increases in x.

General equation:

y = a(bˣ)

Example:

Calculate consecutive ratios:

12 / 4 = 3

36 / 12 = 3

108 / 36 = 3

The constant ratio is 3.

Therefore:

y = 4(3ˣ)

Quick Comparison

These methods identify exact patterns in tables with equally spaced inputs. Actual measurements may contain variation, so an appropriate model should reflect the overall pattern rather than requiring perfect agreement.

9. Interpolation and Extrapolation

Interpolation

Interpolation means predicting a value inside the observed input range.

Example:

A model uses observations with x-values from 4 to 16.

Predicting the output at x = 10 is interpolation.

Extrapolation

Extrapolation means predicting a value outside the observed input range.

Using the same model to predict at x = 40 is extrapolation.

Extrapolation may be less reliable because the relationship might change outside the measured range.

10. Association Does Not Prove Causation

An association between two variables does not necessarily prove that one causes the other.

For example, students who study more hours may tend to obtain higher test scores.

This demonstrates a positive association.

However, other factors, including prior knowledge and study quality, may also influence performance.

A scatterplot alone cannot establish a cause-and-effect relationship.

11. Worked SAT-Style Examples

Example 1 — Positive Association

Problem: A scatterplot contains the points (1, 20), (2, 25), (3, 30), and (4, 35). Describe the association.

Solution:

Step 1: The x-values increase by 1.

Step 2: The y-values increase by 5.

Step 3: The points follow an upward straight-line pattern.

Answer: Positive linear association.

Example 2 — Line of Best Fit

Problem: A fitted line passes through (3, 17) and (7, 29). Find its equation.

Solution:

Step 1: Calculate the slope.

m = (29 − 17) / (7 − 3)

m = 12 / 4 = 3

Step 2: Write y = 3x + b.

Step 3: Substitute (3, 17).

17 = 3(3) + b

17 = 9 + b

b = 8

Answer: y = 3x + 8

Example 3 — Predicting Values

Problem: A model is given by S(t) = 25t + 180. Find S(12).

Solution:

S(12) = 25(12) + 180

S(12) = 300 + 180

Answer: 480

Example 4 — Residual

Problem: A fitted line is ŷ = 4x + 6. The observed point is (8, 41). Calculate the residual.

Solution:

Predicted y = 4(8) + 6 = 38

Observed y = 41

Residual = 41 − 38

Answer: +3

Example 5 — Quadratic Pattern

Problem: The data contain (0, 2), (1, 5), (2, 12), and (3, 23). Identify the model type.

Solution:

First differences:

3, 7, 11

Second differences:

4, 4

Since the second differences are constant, the model is quadratic.

One equation fitting the data is:

y = 2x² + x + 2

Example 6 — Exponential Pattern

Problem: The table contains (0, 10), (1, 20), (2, 40), and (3, 80). Identify the model.

Solution:

20 / 10 = 2

40 / 20 = 2

80 / 40 = 2

The output doubles for each one-unit increase in x.

Answer: y = 10(2ˣ)

Example 7 — Evaluating a Prediction

Problem: A model uses data with x-values from 4 to 16. A student predicts the output when x = 40. Explain why this prediction may be unreliable.

Solution:

Step 1: The observed range is from 4 to 16.

Step 2: The input 40 lies outside this range.

Step 3: Therefore, the prediction is extrapolation.

Step 4: The trend may not continue unchanged.

Answer: It is an extrapolation that may be unreliable.

12. Common SAT Mistakes

• Confusing positive and negative associations.

• Assuming all strong relationships must be linear.

• Mistaking an outlier for the main trend.

• Using two arbitrary observed points instead of points on a fitted line to calculate its slope.

• Forgetting to include the y-intercept when making predictions.

• Confusing an observed value with a predicted value.

• Reversing the residual formula.

• Forgetting that a negative residual is valid.

• Using output differences alone when input intervals are greater than 1.

• Confusing constant first differences with constant ratios.

• Assuming association proves causation.

• Treating an extrapolated value as guaranteed accurate.

• Ignoring units when interpreting slope.

• Selecting a linear model when the data clearly follow a curved trend.

13. Final Revision Checklist

Before attempting SAT practice questions, make sure you can:

• Recognize the direction and strength of an association.

• Identify an outlier.

• Calculate slope from two points on a fitted line.

• Find the y-intercept and write a linear model.

• Use the model to predict an output.

• Calculate and interpret a residual.

• Recognize linear, quadratic, and exponential patterns.

• Distinguish interpolation from extrapolation.

• Explain why association does not necessarily establish causation.

SAT Strategy: Read the question carefully, identify the relevant quantities, choose the appropriate mathematical model, show the necessary working, and interpret the result in context.