13.1 Two-Variable Data: Models and Scatterplots

Problem-Solving and Data Analysis on the PSAT 8/9 includes two-variable data: situations in which you track two measurements on the same people, plants, days, or objects. One-variable work — a single list of scores with a mean or a range — is covered in One-Variable Data: Center and Spread. Here the question is different: how do the two measurements move together?

PSAT 8/9 practice questionsPractice questions with detailed explanations

What a scatterplot shows

A scatterplot displays paired data as points. The explanatory (input) value is usually on the horizontal axis as x. The response (output) value is on the vertical axis as y. Each pair (x, y) becomes one dot.

Before you interpret the cloud of points, read the title, axis labels, and units. A graph titled 'Plant height (cm) vs. weeks after planting' is not the same as 'weeks vs. height.' Mixing up the axes is a common way to get a fluent but wrong answer.

Three trend types

• Positive association: as x increases, y tends to increase. The cloud rises left to right.
• Negative association: as x increases, y tends to decrease. The cloud falls left to right.
• No association: knowing x does not help you predict y. The cloud looks like a shapeless spray, not a slanted band.

You can also describe how tightly the points follow a line. A narrow band hugging a line is a strong linear association. A wide, messy band with the same overall tilt is a weaker linear association. PSAT 8/9 items do not ask you to compute a correlation coefficient. They do ask you to recognize the direction of the trend and to use a given line.

A point that sits far from the rest of the cloud is an outlier in the two-variable picture. Do not let one stray point rewrite the trend if the other points clearly rise or fall together. You are not asked on this exam to run formal outlier tests or to infer from a sample to a whole population.

A worked data set: practice hours and free-throw percentage

A basketball coach records how many hours each of five players practiced in a week and each player's free-throw percentage that week.

Plotting these pairs produces a rising cloud: more practice hours go with higher percentages. That is a positive association. The points are not perfectly in a line — 3 hours matches 57%, a little above a perfectly even pattern — but a straight line still captures the trend.

A linear model that follows the cloud

A line of best fit is a straight line drawn through a scatterplot so that it follows the overall pattern. Some points sit above the line and some sit below. You are not expected to run a calculator regression from scratch on this exam. Items typically give you a model, show a line on the graph, or ask which equation matches the trend.

For the free-throw data, this linear model stays close to every listed point:

y = 6x + 38

Check each row:

• 1 hour: 6(1) + 38 = 44 (exact)
• 2 hours: 6(2) + 38 = 50 (exact)
• 3 hours: 6(3) + 38 = 56 (actual 57, 1 percentage point above the line)
• 4 hours: 6(4) + 38 = 62 (exact)
• 5 hours: 6(5) + 38 = 68 (actual 69, 1 percentage point above the line)

The differences from the line are small. The equation is a fair summary of the table.

Slope and intercept in context

The slope is 6. In this story, each additional practice hour is associated with about 6 more percentage points on free throws. Slope language should name both variables: 'about 6 percentage points per hour,' not a bare '6.'

The y-intercept is 38. Algebraically it is the modeled percentage when x = 0. In context that would be 0 hours of practice. The table never measured 0 hours, so 38% is an extrapolation. It might be a reasonable baseline, or it might not match real players who skip practice. On the exam, use the intercept when the question asks for it, but do not treat it as a measured data point.

This slope-intercept reading is the same algebra you use in Linear Functions and Linear Equations in Two Variables. The new skill is attaching those numbers to a scatterplot.

Using the model to estimate a value

Example. According to y = 6x + 38, what percentage does the model predict for 3.5 hours of practice?

Substitute: y = 6(3.5) + 38 = 21 + 38 = 59.

3.5 hours sits between 1 and 5 hours in the table. An estimate inside the span of the data is interpolation. Interpolation is usually more trustworthy than stretching the line far past the last point.

Example. What does the same model predict for 8 hours?

y = 6(8) + 38 = 48 + 38 = 86.

Eight hours is outside the 1-to-5-hour window. That is extrapolation. The arithmetic is still 86, but a question may ask whether that prediction is as reliable as the 3.5-hour estimate. For grades 8–9, the safe stance is: the formula still produces a number, but the pattern was only observed nearby, so the far prediction can drift.

If an item gives a target y and asks for x, reverse the substitution.

Example. According to the model, how many practice hours correspond to a 74% free-throw rate?

74 = 6x + 38

36 = 6x

x = 6 hours

A negative trend, for contrast

Hot-chocolate sales often fall as outdoor temperature rises.

A line that tracks this table is y = −0.8x + 72:

• 30°F: −0.8(30) + 72 = −24 + 72 = 48
• 40°F: −0.8(40) + 72 = −32 + 72 = 40
• 50°F: −0.8(50) + 72 = −40 + 72 = 32 (actual 31)
• 60°F: −0.8(60) + 72 = −48 + 72 = 24
• 70°F: −0.8(70) + 72 = −56 + 72 = 16

The slope −0.8 means that each extra degree Fahrenheit is associated with about 0.8 fewer cups. Estimate sales at 55°F: −0.8(55) + 72 = −44 + 72 = 28 cups.

A no-association picture would look different: for example, shoe size versus spelling-quiz score for students in one classroom. The points would not rise or fall as a band. Knowing shoe size would not help you predict the quiz score.

Association is not causation

A scatterplot can show that two variables move together. It does not, by itself, prove that one causes the other.

Classic grade 8–9 trap: ice-cream sales and community-pool attendance both rise in June and July. The points would show a positive association. The hidden driver is warm weather. Selling more cones does not fill the pool, and opening the pool does not freeze the ice-cream machine.

Another version: the number of firefighters sent to a fire is associated with the dollar damage. Bigger fires bring more crews and destroy more property. The association is real; 'firefighters cause damage' is not.

When a question asks what you can conclude from a scatterplot alone, prefer:

• the direction of the association
• a prediction from a given model
• a statement that cause is not established just because the points trend

Do not jump to 'x makes y happen' unless the problem's science story already supplies a mechanism and the question is only asking you to use the numbers.

Exam habits that save points

1. Name the axes in words before you compute. 'Hours → percentage' is not 'percentage → hours.'
2. Substitute into the given equation. Do not invent a new slope from two messy points unless the item asks you to.
3. Check one table row against the model. If the line is supposed to fit, a listed point should land on or near it.
4. Watch units. A slope of 6 percentage points per hour is not 6 hours per percentage point.
5. Stay inside what PSAT 8/9 tests. You will use scatterplots and linear models. You will not be asked to infer from a random sample to a whole population, compute a margin of error, or judge whether an experiment was well designed. Those skills appear later in the SAT Suite, not on this test.

PSAT 8/9 practice questionsPractice questions with detailed explanations