Lesson 9.1 · Bivariate Data
Scatter plots
Does studying longer really lead to better test scores? Do older cars sell for less? To answer questions like these, you need data that comes in pairs: two measurements for each person or object. A scatter plot turns those pairs into a picture, so you can see a pattern at a glance.
Paired data
Most data you have worked with so far had one number per person: a height, a score, an age. Bivariate data has two numbers for each person or object, and the question is how they relate.
Definition
Bivariate data
Data made of pairs of values, one pair for each individual. Each pair can be written as an ordered pair .
Here is a small data set. Twelve students recorded how many hours they studied for a science test and the score they earned.
| hours studied | 0 | 1 | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 6 | 6 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| test score | 55 | 60 | 66 | 65 | 72 | 70 | 78 | 80 | 76 | 85 | 88 | 92 |
Each column is one student. The first student studied hours and scored , which gives the ordered pair .
Making a scatter plot
To make a scatter plot:
- Choose which variable goes on each axis. Usually the variable you think explains the other goes on the -axis (here, hours studied). The one you want to predict goes on the -axis (test score).
- Pick a scale for each axis that fits all the data. The scores run from to , so the -axis can start at instead of .
- Plot one point for each pair. Don't connect the points.
The picture shows something the table hid: as the hours go up, the scores tend to go up too. The dots drift upward from left to right.
Describing association
When one variable tends to change in a predictable way as the other changes, the variables have an association.
Types of association
- Positive association: as increases, tends to increase. The dots rise from left to right.
- Negative association: as increases, tends to decrease. The dots fall from left to right.
- No association: knowing doesn't help you predict . The dots are scattered with no clear direction.
An association is linear if the dots cluster around a straight line, and nonlinear if they follow a curve.
The word "tends" matters. In the study data, the student who studied hours and scored did worse than one who studied hours and scored . Not every point follows the pattern, but the overall trend is still clearly upward.
Worked example: Negative association
A dealership lists ten used cars of the same model. The table shows each car's age and its price, in thousands of dollars.
| age (years) | 1 | 2 | 3 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| price (thousands of dollars) | 24 | 21 | 19 | 17 | 16 | 13 | 12 | 9 | 8 | 6 |
Describe the association.
Solution. The dots fall from left to right, and they lie close to a straight line. There is a negative linear association: older cars tend to cost less.
Worked example: No association
A class records the last digit of each student's phone number and the student's height in inches. The scatter plot is below.
Is there an association?
Solution. The dots bounce up and down with no upward or downward drift. A phone number tells you nothing about height, so there is no association.
Nonlinear patterns
Sometimes the dots follow a curve instead of a line. A student measured the height of a ball, in feet, every half second after tossing it upward:
The heights rise and then fall. There is a clear pattern, but it is nonlinear: no single straight line fits these points well.
Outliers and clusters
Definition
Outlier
A data point that is far away from the overall pattern of the other points.
A cluster is a group of points bunched close together, often with a gap separating it from other groups.
Worked example: Spotting an outlier
Ten people measured their arm span and height, in inches.
Describe the association and name any outlier.
Solution. Nine of the points rise steadily from left to right, close to a line, so there is a positive linear association. The point sits far above the others: a person with a -inch arm span would be expected to be about inches tall, not . That point is an outlier. It could be a measuring mistake, or just an unusual person.
Common mistake
An association does not prove that one variable causes the other. Cities with more ice cream shops tend to have more swimming pools, but ice cream doesn't build pools. Both are more common in large, warm cities. A scatter plot shows that two variables move together, not why.
Tip
To check the direction quickly, cover the middle of the plot with your hand and compare the left side with the right. If the right-side dots are generally higher, the association is positive. If they are lower, it is negative.
Practice
On a scatter plot, the points drift downward from left to right and lie close to a straight line. Which describes the association?
Which pair of variables would most likely have a positive association?
Use the study-time table at the start of this lesson. How many students studied more than hours and scored above ?
Enter a number. Fractions like 3/4 and sqrt(2) are OK.
The scatter plot shows the number of hours nine students slept and the number of math problems they got right on a quiz. Which point is an outlier? Give it as an ordered pair.
Enter a point like (2, -3)
The scatter plot shows the number of people at a pool at different times of day, where is the hour ( means 9 a.m. and means 6 p.m.).
Which best describes the pattern?
On the used-car scatter plot in this lesson, what does the point represent?
A scatter plot of towns shows a positive association between the number of umbrellas sold and the number of car accidents on a given day. Which conclusion is best?