Do taller people really have longer arms? Does more studying lead to better scores? To answer questions like these, you need to look at two pieces of information about each person or object at the same time. In this chapter you will draw scatter plots, spot patterns, build a line that models the data, and organize categories in two-way tables.
1. Bivariate data and scatter plots
When you record two numerical measurements for each individual (for example, hours studied and test score), you have bivariate data. Each individual gives an ordered pair \((x, y)\).
A scatter plot is a graph in the coordinate plane where each ordered pair of bivariate data is shown as one point. The independent variable (the one you think may influence the other) goes on the horizontal axis, and the dependent variable goes on the vertical axis.
- Choose the variable for each axis.
- Pick a scale that fits the smallest and largest values, with equal spacing between tick marks.
- Plot one point for each pair. Do not connect the points.
- Label both axes with names and units, and give the graph a title.
Seven students recorded hours studied for a quiz and their score out of 100.
| Hours studied \(x\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Score \(y\) | 59 | 66 | 69 | 77 | 80 | 86 | 89 |
The first point is \((1, 59)\): move 1 unit right and 59 units up. The seven points are shown below.
2. Patterns: positive, negative, or no association
The shape of the cloud of points tells you how the two variables are related. This relationship is called an association.
- Positive association: as \(x\) increases, \(y\) tends to increase.
- Negative association: as \(x\) increases, \(y\) tends to decrease.
- No association: the points show no clear trend.
If the points stay close to a straight line, the association is linear. If they follow a curve instead, it is nonlinear. The closer the points are to the line, the stronger the association.
An association does not prove that one variable causes the other. Ice cream sales and sunburns both rise in summer, but ice cream does not burn anyone. Hot, sunny weather affects both.
3. Outliers and clusters
An outlier is a point that is far away from the overall pattern of the other points. A cluster is a group of points that sit close together, often separated from other groups by a gap.
In the graph above, six points rise steadily, but the point \((7, 3)\) falls far below the trend. It is an outlier. Always ask why: a measuring mistake, a special event, or a truly unusual individual?
An outlier can pull a trend line toward itself and make predictions less accurate. Clusters can reveal that your data comes from two different groups.
4. The line of best fit
When a scatter plot shows a linear association, you can draw a straight line that models the trend. This is called a line of best fit or trend line.
- Use a clear straightedge.
- Place it so it follows the direction of the points.
- Try to have about the same number of points above and below the line.
- The line does not have to pass through any data point, and it does not have to start at the origin.
In the graph of Example 1, the orange line passes through \((2, 65)\) and \((6, 85)\). These two points are on the line, so you can use them to find its equation.
Slope: \(m = \dfrac{85 - 65}{6 - 2} = \dfrac{20}{4} = 5\).
Use \(y = 5x + b\) with the point \((2, 65)\): \(65 = 5(2) + b\), so \(b = 55\). The model is \(y = 5x + 55\).
5. Using a linear model to predict
Once you have an equation, substitute a value of \(x\) to predict \(y\).
Predict the score of a student who studies 4.5 hours: \(y = 5(4.5) + 55 = 77.5\). The model predicts about 77.5 points.
Predicting inside the range of your data is called interpolation and is usually reliable. Predicting far outside the range is extrapolation, which can be risky. For 10 hours, the model gives \(5(10) + 55 = 105\), which is impossible on a test scored out of 100.
6. Interpreting slope and intercept in context
In a model \(y = mx + b\), the numbers have a meaning in the real situation.
- The slope \(m\) is the predicted change in \(y\) for each increase of 1 unit in \(x\).
- The y-intercept \(b\) is the predicted value of \(y\) when \(x = 0\).
For \(y = 5x + 55\): each extra hour of study is associated with about 5 more points. A student who studied 0 hours is predicted to score about 55. Be careful: sometimes the intercept does not make sense in context, for example when \(x = 0\) is far from the data.
On my planet we write units next to every number. Say "5 points per hour," not just "5." The units make your interpretation clear.
7. Two-way frequency tables
Scatter plots show two numerical variables. To study two categorical variables (such as yes/no answers), use a two-way frequency table. Rows show one variable, columns show the other, and each cell counts the individuals in both categories.
One hundred twenty eighth graders were asked how long they sleep and whether they use a phone in bed.
| Phone in bed | No phone | Total | |
|---|---|---|---|
| Sleeps under 8 hours | 45 | 15 | 60 |
| Sleeps 8 hours or more | 20 | 40 | 60 |
| Total | 65 | 55 | 120 |
Each total (marginal frequency) is the sum of its row or column. The number 45 is a joint frequency: the students who sleep under 8 hours and use a phone in bed.
8. Relative frequencies and association
A relative frequency is a count divided by a total, usually written as a percent. To check for association, compare relative frequencies within each row (or each column).
Among the 60 students who sleep under 8 hours, \(\dfrac{45}{60} = 75\%\) use a phone in bed. Among the 60 who sleep 8 hours or more, \(\dfrac{20}{60} \approx 33.3\%\) do. These percents are very different, so there is an association between sleeping less and using a phone in bed.
If the row relative frequencies are about equal, there is no association. If they differ noticeably, there is an association.
Divide by the total of the group you are asking about. "What percent of phone users sleep under 8 hours?" divides by the column total 65, giving \(\dfrac{45}{65} \approx 69.2\%\), not \(\dfrac{45}{120}\).
Key takeaways
- A scatter plot shows pairs \((x, y)\) as points; it is not connected.
- Association can be positive, negative, or none; linear or nonlinear.
- An outlier is far from the pattern; a cluster is a tight group of points.
- A line of best fit follows the trend; find its equation from two points on the line.
- Slope is the change in \(y\) per 1 unit of \(x\); the intercept is the value of \(y\) when \(x = 0\).
- Predict inside the data range; be careful when extrapolating.
- In a two-way table, compare relative frequencies to look for association.
- Association does not prove causation.
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