Bivariate Data: Exploring Two Measurements Together
From Pairs to Patterns: Understanding the Basics
Bivariate data comes in pairs. Think of it as a team of two numbers for each item or person you are studying. For example, if you measure the height and arm span of each student in your class, you are creating bivariate data. The pair (Height, Arm Span) belongs to one student.
These two pieces of information are called variables. Usually, one variable is thought of as the independent or explanatory variable (often labeled $x$), and the other is the dependent or response variable (often labeled $y$). In our example, you might choose height as $x$ to see if it can explain or predict arm span, which would be $y$.
Seeing the Relationship: The Scatter Plot
A table of numbers can be hard to interpret. The most important and effective tool for visualizing bivariate data is the scatter plot. To create one, you draw two perpendicular number lines, called axes. The horizontal axis (x-axis) represents the $x$ variable, and the vertical axis (y-axis) represents the $y$ variable. Each pair of data $(x, y)$ is then plotted as a single point on this graph.
For instance, a student who is 160 cm tall ($x=160$) and has an arm span of 158 cm ($y=158$) would be represented by a point at the coordinates (160, 158). Once all points are plotted, you can look at the overall cloud of points to see if a pattern emerges.
Describing the Pattern: Correlation and Direction
The pattern, or relationship, between the two variables in a scatter plot is called correlation. Correlation tells us two main things: the direction and the strength of the relationship.
Direction:
- Positive Correlation: As $x$ increases, $y$ tends to also increase. The points slope upward from left to right. Example: Study time and test scores.
- Negative Correlation: As $x$ increases, $y$ tends to decrease. The points slope downward from left to right. Example: Time spent playing video games and test scores.
- No Correlation: There is no apparent pattern; the points are scattered randomly. Example: Shoe size and IQ.
Strength: This describes how closely the points follow a clear linear pattern (a straight line). If the points are tightly clustered around an imaginary line, the correlation is strong. If they are widely scattered around that line, the correlation is weak.
| Correlation Type | What it Looks Like | Real-World Example |
|---|---|---|
| Strong Positive | Points form a clear, tight line sloping upwards. | Age (in months) and height of a young child. |
| Weak Positive | Points slope upwards but are very spread out. | Age (in years) and height of adults. |
| Strong Negative | Points form a clear, tight line sloping downwards. | Car speed and time to travel a fixed distance. |
| Weak Negative | Points slope downwards but are very spread out. | Hours spent on social media and self-reported happiness. |
| No Correlation | Points show no directional pattern, like a cloud. | Number of pets owned and math grade. |
Making Predictions: The Line of Best Fit
When a scatter plot shows a linear pattern, we can summarize that pattern with a straight line called the line of best fit or trend line. This line is drawn to pass as close as possible to all the data points, minimizing the overall distance between the points and the line. It is a model that represents the general trend.
The equation of a line is usually written as $y = mx + c$.
- $m$ is the slope. It tells you how much $y$ changes, on average, when $x$ increases by 1 unit. A positive slope indicates a positive correlation; a negative slope indicates a negative correlation.
- $c$ is the y-intercept. It is the predicted value of $y$ when $x = 0$.
It is crucial to remember that predictions made using the line of best fit are estimates, not certainties. The line shows an average relationship.
A Classroom Investigation: Study Time vs. Test Score
Let's walk through a complete, simple investigation using bivariate data. Imagine a teacher wants to know if there is a relationship between the time students spend studying for a math test and their score on that test. She collects data from 10 students, recording study time (in hours) and test score (out of 100). Here is the data she collected:
| Student | Study Time (x) in hours | Test Score (y) |
|---|---|---|
| 1 | 0.5 | 55 |
| 2 | 1.0 | 60 |
| 3 | 1.5 | 65 |
| 4 | 2.0 | 70 |
| 5 | 2.5 | 75 |
| 6 | 3.0 | 78 |
| 7 | 3.5 | 82 |
| 8 | 4.0 | 85 |
| 9 | 4.5 | 88 |
| 10 | 5.0 | 92 |
Step 1: Create a Scatter Plot. Plot each pair: (0.5, 55), (1.0, 60), ..., (5.0, 92).
Step 2: Describe the Correlation. Looking at the plot, the points roughly form a line sloping upwards. This indicates a positive correlation. The points are fairly close to an imaginary straight line, so we can say it's a moderately strong positive correlation.
Step 3: Draw a Line of Best Fit. By eye, we can draw a straight line that goes through the middle of the point cloud. For this data, a reasonable line might pass near points (1, 60) and (4, 85).
Step 4: Make a Prediction. Using our line, we could estimate that a student who studies for 6 hours might score around 98. This is called extrapolation (predicting outside the range of our data), and it can be risky. A safer interpolation (predicting within the data range) would be to estimate that a student studying 2.2 hours might score around 72.
This investigation shows a likely relationship, but it's important to note that correlation does not imply causation. While more study time is associated with higher scores here, other factors (like prior knowledge or sleep) also influence test scores.
Important Questions
Q1: What is the difference between univariate and bivariate data?
A1: Univariate data involves measurements of only one variable or attribute (e.g., just the heights of students). Bivariate data involves measurements of two variables for each subject, taken at the same time, so we can study the possible relationship between them (e.g., heights and arm spans of students).
Q2: If two variables are correlated, does that mean one causes the other?
A2: Absolutely not. This is the most common and important mistake to avoid. Correlation only means there is a relationship or association. There could be a third, unseen variable causing the change in both, or it could be a complete coincidence. For example, there is a positive correlation between ice cream sales and drowning incidents. This doesn't mean ice cream causes drowning. A third variable—hot summer weather—causes both to increase.
Q3: What is an outlier in a scatter plot, and what should we do about it?
A3: An outlier is a data point that falls far outside the overall pattern of the other points. For example, in the study time vs. score data, if one student studied for 1 hour but scored 95, that point would be an outlier. Outliers should be noted and investigated—they could be a data entry error, or they could represent a truly unusual case. They can have a strong effect on the position of the line of best fit, so it's important to consider their impact on your conclusions.
Footnote
1 Variable: A characteristic or attribute that can be measured or counted, and that can vary from one observation to another (e.g., height, temperature, time).
2 Correlation: A statistical measure that describes the extent to which two variables change together. It indicates the direction (positive or negative) and strength (weak or strong) of their linear relationship.
3 Extrapolation: The process of estimating a value of the dependent variable ($y$) for an independent variable ($x$) that lies outside the range of the observed data. It is often less reliable than interpolation.
4 Interpolation: The process of estimating a value of the dependent variable ($y$) for an independent variable ($x$) that lies within the range of the observed data.