English:Scatter Plots and Associations
Scatter Plots and Associations
Introduction
A scatter plot is a graph that helps you investigate whether two numerical variables are related. Each point represents one case, person, object, or event. Its horizontal position shows one variable, and its vertical position shows the other variable.
In Grades 7–8, scatter plots are especially useful because they let you move beyond single-number summaries. Instead of asking only “What is the average?”, you can ask questions such as: Do larger values of one variable tend to go with larger values of another? Is the pattern strong or weak? Are there unusual points? Can the pattern help us make a sensible prediction?
The image above compares three ways of displaying the same paired data. In a scatter plot, the points themselves are the main evidence. You usually do not connect each point to the next point.
Learning Goals
By the end of this aiMOOC, you should be able to:
- Construct a scatter plot: Plot paired numerical data correctly on labeled axes.
- Describe an association: Identify direction, form, strength, clusters, and outliers.
- Use a line of best fit: Draw an informal trend line and use it for reasonable estimates.
- Make predictions: Distinguish interpolation from extrapolation and judge when a prediction is sensible.
- Reason carefully: Explain why an association does not automatically prove that one variable causes the other.
Understanding Bivariate Data
Bivariate data contain measurements of two variables for the same cases. Imagine that you record the number of minutes each student studies and the score each student earns on a quiz. Each student contributes one ordered pair, such as 35 minutes and 78 points.
A scatter plot is suitable when both variables are numerical. Examples include height and arm span, outside temperature and ice-cream sales, hours of sleep and reaction time, or distance traveled and fuel used.
A scatter plot is usually not the best graph when your main variable is categorical, such as favorite color or type of pet. In that situation, a bar chart or another categorical display may be more appropriate.
How to Construct a Scatter Plot
Suppose a class records study time and quiz score. You can construct a scatter plot by:
- Choosing axes: Put one numerical variable on the horizontal axis and the other on the vertical axis.
- Choosing useful scales: Select intervals that cover the data without wasting most of the graph.
- Labeling both axes: Include variable names and units when units are available.
- Plotting each pair: One case becomes one point whose coordinates match its two measurements.
- Checking the display: Make sure every pair is represented and no point has been accidentally shifted.
Do not connect the points in time order unless the graph is specifically meant to show a sequence. The purpose of a scatter plot is to reveal a relationship between two numerical variables.
Describing Associations
When you interpret a scatter plot, look for a combination of features rather than one detail. A useful description often includes direction, form, strength, clusters, and outliers.
The small numbers shown on this image are correlation coefficients used in more advanced statistics. You do not need to calculate them here. Instead, compare the overall direction, shape, and tightness of the point clouds.
Direction
A positive association means that larger values of one variable tend to occur with larger values of the other variable. The overall cloud of points rises from left to right.
A negative association means that larger values of one variable tend to occur with smaller values of the other variable. The overall cloud of points falls from left to right.
If the points show no consistent rise or fall, there may be little or no association.
Direction describes an overall tendency. A positive association does not mean that every single point is higher than the point before it.
Form
The form describes the shape of the pattern. A linear association looks roughly like a band of points around a straight line. A nonlinear association bends or curves.
For Grades 7–8, you should first ask whether a straight-line model is a reasonable summary. If the points clearly curve, a straight line may hide important structure.
Strength
The strength of an association describes how tightly the points follow the overall pattern. If the points lie close to an imagined line or curve, the association is stronger. If they are widely scattered around the pattern, it is weaker.
You do not need a formal correlation coefficient to make a useful visual comparison. Your description should be supported by what you can actually see in the graph.
Clusters and Outliers
A cluster is a group of points gathered in one region of a graph. Clusters can suggest that the data contain subgroups. For example, a class might include students from two different training programs whose results form separate groups.
An outlier is a point that lies noticeably away from the overall pattern. An outlier can be a genuine unusual case, a measurement error, or a data-entry error. You should investigate it rather than automatically deleting it.
In the image above, the overall association is positive, but a few points sit away from the main pattern. Those points matter because they can affect how you judge the trend.
Real-World Interpretation
Scatter plots become more meaningful when you connect the axes to a real situation.
This example compares class attendance with class marks. The visible upward tendency can be described as a positive association. However, a graph alone does not prove that attendance by itself causes a particular mark. Other factors may also matter, such as prior knowledge, health, study habits, teaching conditions, or access to learning resources.
A careful interpretation separates three questions:
- What pattern is visible?: Describe what the points show.
- What estimate is reasonable?: Use the pattern only within a sensible range.
- What causes what?: Do not claim cause and effect unless the study design and evidence support that conclusion.
Lines of Best Fit
When a scatter plot has a roughly linear association, you can draw an informal line of best fit, also called a trend line. The goal is not to pass through every point. Instead, the line should follow the center of the data cloud, with points reasonably balanced above and below it.
A line of best fit can help you estimate a value. For example, if a graph relates hours of practice to a performance score, you might use the line to estimate the score for a practice time that falls inside the observed data range.
Interpolation and Extrapolation
Interpolation means making a prediction inside the range of observed data. It is often more trustworthy because the prediction is supported by nearby points.
Extrapolation means making a prediction outside the observed range. It can be risky because the relationship may change beyond the data you collected.
For example, if your data cover temperatures from 10°C to 30°C, predicting at 22°C is interpolation. Predicting at 60°C is extrapolation and should be treated much more cautiously.
Association Does Not Automatically Mean Causation
Two variables can be associated without one directly causing the other. A third variable may influence both.
Imagine that sunglasses sales and ice-cream sales both rise during warm months. Buying sunglasses does not cause people to buy ice cream. A lurking factor, such as warmer weather, can help explain why both variables rise at the same time.
This idea is important whenever you read graphs in news reports, advertisements, science articles, or social media. Ask what was measured, how the data were collected, whether other variables could matter, and whether the graph supports the claim being made.
Why Looking at the Plot Matters
Two data sets can have similar numerical summaries but very different shapes. That is one reason data visualization matters.
Anscombe's quartet is a famous example in which four data sets share the same means, variances, correlation, and fitted regression line even though their visual patterns are very different. At your level, the key lesson is simple: do not rely on one number alone when a graph can reveal shape, unusual points, or other structure.
A Practical Description Checklist
When you describe a scatter plot, try to write a complete statement such as: “The graph shows a moderately strong positive linear association between the two variables, with one possible outlier near the upper-left part of the plot.”
Use these questions:
- Direction: Is the association positive, negative, or not clear?
- Form: Is the pattern roughly linear or curved?
- Strength: Are points tightly grouped around the pattern or widely scattered?
- Clusters: Are there separate groups of points?
- Outliers: Are any points unusually far from the main pattern?
- Context: What do the variables and units actually mean?
Common Mistakes and How to Avoid Them
A scatter plot is easy to draw, but careful reasoning is still important.
- Misleading scales: Check whether the chosen scales make the pattern look stronger or weaker than it really is.
- Causal claims: An association alone does not establish cause and effect.
- Wild extrapolation: Predictions far beyond the observed data range may fail.
- Ignoring unusual points: An outlier may contain important information.
- Confusing variables: State clearly which quantity belongs on each axis and include units.
- Overgeneralizing: A small or biased sample may not represent a larger population.
Interactive Tasks
Quiz: Test Your Knowledge
What does one point in a scatter plot usually represent? (One case with two numerical measurements) (!One category with no numerical value) (!The average of every value in the data set) (!A line connecting two different graphs)
Which description best matches a positive association? (Larger values of one variable tend to occur with larger values of the other) (!Larger values of one variable always cause larger values of the other) (!The points must form a perfect straight line) (!The variables must have the same units)
Which pattern suggests a negative association? (The points tend to fall from left to right) (!The points tend to rise from left to right) (!All points have the same horizontal coordinate) (!Every point lies on the horizontal axis)
What does the strength of an association describe? (How closely the points follow the overall pattern) (!How large the axis labels are) (!How many colors are used in the graph) (!How far the graph is from the page edge)
What is an outlier in a scatter plot? (A point noticeably away from the main pattern) (!The first point plotted) (!A label written beside an axis) (!The center of every cluster)
What is the purpose of an informal line of best fit? (To summarize the center of a roughly linear trend) (!To connect every point in order) (!To force all points onto one straight line) (!To replace the original data values)
Which prediction is an example of interpolation? (A prediction made inside the observed data range) (!A prediction made far beyond the observed data range) (!A prediction made without looking at any data) (!A prediction based only on a category name)
Why can extrapolation be risky? (The relationship may change outside the observed range) (!Scatter plots cannot contain numerical values) (!All extrapolated values must be negative) (!The horizontal axis disappears outside the range)
What can you conclude from association alone? (The two variables show a pattern of relationship) (!One variable definitely causes the other) (!No other variable can affect the results) (!Every future case will follow the pattern exactly)
Why is it useful to inspect a scatter plot even when summary numbers are available? (The plot can reveal shape clusters and unusual points) (!A plot proves that every association is causal) (!A plot removes all measurement error) (!A plot guarantees perfect predictions)
Memory Game
| Positive association | Larger values tend to occur with larger values |
| Negative association | Larger values tend to occur with smaller values |
| Outlier | A point noticeably separated from the main pattern |
| Cluster | A group of points gathered in one region |
| Interpolation | A prediction made inside the observed data range |
| Extrapolation | A prediction made outside the observed data range |
| Line of best fit | A straight line that informally summarizes a linear trend |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Points tend to rise from left to right | Positive association |
| Points tend to fall from left to right | Negative association |
| Points gather in a separate group | Cluster |
| A point lies far from the main pattern | Outlier |
| An estimate is made within the data range | Interpolation |
...
Crossword Puzzle
| Scatterplot | What graph displays paired numerical data as points? |
| Association | What word describes a relationship or pattern between two variables? |
| Positive | What direction has a pattern that generally rises from left to right? |
| Negative | What direction has a pattern that generally falls from left to right? |
| Outlier | What is an unusual point far from the main pattern? |
| Prediction | What do you call an estimate of an unknown value based on a pattern? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Scatter plot sketch: Collect five to eight paired measurements from your classroom, such as hand span and height, and draw a scatter plot with clear axis labels.
- Association description: Find a scatter plot in a textbook or trusted website and write two sentences describing its direction and strength.
- Outlier hunt: Create a small data set that contains one unusual point, plot it, circle the outlier, and explain why it stands out.
- Graph explanation video: Record a one-minute video in which you explain how one point on a scatter plot represents two measurements.
Standard
- Class survey: Design a short survey that collects two numerical variables from at least ten volunteers, create a scatter plot, and describe the association without claiming causation.
- Line of best fit investigation: Plot a data set with a roughly linear pattern, draw an informal line of best fit, and use it to make two reasonable estimates.
- Media graph review: Locate a scatter plot in a news article or public report, identify the variables, and evaluate whether the written claim matches the visual evidence.
- Interview about data: Interview a teacher, coach, technician, or another adult about two quantities they compare in their work, then propose how a scatter plot could help them.
Advanced
- Interpolation and extrapolation experiment: Collect paired data from a simple experiment, build a scatter plot, make one interpolation and one extrapolation, and compare how confident you are in each estimate.
- Causation challenge: Create a poster or infographic showing a plausible association that could be explained by a third variable, and clearly separate association from causation.
- Outlier impact study: Create or find a data set with one influential outlier, draw a line of best fit with and without that point, and explain how your interpretation changes.
- Data story project: Produce a short written report, slide deck, or video that combines a question, data collection method, scatter plot, association description, prediction, limitations, and conclusion.
Learning Assessment
- Constructing scatter plots: Given a table of paired measurements, create a correctly scaled and labeled scatter plot and explain two choices you made while constructing it.
- Interpreting associations: Compare two scatter plots and justify which shows the stronger association using direction, form, spread, clusters, and outliers.
- Evaluating predictions: Use a line of best fit to make a prediction, then decide whether the prediction is interpolation or extrapolation and explain why that distinction matters.
- Reasoning about causation: Read a claim based on an observed association and write a response that identifies at least one alternative explanation or lurking variable.
- Critiquing data displays: Examine a scatter plot with unusual scaling, missing labels, or an outlier and explain how those features could affect interpretation.
- Transferring statistical thinking: Choose a real question from science, sports, school life, or technology and explain what paired data you would collect to investigate it with a scatter plot.
Evidence of Learning
Important evidence of learning includes:
- Statistical knowledge: You can explain bivariate data, association, direction, form, strength, clusters, outliers, interpolation, extrapolation, and causation in context.
- Graphing skill: You can construct a readable scatter plot from paired numerical data with appropriate axes, scales, labels, and units.
- Interpretation skill: You can describe a visible pattern using evidence from the graph rather than vague statements.
- Modeling skill: You can draw and use an informal line of best fit when a linear model is reasonable.
- Communication product: You can produce a graph, written explanation, poster, presentation, or video that accurately communicates a bivariate-data investigation.
- Transfer achievement: You can evaluate scatter plots in unfamiliar contexts and recognize when claims, predictions, or causal conclusions go beyond the evidence.
OERs on the Topic
Useful related learning topics include Scatter plot, Bivariate data, Correlation, Line of best fit, Outlier, Data visualization, and Statistics.
Linked Learning Areas