The Central Idea
Suppose each member of a group has two numerical measurements. For one student, we might record \(4\) hours studied and a score of \(72\). That paired observation becomes the point
\[ (4,72). \]Plotting every pair produces a scatterplot. The collection of points lets us investigate whether larger values of one variable tend to occur with larger values, smaller values, or no consistent change in the other variable.
The key principle is
\[ \boxed{\text{one case} \longleftrightarrow \text{one ordered pair} \longleftrightarrow \text{one point}.} \]Paired Quantitative Data
A scatterplot is appropriate when the data contain two quantitative variables measured on the same cases. A case may be a person, business, city, machine, day, or any other unit being studied.
If the variables are \(x\) and \(y\), the data have the form
\[ (x_1,y_1),\ (x_2,y_2),\ldots,(x_n,y_n). \]The subscripts preserve the pairing: \(x_i\) and \(y_i\) must come from the same case.
Example: Study Time and Score
Consider six students with the following measurements.
| Student | Study time (hours) | Test score |
|---|---|---|
| A | 1 | 58 |
| B | 2 | 64 |
| C | 3 | 67 |
| D | 4 | 72 |
| E | 5 | 81 |
| F | 6 | 85 |
The plotted points are
\[ (1,58),\ (2,64),\ (3,67),\ (4,72),\ (5,81),\ (6,85). \]The points generally rise from left to right. Students who studied longer tended to have higher scores in this group.
The Explanatory and Response Variables
When one variable is used to explain or predict another, the variables receive different roles.
- The explanatory variable is usually plotted on the horizontal axis and denoted by \(x\).
- The response variable is usually plotted on the vertical axis and denoted by \(y\).
For study time and score, study time is a natural explanatory variable and score is a natural response variable. These names describe analytical roles; they do not by themselves prove a cause-and-effect relationship.
When the Roles Are Symmetric
Sometimes neither variable naturally explains the other. For example, a researcher may compare the lengths of a person's left and right hands. Either variable could be placed on either axis.
In a symmetric comparison, changing the axes changes the appearance of the graph but not the fact that the variables are associated. The axis choice should still be stated clearly.
Constructing a Scatterplot
- Confirm that both variables are quantitative and properly paired.
- Choose the explanatory variable for the horizontal axis when roles exist.
- Choose scales that cover the observed values without distorting the pattern.
- Label each axis with the variable name and its units.
- Plot one point for every case.
- Add a concise title or caption that identifies the context.
Points are not normally connected in a scatterplot. Connecting them would suggest an order or continuous path that may not exist.
Choose Honest Scales
Axis intervals should be consistent, readable, and wide enough to include all observations. A scatterplot does not always need to begin at zero, but a restricted range can make small differences look visually dramatic.
Before interpreting a graph, inspect
- where each axis begins and ends;
- the size of each tick interval;
- whether either axis uses a transformed scale;
- whether units are identified.
What Is Association?
Two variables are associated when values of one variable tend to occur with particular values of the other variable.
Association concerns an overall pattern, not a rule that every point must follow. The study-time data show an upward tendency even if one student who studied longer scored below another student who studied less.
How to Describe a Scatterplot
A complete first description addresses four features:
- Direction: positive, negative, or neither.
- Form: linear, curved, clustered, or another shape.
- Strength: how closely the points follow the form.
- Unusual features: outliers, gaps, clusters, or influential points.
Every description should name the variables and interpret the pattern in context.
Positive Association
A scatterplot has a positive association when larger values of \(x\) tend to occur with larger values of \(y\). The overall pattern rises from left to right.
Examples may include height and weight among adults, advertising spending and sales across comparable markets, or outside temperature and electricity demand for air conditioning.
Positive does not mean beneficial. It describes direction only.
Negative Association
A scatterplot has a negative association when larger values of \(x\) tend to occur with smaller values of \(y\). The overall pattern falls from left to right.
Examples may include the age of a used vehicle and its resale value, or distance from a wireless router and signal strength under controlled conditions.
Negative does not mean undesirable. It also describes direction only.
No Clear Direction
A plot has no clear positive or negative direction when high and low values of \(y\) appear throughout the range of \(x\) without a consistent upward or downward tendency.
This does not prove that the variables are completely unrelated. A curved pattern, separate subgroups, measurement error, or a limited range of data may conceal a relationship that a simple directional description misses.
Linear Form
An association is linear when its point cloud follows an approximately straight path. A line could summarize the main tendency without systematically missing the pattern.
The points do not need to lie exactly on a line. Real data normally vary around the underlying pattern.
Nonlinear Form
An association is nonlinear when the typical value of \(y\) changes with \(x\) along a curve or another non-straight form.
For example, plant growth may improve as fertilizer increases to a useful level and then decline when excessive fertilizer damages the plant. The resulting pattern may rise and then fall.
A strong nonlinear association can have little overall upward or downward direction. Form must therefore be inspected before a straight-line method is applied.
Strength of Association
Strength describes how tightly the points follow the stated form.
- A strong association has points close to the overall pattern.
- A moderate association has a visible pattern with noticeable scatter.
- A weak association has a faint pattern and substantial scatter.
These visual labels are approximate. Later lessons develop numerical tools for measuring linear association and modeling prediction.
Perfect Association
A perfect linear association occurs when every point lies exactly on one straight line. If that line rises, the association is perfectly positive; if it falls, the association is perfectly negative.
Perfect relationships are unusual in observational data. They may occur when one measurement is computed directly from another, such as converting temperatures from Celsius to Fahrenheit:
\[ F=\frac{9}{5}C+32. \]Strength Is Not Slope
A steep pattern is not necessarily strong, and a nearly horizontal pattern is not necessarily weak. Slope describes the rate and direction of change; strength describes how closely points follow the form.
Rescaling an axis can make a pattern look steeper or flatter without changing how tightly the points cluster around it. Always keep units and scale separate from strength.
Variation Around a Pattern
Even when two variables are associated, cases with the same or similar \(x\)-values may have different \(y\)-values. Other variables, natural variation, and measurement error can all contribute to this vertical spread.
A scatterplot therefore supports statements about tendencies: “stores with more floor space tended to have higher weekly sales.” It does not support an exact rule unless the data actually show one.
Clusters
A cluster is a group of observations concentrated in one region of the plot. Multiple clusters may indicate that the data contain distinct populations or operating conditions.
For example, a plot of vehicle weight against fuel consumption might form separate clusters for passenger cars and commercial trucks. A useful next step would be to identify the vehicle type of each point.
Gaps
A gap is a region containing few or no observations. A gap may result from natural separation, a sampling rule, unavailable cases, or an error in data collection.
Gaps matter because conclusions drawn on one side of a gap may not extend reliably across it.
Outliers
An outlier is an observation that departs noticeably from the overall pattern. In a plot of house size against sale price, a very large house may not be an outlier if its price follows the main trend. A modest house with an exceptionally high price may be an outlier relative to the relationship.
An outlier should be investigated, not automatically deleted. It may represent
- a recording or measurement error;
- a valid but unusual case;
- a missing explanatory variable;
- a different population or process.
High-Leverage and Influential Observations
A point with an \(x\)-value far from the rest of the data has high leverage. Because of its horizontal separation, it can strongly affect the apparent direction or fitted line.
An observation is influential when including or excluding it materially changes a statistical conclusion. High leverage creates the opportunity for influence, but a high-leverage point that follows the main pattern may have less disruptive effect than one that does not.
Example: One Point Changes the Story
Suppose five observations form a diffuse cloud with no visible direction, and a sixth point lies far to the upper right. Taken together, the six points may appear strongly positive.
A responsible analysis should describe both views: the complete data show an apparent positive association, but that appearance depends heavily on one distant observation. The case should be verified and interpreted in context before a general conclusion is made.
Overplotting
Different cases can share the same or nearly the same coordinates. Their points may overlap, making a dense region appear to contain fewer cases than it does. This problem is called overplotting.
Smaller markers, partial transparency, slight jitter, or counts within small regions can reveal hidden density. Any adjustment should be stated so that readers understand what the display represents.
Subgroups Can Hide or Reverse Patterns
A single scatterplot may combine groups that have different relationships. The overall association can be weaker, stronger, or even opposite in direction from the association within each group.
Color, symbol, or separate panels can identify a relevant categorical variable. When groups are present, describe both the within-group patterns and the combined pattern.
Restricted Range
An association can be difficult to see when the sampled cases cover only a narrow range of one variable. Among adults of nearly the same height, for example, a height-and-weight association may appear weaker than it would in a population with a broader height range.
Conclusions should be limited to the population and range actually represented by the data.
Time Can Create Apparent Association
Two variables that both rise over time may show a positive association even if neither directly affects the other. A plot of annual internet use against annual healthcare spending could rise largely because both have long-term trends.
When observations occur across time, inspect the time order and consider whether trend, seasonality, or other temporal structure explains the pattern.
Association Is Not Causation
A scatterplot can show that variables move together, but it cannot by itself establish that changing \(x\) will cause a change in \(y\).
An observed association may arise from
- a causal effect of \(x\) on \(y\);
- a causal effect of \(y\) on \(x\);
- a third variable affecting both;
- selection or measurement bias;
- chance variation.
Causal claims require evidence from study design and subject-matter reasoning, not merely a visible pattern.
Example: A Lurking Variable
Ice-cream sales and sunburn cases may be positively associated. Buying ice cream does not ordinarily cause sunburn. Hot, sunny weather increases both outdoor exposure and demand for ice cream.
Weather is a lurking variable: an unexamined variable that helps explain the observed relationship. Identifying plausible lurking variables is part of responsible interpretation.
Do Not Extrapolate Carelessly
Extrapolation uses a pattern observed in one range to make claims beyond that range. A relationship that is approximately linear within the data may flatten, curve, or reverse outside them.
If the observed study times range from \(1\) to \(6\) hours, the scatterplot does not justify predicting the score of a student who studies \(30\) hours. Predictions are generally more defensible within the observed range, where they are called interpolations.
Describing a Pattern in Context
Avoid a bare statement such as “the graph is positive.” A useful description names the variables, the cases, and the practical meaning.
For the study-time data, an appropriate description is:
Among these six students, study time and test score have a strong, positive, approximately linear association. Students who studied longer tended to earn higher scores, and the plot contains no obvious outlier.
The wording reports association without claiming that the graph alone proves causation.
A Reliable Interpretation Procedure
- Identify the cases and the two quantitative variables.
- Read the axis labels, units, scales, and observed ranges.
- Describe direction.
- Describe form.
- Describe strength.
- Identify clusters, gaps, outliers, and potentially influential points.
- Translate the pattern into the context.
- State important limitations, especially for causal or extrapolated claims.
Common Error: Using a Scatterplot for Categorical Data
A scatterplot requires two quantitative variables. If one variable is a category such as department, region, or product type, placing category codes \(1,2,3\) on a numerical axis creates artificial distances and order.
Use a display designed for categorical comparisons, or use color or symbol to show the category while two genuinely quantitative variables remain on the axes.
Common Error: Breaking the Pairing
Sorting the \(x\)-values and \(y\)-values independently destroys the cases that produced them. The new ordered pairs no longer represent real observations and can manufacture a misleading association.
Rows may be reordered, but the two measurements for each case must always move together.
Common Error: Looking Only at Direction
Calling a relationship “positive” is incomplete. A positive pattern may be linear or curved, strong or weak, uniform or driven by one point.
Always inspect direction, form, strength, and unusual features as a set.
Common Error: Treating Every Point as a Trend
Association summarizes the point cloud. It does not mean that \(y\) must increase whenever \(x\) increases from one selected case to another. Individual reversals are compatible with a positive overall association.
Common Error: Deleting an Inconvenient Point
Removing an unusual observation merely because it weakens the desired pattern biases the analysis. Verify the record, investigate its source, and report how conclusions change with and without it when appropriate.
Practice 1: Identify the Cases and Variables
A company records monthly advertising spending and monthly revenue for each of its \(24\) regional offices. Identify the cases, the explanatory variable, and the response variable for a prediction-oriented scatterplot.
Practice 2: Interpret Positive Association
A scatterplot of apartment floor area and monthly rent rises from left to right with moderate scatter. Describe the association in context without making a causal claim.
Practice 3: Interpret Negative Association
A plot of a battery's number of completed charge cycles against its maximum remaining capacity follows a tight downward path. Describe its direction, form, and strength.
Practice 4: Recognize Nonlinearity
Crop yield rises as irrigation increases from low to moderate levels, then levels off. Why would a single straight-line description be incomplete?
Practice 5: Investigate an Outlier
In a scatterplot of employee training hours and assessment scores, one record shows \(500\) training hours while all others fall between \(0\) and \(40\). Give two appropriate actions before deciding whether to retain the point.
Practice 6: Explain a Cluster
A plot of package weight against delivery cost forms two distinct clusters. What question should an analyst ask next?
Practice 7: Protect the Pairing
An analyst sorts the \(x\)-column from smallest to largest but leaves the \(y\)-column unchanged. Explain why the resulting scatterplot is invalid.
Practice 8: Association and Causation
Cities with more firefighters tend to have greater fire damage. Does this show that firefighters cause damage? Give a more plausible explanation.
Practice 9: Extrapolation
A roughly linear relationship is observed for engine speeds from \(1{,}000\) to \(4{,}000\) revolutions per minute. Why should a prediction at \(9{,}000\) revolutions per minute be treated cautiously?
Practice 10: Write a Complete Description
A scatterplot of outdoor temperature and household heating use has a strong, curved, downward pattern with one isolated point. Write the elements that a complete description should include.
Practice Answers
Practice 1
The cases are the \(24\) regional offices. Advertising spending is the explanatory variable, and revenue is the response variable. Each point represents one office's paired monthly measurements for the specified period.
Practice 2
Apartment floor area and monthly rent have a moderate positive association: larger apartments tend to have higher monthly rents in the observed data.
Practice 3
Charge cycles and maximum remaining capacity have a strong, negative, approximately linear association. Batteries with more completed cycles tend to have lower remaining capacity.
Practice 4
The rate of increase changes. A straight line would miss the leveling-off behavior and could overstate the gain from additional irrigation at high levels.
Practice 5
Check the source record for a misplaced decimal, incorrect unit, duplicate, or other entry error. Then identify the employee and determine whether the value is a valid unusual case, a different training program, or a member of another population. The point should not be removed merely because it is inconvenient.
Practice 6
Ask whether a categorical variable—such as shipping method, destination zone, carrier, or service level—separates the packages into two groups.
Practice 7
Independent sorting breaks the original case-level pairing. Each new point may combine an \(x\)-measurement from one case with a \(y\)-measurement from another, so the plotted observations are not real.
Practice 8
No. Fire severity is a plausible lurking variable: larger fires require more firefighters and also produce more damage. The association does not show that the firefighters caused the damage.
Practice 9
\(9{,}000\) revolutions per minute lies far outside the observed range. The relationship may change because of mechanical limits, heat, control systems, or failure, so the within-range linear pattern may not continue.
Practice 10
State that outdoor temperature and heating use have a strong, negative, nonlinear association: heating use tends to fall as temperature rises, but the rate of decline changes. Identify and investigate the isolated point, give the cases and units, and avoid claiming causation unless the study design supports it.
What to Remember
A scatterplot represents paired quantitative data:
\[ \boxed{(x_i,y_i)\text{ represents one case}.} \]To interpret the plot, describe
- direction: positive, negative, or neither;
- form: linear, curved, clustered, or another pattern;
- strength: how closely the points follow the form;
- unusual features: outliers, gaps, clusters, and influential points.
Interpret every feature in context, preserve the pairing of observations, and remember that association alone does not establish causation.
Where We Go Next
A scatterplot provides the essential visual analysis of paired quantitative data. When the form is approximately linear, we can supplement that visual judgment with a numerical measure of direction and strength.
The next lesson, Correlation, develops the correlation coefficient, explains what it measures, and examines the conditions and limitations that govern its interpretation.