The Central Idea
Bivariate data consist of paired observations on two variables.
For each case, we record
\[ (x_i,y_i). \]The central question is:
How does one variable tend to change as the other changes?
From Univariate to Bivariate Data
A univariate analysis studies one variable at a time.
A bivariate analysis studies two variables together.
For example:
- univariate: study student exam scores;
- bivariate: study exam scores together with hours studied.
Paired Observations
The pairing is essential.
Suppose five students have study times and exam scores:
| Student | Hours Studied \(x\) | Exam Score \(y\) |
|---|---|---|
| A | 2 | 68 |
| B | 4 | 74 |
| C | 5 | 80 |
| D | 7 | 86 |
| E | 9 | 91 |
The meaningful observations are the pairs
\[ (2,68),\ (4,74),\ (5,80),\ (7,86),\ (9,91). \]Why Pairing Matters
If the study times and exam scores were listed separately without preserving which values belong to the same student, the relationship between the variables would be lost.
Bivariate analysis depends on matching the two measurements for each case.
Explanatory and Response Variables
In many studies, one variable is treated as an explanatory variable and the other as a response variable.
The explanatory variable is often denoted by
\[ x, \]and the response variable by
\[ y. \]Example: Explanatory and Response Roles
If we study whether hours of training help explain task-completion time, then:
- hours of training may be the explanatory variable;
- task-completion time may be the response variable.
These roles are contextual rather than automatically determined by the mathematical values themselves.
Not Every Bivariate Study Has a Natural Direction
Sometimes the goal is simply to study association between two variables, without designating one as explanatory and the other as response.
For example, a study of height and arm span may focus on their relationship without treating one as a causal predictor of the other.
Scatterplots
A scatterplot displays paired quantitative observations as points in the coordinate plane.
Each case produces one point
\[ (x_i,y_i). \]The horizontal axis represents one quantitative variable, and the vertical axis represents the other.
Constructing a Scatterplot
To construct a scatterplot:
- place the explanatory variable, if there is one, on the horizontal axis;
- place the response variable on the vertical axis;
- label both axes and units;
- plot one point for each paired observation.
What to Look for in a Scatterplot
A useful scatterplot description addresses:
- direction;
- form;
- strength;
- clusters;
- gaps;
- unusual observations.
Positive Association
A relationship has positive association when larger values of one variable tend to occur with larger values of the other.
In a scatterplot, the overall pattern tends to rise from left to right.
Example: Positive Association
Suppose advertising expenditure and sales revenue tend to increase together.
A scatterplot may show points rising from lower-left toward upper-right.
This suggests positive association.
Negative Association
A relationship has negative association when larger values of one variable tend to occur with smaller values of the other.
In a scatterplot, the overall pattern tends to fall from left to right.
Example: Negative Association
Suppose product price and quantity demanded tend to move in opposite directions within a particular market setting.
A scatterplot may slope downward overall, suggesting negative association.
No Clear Association
If points show no systematic upward, downward, or structured pattern, the variables may have little visible association.
This does not prove that the variables are completely unrelated.
A relationship may be weak, nonlinear, obscured by noise, or dependent on another variable.
Linear Form
A relationship is approximately linear when the points cluster around a straight-line pattern.
The line need not pass through every point.
Linear structure means that a straight line captures the main trend reasonably well.
Nonlinear Form
A relationship is nonlinear when the main pattern bends or follows some other systematic form.
Possible forms include:
- curved growth;
- decay;
- U-shaped relationships;
- threshold effects;
- periodic patterns.
Why Form Matters
A numerical summary designed for linear association may fail to describe a strong nonlinear pattern.
Always inspect a scatterplot before relying on a single association statistic.
Strength of Association
Strength describes how tightly the points follow the underlying pattern.
A strong relationship has points concentrated near a recognizable form.
A weak relationship has greater scatter around the underlying form.
Strong Does Not Mean Steep
The steepness of a relationship and its strength are different concepts.
A shallow linear relationship can be very strong if the points lie close to a line.
A steep relationship can be weak if the points are widely scattered.
Clusters in Bivariate Data
A scatterplot may contain clusters of points.
Clusters can suggest that the data contain distinct subgroups.
For example, a plot of engine size and fuel consumption may contain separate clusters for passenger cars and heavy trucks.
Hidden Group Structure
If a scatterplot contains several clusters, a third categorical variable may help explain the pattern.
Examples include:
- region;
- product type;
- age group;
- machine model;
- treatment group.
Gaps in a Scatterplot
A gap is a region of the coordinate plane with few or no observations.
Gaps may result from:
- natural limits;
- sampling structure;
- different subgroups;
- threshold rules;
- missing ranges of the explanatory variable.
Bivariate Outliers
An observation may be unusual because its pair
\[ (x_i,y_i) \]does not fit the overall bivariate pattern.
Such a point may be ordinary in each variable separately but unusual when the two values are considered together.
Example: A Bivariate Outlier
Suppose study time and exam score show a strong positive relationship, but one student studies many hours and receives an unusually low score.
The student's study time and score may each fall within ordinary ranges, yet the pair may lie far from the main trend.
Influential Observations
Some points can strongly affect numerical summaries or fitted models.
Such observations are called influential when including or removing them substantially changes the analysis.
Influence is especially important in later regression work.
Association Is Not Causation
A statistical association between two variables does not automatically imply that changes in one variable cause changes in the other.
Association can arise from:
- direct causation;
- reverse causation;
- a third variable;
- selection effects;
- common trends over time;
- chance variation.
Example: Association Without Direct Causation
Ice-cream sales and sunburn cases may both rise during hot weather.
Their association does not mean that ice-cream purchases cause sunburn.
Temperature is a plausible third variable influencing both.
Lurking Variables
A lurking variable is a variable not included in the analysis that may help explain an observed relationship.
Lurking variables can create, weaken, strengthen, or reverse apparent associations.
Confounding
Confounding occurs when the effects of variables are mixed together so that their individual contributions cannot be cleanly separated.
Observational studies are particularly vulnerable to confounding.
Observational Studies
In an observational study, researchers observe variables without assigning treatments.
Such studies can identify association, but causal conclusions usually require additional assumptions and careful design.
Experiments
In a well-designed experiment, researchers deliberately assign treatments or conditions.
Random assignment can help separate treatment effects from confounding variables, making causal conclusions more defensible.
Time Can Create Apparent Association
Two variables may both increase over time and therefore appear strongly associated even if they do not directly affect one another.
For example, over many decades:
- population may increase;
- total energy consumption may increase.
Their common time trend can contribute to the observed association.
Range Restriction
The observed relationship can depend on the range of values included.
If a study samples only a narrow band of the explanatory variable, a relationship that is strong across the full population may appear weak within the restricted sample.
Example: Restricted Range
Height and weight may show a clear positive association across all adults.
If a sample includes only people between \(170\) cm and \(172\) cm tall, the relationship may appear much weaker because the height range is so narrow.
Extrapolation
Extrapolation means extending a relationship beyond the range of observed data.
This can be risky because the relationship may change outside the observed range.
A pattern that is approximately linear over one interval may curve, flatten, or reverse elsewhere.
Marginal Versus Joint Information
The separate distributions of \(x\) and \(y\) are called their marginal distributions.
Bivariate analysis also studies the joint structure of the paired observations.
Two datasets can have the same marginal distributions but different pairings and therefore very different bivariate relationships.
Example: Same Marginals, Different Pairing
Suppose
\[ x: 1,\ 2,\ 3,\ 4 \]and
\[ y: 10,\ 20,\ 30,\ 40. \]Pairing them as
\[ (1,10),\ (2,20),\ (3,30),\ (4,40) \]creates a strong positive pattern.
Pairing them as
\[ (1,40),\ (2,30),\ (3,20),\ (4,10) \]creates a strong negative pattern.
The separate \(x\) and \(y\) values are unchanged; only the pairing changed.
Standardization in Bivariate Analysis
Each variable can be standardized separately:
\[ z_x=\frac{x-\bar{x}}{s_x}, \] \[ z_y=\frac{y-\bar{y}}{s_y}. \]Standardized values place both variables on unit-free scales and prepare the way for correlation.
Quadrants Relative to the Means
Compare each observation with the variable means.
A point may lie:
- above both means;
- below both means;
- above the \(x\)-mean but below the \(y\)-mean;
- below the \(x\)-mean but above the \(y\)-mean.
This structure will connect directly to the sign of covariance and correlation.
Concordant Direction Around the Means
If
\[ x_i-\bar{x} \]and
\[ y_i-\bar{y} \]have the same sign, their product is positive:
\[ (x_i-\bar{x})(y_i-\bar{y})>0. \]Such points support positive association.
Opposite Direction Around the Means
If the deviations have opposite signs, then
\[ (x_i-\bar{x})(y_i-\bar{y})<0. \]Such points support negative association.
This idea leads naturally to covariance.
Describing a Scatterplot Systematically
A useful description might say:
The scatterplot shows a strong positive approximately linear association, with one possible outlier at high \(x\) and low \(y\), and no obvious clusters.
This is more informative than simply saying "the variables are related."
Example: Strong Positive Linear Association
Suppose points lie close to an upward-sloping straight-line pattern.
A reasonable description is:
- direction: positive;
- form: approximately linear;
- strength: strong;
- unusual features: none obvious.
Example: Weak Negative Association
Suppose points show a broad downward tendency but are widely scattered.
A reasonable description is:
- direction: negative;
- form: approximately linear;
- strength: weak;
- unusual features: inspect any isolated points separately.
Example: Strong Nonlinear Association
Suppose points follow a tight U-shaped curve.
The relationship can be strong even though it is not linear.
This is a key reason to inspect a scatterplot before using linear statistics.
Common Error: Ignoring Pairing
Computing separate summaries of \(x\) and \(y\) does not reveal how the variables move together.
Preserve the paired structure.
Common Error: Calling Any Upward Pattern Causal
An upward scatterplot shows positive association.
It does not by itself establish that increasing \(x\) causes \(y\) to increase.
Common Error: Describing Only Direction
Saying "positive association" leaves out form, strength, clusters, and unusual observations.
A complete description should address all major structural features.
Common Error: Assuming No Linear Pattern Means No Relationship
A strong curved relationship may have little linear appearance.
Always inspect for nonlinear structure before concluding that no association exists.
Common Error: Treating Strength as Slope
Strength concerns how tightly points follow a pattern.
Slope concerns how rapidly one variable changes relative to the other.
They are different properties.
Common Error: Extrapolating Far Beyond the Data
A relationship observed for
\[ 10\le x\le20 \]may not continue unchanged at
\[ x=100. \]Predictions far outside the observed range require strong justification.
Common Error: Ignoring Clusters
If several groups form distinct clusters, one overall relationship may hide important subgroup differences.
Investigate whether another variable explains the grouping.
Common Error: Removing an Unusual Point Automatically
An unusual bivariate observation may be important rather than erroneous.
Investigate:
- data-entry accuracy;
- measurement conditions;
- case identity;
- subgroup membership;
- its effect on later numerical summaries.
A Reliable Procedure for Bivariate Analysis
- Identify the two variables and their units.
- Preserve the paired observations.
- Identify explanatory and response roles when appropriate.
- Construct or inspect a scatterplot.
- Describe direction.
- Describe form.
- Describe strength.
- Identify clusters, gaps, and unusual observations.
- Consider lurking variables and study design.
- Separate association from causal claims.
Practice 1: Identify the Pair
A study records hours studied and exam score for each student.
If one student studies \(6\) hours and scores \(84\), write the paired observation.
Practice 2: Positive or Negative?
As outdoor temperature increases, heating-fuel use tends to decrease.
What direction of association is suggested?
Practice 3: Describe Form
Points lie close to a straight line that rises from left to right.
Describe the direction and form.
Practice 4: Strength
Two scatterplots have the same general upward slope. In Plot A the points lie very close to a line. In Plot B they are widely scattered.
Which plot shows the stronger association?
Practice 5: Nonlinear Relationship
Points lie close to a U-shaped curve.
Is the relationship weak simply because it is not linear?
Practice 6: Association Versus Causation
A study finds that students who sleep more also tend to earn higher exam scores.
Does this observational association prove that additional sleep causes the higher scores?
Practice 7: Lurking Variable
Sunglasses sales and cold-drink sales rise together.
Name one plausible lurking variable.
Practice 8: Bivariate Outlier
Most observations show higher training time associated with faster task completion, but one heavily trained worker has an unusually slow completion time.
What feature should be investigated?
Practice 9: Extrapolation
A linear trend is observed only for
\[ 1\le x\le10. \]Why should prediction at \(x=100\) be treated cautiously?
Practice 10: Full Scatterplot Description
A scatterplot rises from left to right, points lie close to a straight-line pattern, and one isolated point lies far below the main trend.
Give a concise description.
Practice Answers
Practice 1
\[ \boxed{(6,84)}. \]Practice 2
The association is negative because larger temperature values tend to occur with smaller heating-fuel use.
Practice 3
The association is positive and approximately linear.
Practice 4
Plot A shows the stronger association because its points are more tightly concentrated around the linear pattern.
Practice 5
No. A relationship can be strong and nonlinear.
Practice 6
No. An observational association does not by itself establish causation. Other variables or selection effects may contribute to the relationship.
Practice 7
A plausible lurking variable is temperature or season.
Practice 8
The isolated pair should be investigated as a possible bivariate outlier or influential observation.
Practice 9
The observed relationship is known only within the range \(1\le x\le10\). The pattern may change outside that range, so prediction at \(x=100\) is extrapolation.
Practice 10
The scatterplot shows a strong positive approximately linear association with one possible outlier below the main trend.
What to Remember
- Bivariate data preserve paired observations on two variables.
- A scatterplot represents each pair as one point \((x_i,y_i)\).
- Describe bivariate relationships by direction, form, strength, and unusual features.
- Positive association means larger values tend to occur together.
- Negative association means larger values of one variable tend to occur with smaller values of the other.
- Strong relationships can be linear or nonlinear.
- Clusters may reveal hidden subgroup structure.
- Bivariate outliers can be unusual even when neither coordinate is individually extreme.
- Association does not automatically imply causation.
- Lurking variables, confounding, restricted ranges, and extrapolation can affect interpretation.
Where We Go Next
Scatterplots reveal direction, form, and strength visually. The next step is to quantify how two quantitative variables vary together.
The next lesson, Covariance and Correlation, develops numerical measures of joint variation and standardized linear association.