Bivariate Data and Regression Models

Scatter Plots and Trend Identification

When we collect data about two variables at the same time, we call it bivariate data. For example, we might record the number of hours a student studies and the score they earn on a quiz. Since the two values belong together, we can graph them as points on a coordinate plane.

A scatter plot is a graph that shows pairs of data as points. It helps us look for patterns, trends, and relationships between the two variables.

Learning to read a scatter plot is important because it helps us answer questions like:

• Do the variables seem related?
• As one variable increases, does the other increase or decrease?
• Is the relationship strong or weak?
• Are there any unusual points that do not fit the pattern?

These ideas are the first step toward building prediction models later.

1. What a scatter plot shows

Each point on a scatter plot represents one data pair, written as \((x, y)\).

• The x-value is placed on the horizontal axis.
• The y-value is placed on the vertical axis.

For example, if a student studies for 3 hours and earns 85 points, that data pair is \((3, 85)\).

If we graph many such pairs, we can often see whether the points follow a pattern.

2. How to make a scatter plot

1. Choose which variable goes on the x-axis and which goes on the y-axis.
2. Draw and label both axes clearly.
3. Choose a scale that fits all the data.
4. Plot each ordered pair as a point.
5. Step back and look for an overall pattern.

Usually, the variable that may help explain or predict the other is placed on the x-axis. For example, study hours would usually go on the x-axis, and test score on the y-axis.

3. Identifying the direction of a trend

One of the first things to look for is the direction of the relationship.

• Positive trend: As \(x\) increases, \(y\) tends to increase.
• Negative trend: As \(x\) increases, \(y\) tends to decrease.
• No clear trend: The points do not seem to rise or fall in any clear way.

If the points move upward from left to right, the trend is positive. If they move downward from left to right, the trend is negative.

Examples of direction

• Hours practiced and free throw accuracy: often a positive trend.
• Speed of a car and time needed to finish a trip: often a negative trend.
• Shoe size and math grade: often no clear trend.

4. Identifying the strength of a trend

After finding the direction, we look at how tightly the points cluster around an imaginary line.

• Strong trend: Points are close together around a clear pattern.
• Weak trend: Points are more spread out, even if they still generally rise or fall.

A strong positive trend means the points rise from left to right and stay fairly close together. A weak positive trend still rises overall, but the points are more scattered.

The same idea applies to negative trends.

5. Clustering and spread

Sometimes points form a cluster, meaning many points are grouped close together in one part of the graph. Clusters can help us notice where most of the data is located.

For example, if most students studied between 2 and 4 hours, many points might appear in that region of the graph. A cluster does not always mean the trend is strong, but it tells us where the data is concentrated.

6. Outliers

An outlier is a point that lies far away from the rest of the data. It does not match the overall pattern.

Outliers matter because they can change how we describe the graph. Sometimes an outlier is caused by a mistake in recording data. Other times it is a real but unusual result.

When reading a scatter plot, always ask:

• Does one point look very different from the others?
• Does it affect the trend I see?

7. Association does not always mean cause

If two variables have a trend, we say they have an association. This means they seem related in the data.

However, this does not automatically mean one variable causes the other. For example, ice cream sales and swimming activity may both increase in summer. That does not mean buying ice cream causes people to swim. A third factor, warmer weather, affects both.

8. Describing a scatter plot clearly

When asked to describe a scatter plot, a good answer often includes:

• the direction of the trend,
• the strength of the trend,
• any clusters,
• any outliers.

For example, you might say:

“The scatter plot shows a moderate positive trend. The points generally rise from left to right, with most points clustered between \(x=2\) and \(x=5\). There is one outlier above the main pattern.”

Worked Example 1: Making and reading a scatter plot

A teacher records the following data for study time and quiz score:

\((1, 65), (2, 70), (3, 74), (4, 81), (5, 87)\)

Step 1: Plot the points.

Place study time on the x-axis and quiz score on the y-axis. Then plot each ordered pair.

Step 2: Look for a trend.

As study time increases from 1 to 5 hours, the quiz score also increases from 65 to 87.

Conclusion: The scatter plot shows a positive trend.

Step 3: Decide strength.

The points rise in a very regular way and stay close to a line.

Final description: This scatter plot shows a strong positive trend.

Worked Example 2: Negative trend

A cyclist tracks speed and travel time for the same distance:

\((10, 30), (15, 22), (20, 17), (25, 13), (30, 10)\)

Step 1: Interpret the variables.

• \(x\): speed in miles per hour
• \(y\): time in minutes

Step 2: Look for direction.

As speed increases, time decreases.

That means the scatter plot has a negative trend.

Step 3: Judge strength.

The points would lie close to a smooth downward pattern.

Final description: This scatter plot shows a strong negative trend.

Worked Example 3: Weak trend with clustering

Suppose the number of minutes students spend exercising each day is compared with hours of sleep:

\((10, 7), (15, 8), (20, 7), (25, 8), (30, 6), (35, 8), (40, 7)\)

Step 1: Look for direction.

As exercise time increases, sleep hours do not clearly rise or fall. The values move between 6, 7, and 8 hours.

Step 2: Look for strength.

There is no tight upward or downward pattern.

Step 3: Look for clusters.

Most y-values are clustered around 7 to 8 hours of sleep.

Final description: This scatter plot shows no clear trend, with a cluster around 7 to 8 hours of sleep.

Worked Example 4: Trend with an outlier

A plant scientist records sunlight hours and plant height:

\((1, 4), (2, 6), (3, 8), (4, 10), (5, 11), (6, 3)\)

Step 1: Look at most of the points.

The first five points show that as sunlight hours increase, plant height generally increases.

Step 2: Identify unusual points.

The point \((6, 3)\) is far below the others. A plant with 6 hours of sunlight would usually be expected to be taller based on the pattern.

Step 3: Describe the graph.

Most points show a positive trend, but there is an outlier at \((6, 3)\).

9. Useful sentence starters

These sentence starters can help when writing about scatter plots:

• “As \(x\) increases, \(y\) tends to increase/decrease.”
• “The points show a strong/moderate/weak positive/negative trend.”
• “There is no clear relationship between the variables.”
• “Most of the data is clustered around...”
• “There appears to be an outlier at...”

10. Common mistakes to avoid

• Looking only at one or two points: Always focus on the overall pattern.
• Forgetting direction: Say whether the trend is positive, negative, or none.
• Ignoring outliers: Unusual points can be important.
• Assuming cause: A trend does not prove one variable causes the other.
• Mixing up axes: Be sure you know which variable is on each axis.

11. Quick checklist for any scatter plot

When you see a scatter plot, ask yourself these questions:

1. What does each axis represent?
2. Do the points go up, go down, or show no pattern?
3. Are the points close together or spread out?
4. Are there clusters?
5. Are there outliers?

If you answer these five questions, you can usually describe the scatter plot correctly.

Summary

A scatter plot graphs pairs of values for two variables. It helps us see whether the variables have a positive trend, a negative trend, or no clear trend.

To describe a scatter plot well, look at the direction, strength, clusters, and any outliers. These ideas help us understand bivariate data and prepare us to make predictions from data later.

Correlation Coefficients (r) help us describe how strongly two variables are related in a linear way.

For example, we might look at:

• hours studied and test score,
• height and arm span,
• outside temperature and hot chocolate sales.

When we collect pairs of data like these, we call it bivariate data because it involves two variables.

The correlation coefficient, written as (r), is a number between -1 and 1.

It tells us both:

• the direction of the relationship, and
• the strength of the relationship.

Direction means whether the variables move together or in opposite directions.

• If r > 0, the association is positive: as one variable increases, the other tends to increase.
• If r < 0, the association is negative: as one variable increases, the other tends to decrease.
• If r = 0, there is no linear correlation.

Strength means how closely the data points follow a straight-line pattern.

• If independent the value of r is close to 1 or -1, the linear relationship is strong.
• If r is close to 0, the linear relationship is weak.

So the sign of r tells direction, and the distance from 0 tells strength.

The possible values of the correlation coefficient are:

Here is what common values mean:

• r = 1: perfect positive linear correlation
• r = -1: perfect negative linear correlation
• r = 0: no linear correlation

Values in between show different levels of strength.

A useful guide is:

• 0.8 to 1.0 or -0.8 to -1.0: very strong linear correlation
• 0.5 to 0.79 or -0.5 to -0.79: moderate linear correlation
• 0.1 to 0.49 or -0.1 to -0.49: weak linear correlation
• close to 0: little to no linear correlation

These ranges are just guides. The exact meaning can depend on the situation, but they are very helpful in school maths.

Main Idea: How to Interpret r

When you are given a value of r, ask yourself two questions:

1. Is it positive or negative?
2. Is it close to 0, or close to 1 or -1?

This gives you the direction and the strength right away.

For example:

• r = 0.92 means a very strong positive linear relationship.
• r = -0.88 means a very strong negative linear relationship.
• r = 0.21 means a weak positive linear relationship.
• r = -0.04 means almost no linear relationship.

Scatter Plots and Correlation

Correlation coefficients are often connected to scatter plots.

A scatter plot shows pairs of data points on a graph. The value of r describes how those points look.

• If the points rise from left to right, r is positive.
• If the points fall from left to right, r is negative.
• If the points are tightly clustered around a line, the correlation is strong.
• If the points are spread out with no clear straight-line pattern, the correlation is weak.

So, r is like a numerical summary of the scatter plot.

Important Note: Correlation Describes Linear Relationships

The correlation coefficient measures linear relationships only.

That means it checks how well the points fit a straight line.

Sometimes two variables are clearly related, but not in a straight-line way. In that case, r might be close to 0 even though a relationship exists.

For example, if data follows a curved pattern, the linear correlation may be weak.

So remember: no linear correlation does not always mean no relationship at all.

Worked Example 1: Interpreting a Positive Correlation

A class collects data on hours of practice and basketball free-throw accuracy. The correlation coefficient is r = 0.84.

Step 1: Look at the sign.

Since r is positive, the relationship is positive.

This means that as practice time increases, free-throw accuracy tends to increase.

Step 2: Look at the size.

0.84 is close to 1, so the relationship is very strong.

Answer: There is a very strong positive linear correlation between practice time and free-throw accuracy.

Worked Example 2: Interpreting a Negative Correlation

A student studies the relationship between the number of absences and a final course grade. The correlation coefficient is r = -0.76.

Step 1: Look at the sign.

Since r is negative, the relationship is negative.

This means that as absences increase, course grade tends to decrease.

Step 2: Look at the size.

|-0.76| = 0.76, which is fairly close to 1.

So the relationship is moderately strong to strong. In 9th Grade maths, it is fine to call this a strong negative linear correlation.

Answer: There is a strong negative linear correlation between absences and final grade.

Worked Example 3: Comparing Correlations

Which relationship is stronger?

• Relationship A: r = 0.67
• Relationship B: r = -0.91

To compare strength, do not focus on the sign first. Instead, compare how far each value is from 0.

Find the absolute values:

Since 0.91 > 0.67, Relationship B is stronger.

Now describe each one:

• r = 0.67 is a moderate positive linear correlation.
• r = -0.91 is a very strong negative linear correlation.

Answer: Relationship B is stronger because its correlation coefficient is closer to -1 than 0.67 is to 1.

Worked Example 4: Correlation Close to Zero

A survey compares shoe size and score on a history quiz. The correlation coefficient is r = 0.03.

Step 1: Look at the sign.

The sign is positive, but that alone does not mean much here.

Step 2: Look at the size.

0.03 is extremely close to 0.

So there is almost no linear correlation.

Answer: Shoe size and history quiz score have no meaningful linear relationship.

What r Does Not Tell You

It is important to understand what the correlation coefficient cannot do.

• It does not prove that one variable causes the other.
• It does not always mean there is a useful relationship in real life.
• It does not describe curved relationships well.

For example, if ice cream sales and sunglasses sales increase together, they may have a positive correlation. But this does not mean buying ice cream causes people to buy sunglasses. A third factor, like hot weather, may affect both.

This idea is often stated as: correlation does not mean causation.

Quick Interpretation Guide

Use this simple process whenever you see a correlation coefficient.

1. If r is positive, say positive correlation.
2. If r is negative, say negative correlation.
3. If r is near 0, say weak or no linear correlation.
4. If r is near 1 or -1, say strong linear correlation.

You can also remember:

• Positive = points trend upward
• Negative = points trend downward
• Closer to 1 or -1 = stronger
• Closer to 0 = weaker

Common Mistakes to Avoid

• Mistake 1: Thinking r = -0.9 is weaker than r = 0.5 because it is negative.
  Actually, -0.9 is stronger because it is farther from 0.
• Mistake 2: Thinking r = 0 means no relationship at all.
  Actually, it means no linear relationship.
• Mistake 3: Thinking a strong correlation proves cause and effect.
  Actually, correlation does not prove causation.

Practice Yourself

Try interpreting these values of r:

• r = 0.58
• r = -0.12
• r = 0.99
• r = -1

Possible answers:

• r = 0.58: moderate positive linear correlation
• r = -0.12: weak negative linear correlation
• r = 0.99: very strong positive linear correlation
• r = -1: perfect negative linear correlation

Summary

The correlation coefficient r is a number from -1 to 1 that describes the direction and strength of a linear relationship between two variables.

A positive value means the variables tend to increase together, while a negative value means one tends to decrease as the other increases.

The closer r is to 1 or -1, the stronger the linear relationship. The closer it is to 0, the weaker the linear relationship.

Always remember that correlation describes association, not cause and effect.

Line of Best Fit and Least Squares Regression

When we collect bivariate data, we are looking at two variables together. For example, we might record the number of hours a student studies and the score they earn on a quiz. One variable may help us predict the other.

A scatter plot is a graph of ordered pairs \\((x, y)\\). It helps us see whether the two variables seem to be related. If the points on the graph follow a general pattern, we can draw a line that represents that pattern. This is called a line of best fit.

The line of best fit is used to model the relationship between the variables. It does not usually pass through every point. Instead, it goes through the middle of the data so that the predictions made by the line are as reasonable as possible.

A more exact way to find this line is called least squares regression. This method chooses the line that makes the total squared prediction errors as small as possible.

1. Understanding the line of best fit

A line of best fit is a straight line that shows the overall trend of the data in a scatter plot. It can help answer questions like:

• As \\((x)\\) increases, does \\((y)\\) usually increase or decrease?
• How quickly does \\((y)\\) change compared to \\((x)\\)?
• What value of \\((y)\\) might we predict for a given value of \\((x)\\)?

If the points rise from left to right, the data show a positive association. If the points fall from left to right, the data show a negative association.

The equation of a linear model is usually written as:

$$y = mx + b$$

Here:

• \\((m)\\) is the slope, which tells how much \\((y)\\) changes when \\((x)\\) increases by 1.
• \\((b)\\) is the y-intercept, which is the value of \\((y)\\) when \\((x = 0)\\).

In regression, we often write the predicted value as \\((\hat{y})\\), read as “y-hat.” So a regression line may be written as:

$$\hat{y} = mx + b$$

This reminds us that the line gives a prediction, not an exact answer.

2. Residuals: how far off a prediction is

For each data point, the line gives a predicted value. The difference between the actual value and the predicted value is called the residual.

$$\text{residual} = y - \hat{y}$$

A residual tells us how far the point is above or below the regression line.

• If the residual is positive, the point is above the line.
• If the residual is negative, the point is below the line.
• If the residual is 0, the point is on the line.

Residuals are important because they measure how well the line fits the data. Smaller residuals mean better predictions.

3. Why use squares?

If we simply added all the residuals, positive and negative values could cancel each other out. A line with large errors might wrongly seem like a good fit.

To avoid this, least squares regression squares each residual:

$$\text{squared residual} = (y - \hat{y})^2$$

Then it adds them all together:

$$\text{sum of squared residuals} = \sum (y - \hat{y})^2$$

The least squares regression line is the line that makes this total as small as possible.

Squaring helps in two ways:

• All squared values are positive, so errors do not cancel out.
• Larger errors count more heavily, so the method tries hard to avoid big misses.

4. What the least squares regression line tells us

The regression line is the linear model that best matches the data overall. It helps us:

• describe the trend in the data,
• make predictions,
• compare actual values to predicted values.

However, a regression line does not mean one variable causes the other. It only shows a pattern in the data.

5. Interpreting slope and intercept in context

When you find a regression equation, always connect the numbers to the situation.

For example, if the regression equation is:

$$\hat{y} = 5x + 42$$

then:

• The slope \\((5)\\) means that for each increase of 1 unit in \\((x)\\), the predicted \\((y)\\) increases by 5 units.
• The intercept \\((42)\\) means the predicted value of \\((y)\\) is 42 when \\((x = 0)\\).

Always ask whether the intercept makes sense in the real-world situation. Sometimes \\((x = 0)\\) is meaningful, and sometimes it is not.

Worked Example 1: Reading a line of best fit

A teacher looks at a scatter plot of hours studied \\((x)\\) and test score \\((y)\\). A line of best fit is given by:

$$\hat{y} = 8x + 50$$

Question 1: What does the slope mean?

The slope is \\((8)\\). This means that for each extra hour studied, the predicted test score goes up by 8 points.

Question 2: What score is predicted for a student who studies 3 hours?

Substitute \\((x = 3)\\) into the equation:

$$\hat{y} = 8(3) + 50$$

$$\hat{y} = 24 + 50 = 74$$

The predicted score is 74.

Question 3: What does the intercept mean?

The intercept is \\((50)\\). It means the predicted test score is 50 for a student who studies 0 hours.

Worked Example 2: Finding a residual

Use the same regression equation:

$$\hat{y} = 8x + 50$$

A student studied 4 hours and actually scored 79.

Step 1: Find the predicted score.

$$\hat{y} = 8(4) + 50 = 32 + 50 = 82$$

Step 2: Find the residual.

$$\text{residual} = y - \hat{y} = 79 - 82 = -3$$

The residual is -3.

This means the actual score was 3 points below the predicted score, so the point lies below the regression line.

Worked Example 3: Comparing two possible lines

Suppose we have the data points:

\\((1, 2)\\), \\((2, 4)\\), \\((3, 5)\\)

We want to compare these two possible lines:

• Line A: \\((\hat{y} = x + 1)\\)
• Line B: \\((\hat{y} = x + 2)\\)

We will find the sum of squared residuals for each line.

For Line A: \\((\hat{y} = x + 1)\\)

At \\((x = 1)\\):

$$\hat{y} = 1 + 1 = 2$$

Residual: \\((2 - 2 = 0)\\), squared residual: \\((0^2 = 0)\\)

At \\((x = 2)\\):

$$\hat{y} = 2 + 1 = 3$$

Residual: \\((4 - 3 = 1)\\), squared residual: \\((1^2 = 1)\\)

At \\((x = 3)\\):

$$\hat{y} = 3 + 1 = 4$$

Residual: \\((5 - 4 = 1)\\), squared residual: \\((1^2 = 1)\\)

Sum of squared residuals for Line A:

$$0 + 1 + 1 = 2$$

For Line B: \\((\hat{y} = x + 2)\\)

At \\((x = 1)\\):

$$\hat{y} = 1 + 2 = 3$$

Residual: \\((2 - 3 = -1)\\), squared residual: \\(((-1)^2 = 1)\\)

At \\((x = 2)\\):

$$\hat{y} = 2 + 2 = 4$$

Residual: \\((4 - 4 = 0)\\), squared residual: \\((0^2 = 0)\\)

At \\((x = 3)\\):

$$\hat{y} = 3 + 2 = 5$$

Residual: \\((5 - 5 = 0)\\), squared residual: \\((0^2 = 0)\\)

Sum of squared residuals for Line B:

$$1 + 0 + 0 = 1$$

Since \\((1 < 2)\\), Line B is the better fit.

This example shows the basic idea of least squares regression: the best line is the one with the smallest total of squared residuals.

Worked Example 4: Using a regression model for prediction

A sports coach records the number of practice sessions \\((x)\\) and the number of free throws made \\((y)\\). The regression equation is:

$$\hat{y} = 3.5x + 12$$

Question 1: Predict the number of free throws made after 6 practice sessions.

$$\hat{y} = 3.5(6) + 12$$

$$\hat{y} = 21 + 12 = 33$$

The model predicts 33 free throws.

Question 2: What does the slope mean in this context?

The slope is \\((3.5)\\). This means each additional practice session is associated with about 3.5 more free throws made, on average.

Question 3: If a player had 6 practice sessions and actually made 30 free throws, what is the residual?

$$\text{residual} = y - \hat{y} = 30 - 33 = -3$$

The residual is -3, so the player made 3 fewer free throws than the model predicted.

6. How to work with line of best fit questions

When solving problems, use this process:

1. Look at the scatter plot or equation.
2. Decide whether the trend is positive or negative.
3. Use the slope to describe how \\((y)\\) changes with \\((x)\\).
4. Substitute a value of \\((x)\\) into the equation to make a prediction.
5. If needed, compare the actual value to the predicted value to find the residual.
6. Remember that the best regression line has the smallest sum of squared residuals.

7. Important ideas to remember

• A line of best fit shows the overall trend of data in a scatter plot.
• A least squares regression line is the line that minimizes the sum of squared residuals.
• A residual is the difference between the actual value and the predicted value: \\((y - \hat{y})\\).
• A small residual means the prediction is close to the actual value.
• The slope tells how much the predicted \\((y)\\) changes for each increase of 1 in \\((x)\\).
• The intercept is the predicted \\((y)\\) value when \\((x = 0)\\).

8. Common mistakes

• Mixing up actual and predicted values: Residual is \\((y - \hat{y})\\), not \\((\hat{y} - y)\\).
• Forgetting to square the residuals when comparing possible best-fit lines.
• Thinking the line must pass through every point: it usually does not.
• Ignoring context: always explain what slope and intercept mean in the real situation.
• Assuming cause: a regression line shows association, not proof that one variable causes the other.

Brief Summary

A line of best fit models the overall pattern in bivariate data. The least squares regression line is the line that makes the sum of squared residuals as small as possible. You can use the equation of this line to describe the trend, make predictions, and compare actual values to predicted values using residuals.

Residual Plots help us decide whether a regression model is a good fit for a set of bivariate data.

When we use a line or curve to predict values, the model will usually not match every data point exactly. A residual tells us how far off the prediction is for each point.

Learning how to read a residual plot helps us answer an important question: Is this model appropriate for the data?

1. What is a residual?

A residual is the difference between the actual value and the predicted value.

The formula is:

$$\text{residual} = \text{observed } y - \text{predicted } y$$

If the actual point is above the regression line, the residual is positive.

If the actual point is below the regression line, the residual is negative.

If the point is exactly on the regression line, the residual is 0.

2. What is a residual plot?

A residual plot is a graph that shows:

• the input values, usually the x-values, on the horizontal axis
• the residuals on the vertical axis

Instead of plotting the original data points, we plot how far each point is from the model.

The horizontal line at residual 0 is very important. It represents perfect predictions.

3. Why do we use residual plots?

A regression equation can look reasonable, but the residual plot gives more information.

Residual plots help us check whether:

• the model fits the data well
• a linear model makes sense
• there is a pattern showing the model is missing something

In general:

• Good fit: residuals are scattered randomly above and below 0
• Poor fit: residuals show a clear pattern, such as a curve or clusters

4. How to calculate a residual

To find a residual for one data point:

1. Use the regression model to predict the value of y.
2. Take the actual value of y.
3. Subtract:

$$\text{residual} = y - \hat{y}$$

Here, \(y\) is the observed value and \(\hat{y}\) is the predicted value.

5. Worked Example 1: Finding one residual

Suppose a line of best fit is:

$$\hat{y} = 2x + 1$$

And one data point is \((3, 8)\).

Step 1: Predict the value when \(x=3\).

$$\hat{y} = 2(3) + 1 = 7$$

Step 2: Compare with the actual value.

The observed value is \(8\).

Step 3: Find the residual.

$$\text{residual} = 8 - 7 = 1$$

Answer: The residual is \(1\).

This means the actual point is 1 unit above the line.

6. Worked Example 2: Positive, negative, and zero residuals

Use the model:

$$\hat{y} = x + 4$$

Find the residual for each point.

a) Point \((2, 7)\)

Predicted value:

$$\hat{y} = 2 + 4 = 6$$

Residual:

$$7 - 6 = 1$$

The residual is positive.

b) Point \((5, 8)\)

Predicted value:

$$\hat{y} = 5 + 4 = 9$$

Residual:

$$8 - 9 = -1$$

The residual is negative.

c) Point \((1, 5)\)

Predicted value:

$$\hat{y} = 1 + 4 = 5$$

Residual:

$$5 - 5 = 0$$

The residual is zero.

What this tells us:

• Positive residual: point is above the model
• Negative residual: point is below the model
• Zero residual: point is on the model

7. How to read a residual plot

Once all residuals are plotted, look at the overall pattern.

A good residual plot has points scattered randomly around the line \(y=0\).

This suggests the model is doing a reasonable job and is not consistently too high or too low.

A bad residual plot has a pattern.

Patterns might include:

• a curve shape
• points mostly above 0 on one side and below 0 on the other side
• groups or clusters

If there is a pattern, the model may not match the relationship very well.

8. Worked Example 3: Interpreting a residual plot

Suppose a student makes a linear regression model for a data set. The residual plot shows points scattered above and below 0 with no obvious pattern.

Interpretation:

This is a sign that the linear model is appropriate.

The errors seem random, so the line fits the data reasonably well.

Now suppose another residual plot has residuals that form a U-shape.

Interpretation:

This is a sign that a linear model is not a good fit.

The pattern suggests the data may follow a curve instead of a straight line.

9. Worked Example 4: Making a small residual table

A regression model is:

$$\hat{y} = 3x + 2$$

Use it for these data points:

• \((1, 6)\)
• \((2, 7)\)
• \((3, 12)\)

Point 1: \((1, 6)\)

Predicted value:

$$\hat{y} = 3(1) + 2 = 5$$

Residual:

$$6 - 5 = 1$$

Point 2: \((2, 7)\)

Predicted value:

$$\hat{y} = 3(2) + 2 = 8$$

Residual:

$$7 - 8 = -1$$

Point 3: \((3, 12)\)

Predicted value:

$$\hat{y} = 3(3) + 2 = 11$$

Residual:

$$12 - 11 = 1$$

We can organize the results in a table:

x-value, observed y, predicted y, residual

• \(1, 6, 5, 1\)
• \(2, 7, 8, -1\)
• \(3, 12, 11, 1\)

On a residual plot, we would graph the points:

• \((1, 1)\)
• \((2, -1)\)
• \((3, 1)\)

10. What patterns mean

Here is a simple guide for interpreting residual plots:

• Random scatter around 0: model is probably a good fit
• Curved pattern: linear model may not be appropriate
• Most points above or below 0: model may be biased
• Large residuals: some predictions are far from the actual values

Remember: residual plots do not usually tell you the exact best model. They help you decide whether your current model makes sense.

11. Common mistakes to avoid

• Mixing up the subtraction order: residual is observed minus predicted, not predicted minus observed
• Confusing the residual plot with the original scatter plot: a residual plot uses residuals on the vertical axis
• Looking at only one point: the overall pattern matters most
• Ignoring a curve pattern: if residuals curve, a straight-line model may not fit well

12. Quick checklist for residual plots

When you see a residual plot, ask yourself:

1. Are the residuals centered around 0?
2. Are the points scattered randomly?
3. Is there a visible pattern, like a curve?
4. Do any residuals seem much larger than the others?

If the answers show random scatter around 0, the model is likely reasonable.

Summary

A residual is the difference between an observed value and a predicted value:

$$\text{residual} = y - \hat{y}$$

A residual plot graphs residuals against the x-values. It helps us judge how well a regression model fits data.

If the residuals are randomly spread around 0, the model is usually a good fit. If the residuals form a pattern, especially a curve, then the model may not be appropriate.

Interpolation vs. Extrapolation Risks

When we collect bivariate data, we record two variables that are related. For example, we might record hours studied and test score, or temperature and ice cream sales. A regression model is an equation or trend line that helps us predict one variable from the other.

But not every prediction from a regression model is equally trustworthy. A prediction made inside the range of the data is called interpolation. A prediction made outside the range of the data is called extrapolation.

Understanding the difference is very important. Interpolation is usually safer because it uses values between data points we have actually observed. Extrapolation is riskier because it assumes the same pattern continues beyond the data, and that may not be true.

1. What is interpolation?

Interpolation means using a model to estimate a value within the domain of the observed data. The domain is the set of input values we have data for.

Suppose a class measured study time from 1 hour to 5 hours. If we use the model to predict a score for 3.5 hours of study, that is interpolation, because 3.5 is between 1 and 5.

Interpolation is generally more reliable because the model is being used in the same area where the data was collected.

• If the input is between the smallest and largest observed values, it is interpolation.
• Interpolation still gives an estimate, not a guaranteed exact answer.
• It is usually reasonable if the data pattern is strong.

2. What is extrapolation?

Extrapolation means using a model to estimate a value outside the domain of the observed data.

If the study-time data only goes from 1 hour to 5 hours, then predicting a score for 8 hours of study is extrapolation. We are moving beyond the data we actually know.

This is dangerous because real-life relationships often change. A line or curve that fits the known data may not keep working forever.

• Patterns may level off.
• Patterns may change direction.
• There may be physical limits, like a score not being able to go above 100.
• Other factors may become important outside the original data range.

3. Why is extrapolation risky?

A regression model is built from the data we have. It describes a pattern in that specific range. When we go far outside that range, we are assuming the pattern stays the same. That assumption can easily be wrong.

For example, if a plant grows steadily for the first few weeks, a linear model might fit that data well. But the plant will not keep growing at the same rate forever. It may slow down or stop growing.

So a model can be useful, but only when we remember its limits.

4. How to decide whether a prediction is interpolation or extrapolation

Follow these steps:

1. Identify the smallest and largest input values in the data.
2. Look at the input value you want to predict for.
3. If it is inside the data range, it is interpolation.
4. If it is outside the data range, it is extrapolation.

For example, if the input values are from 10 to 30:

• Predicting at 18 is interpolation.
• Predicting at 30 is still within the data range, so it is interpolation if 30 was observed.
• Predicting at 35 is extrapolation.
• Predicting at 8 is extrapolation.

5. Worked Example 1: Identifying interpolation and extrapolation

A data set shows the relationship between hours of exercise and calories burned. The hours recorded are 1, 2, 3, 4, and 5.

Classify each prediction:

• Predict calories burned after 3.5 hours.

  Since 3.5 is between 1 and 5, this is interpolation.

• Predict calories burned after 6 hours.

  Since 6 is greater than 5, this is extrapolation.

• Predict calories burned after 0.5 hours.

  Since 0.5 is less than 1, this is extrapolation.

The key idea is to compare the input value to the range of the data.

6. Worked Example 2: Using a linear model for interpolation

A regression line for a set of data is

Suppose the data was collected for values of \(x\) from 2 to 8. Predict \(y\) when \(x = 5\), and decide whether the prediction is interpolation or extrapolation.

Step 1: Check the range.

The data range is from 2 to 8. Since \(5\) is between 2 and 8, this is interpolation.

Step 2: Substitute into the equation.

Answer: The predicted value is \(30\), and the prediction is interpolation.

This prediction is usually considered more reliable because it stays inside the observed data range.

7. Worked Example 3: Using a linear model for extrapolation

Use the same model:

Again, the data was collected for values of \(x\) from 2 to 8. Predict \(y\) when \(x = 12\), and decide whether the prediction is interpolation or extrapolation.

Step 1: Check the range.

The data range is from 2 to 8. Since \(12\) is greater than 8, this is extrapolation.

Step 2: Substitute into the equation.

Answer: The model predicts \(58\), but this prediction is extrapolation, so it should be used carefully.

Even though we can calculate the value, that does not mean the prediction is trustworthy. The real relationship might not continue in the same way up to \(x = 12\).

8. Worked Example 4: Why extrapolation can give unrealistic results

A student's test scores and study hours are modeled by

where \(x\) is hours studied and \(y\) is predicted test score. Suppose the data only included study times from 1 to 6 hours.

Predict the score for \(x = 7\), and explain the risk.

Step 1: Classify the prediction.

Since 7 is outside the data range 1 to 6, this is extrapolation.

Step 2: Calculate the prediction.

The model predicts a score of \(104\).

Step 3: Explain why this is risky.

A test score of \(104\) may not make sense if the highest possible score is 100. This shows that an extrapolated prediction can be unrealistic, even if the math is done correctly.

This is one of the biggest warnings about extrapolation: the model may produce values that do not fit the real situation.

9. Important ideas to remember about regression models

• A regression model is based on observed data.
• It is strongest near the data used to create it.
• Interpolation is generally safer than extrapolation.
• The farther you extrapolate, the less reliable the prediction usually becomes.
• Always think about whether the answer makes sense in the real world.

10. Common mistakes

• Mistake 1: Thinking every value from a regression equation is equally reliable.
  Not true. Predictions outside the data range are more risky.
• Mistake 2: Forgetting to check the domain of the data.
  Always compare the input value to the smallest and largest observed values.
• Mistake 3: Accepting unrealistic answers.
  A model might predict impossible values, like a negative height or a test score above 100.
• Mistake 4: Assuming a trend continues forever.
  Many real-world patterns do not keep the same shape outside the collected data.

11. Quick comparison

• Interpolation: prediction inside the data range; usually more reliable.
• Extrapolation: prediction outside the data range; usually less reliable and more risky.

You can think of it like this: interpolation is filling in a gap between known points, while extrapolation is stretching the pattern beyond the known points.

12. Brief summary

In bivariate data and regression models, interpolation means predicting within the observed data range, while extrapolation means predicting outside it. Interpolation is usually safer because it stays close to known data. Extrapolation is riskier because the pattern may change, and the model can give unrealistic answers. Always check the data range before trusting a prediction.

Correlation vs. Causation

When we study bivariate data, we look at two variables at the same time. For example, we might compare hours studied and test scores, or temperature and ice cream sales.

Sometimes the two variables seem to change together. When that happens, we say there is a correlation. But just because two things are connected does not always mean that one causes the other. That is the big idea of this lesson.

Correlation means that two variables show a pattern or relationship. As one variable changes, the other tends to change too.

• Positive correlation: both variables tend to increase together, or decrease together.
• Negative correlation: as one variable increases, the other tends to decrease.
• No correlation: there is no clear pattern between the variables.

Causation means that a change in one variable directly causes a change in the other variable.

For example, if the number of hours a student studies increases, their test score may increase. In many cases, studying more can help cause better performance. That is a possible example of causation.

But if two variables are correlated, we must be careful. A correlation alone does not prove causation.

Important idea:

$$\text{Correlation} \neq \text{Causation}$$

This means “correlation is not the same as causation.” Two things can happen together without one making the other happen.

There are several reasons why correlation does not always mean causation.

1. A third variable may affect both.
2. The relationship may be a coincidence.
3. The direction of cause may be unclear.

Let’s look at each of these.

1. A third variable may affect both variables.

Suppose ice cream sales and the number of people swimming both increase during the same months. These variables are positively correlated.

Does buying ice cream cause people to go swimming? Probably not. A third variable, hot weather, affects both. When it is hotter, more people buy ice cream and more people go swimming.

2. The relationship may be a coincidence.

Sometimes two variables appear connected just by chance. If we collect enough data, some patterns can happen even when there is no real cause.

That is why mathematicians and scientists do not make strong conclusions from a pattern alone.

3. The direction of cause may be unclear.

If two variables are related, it may be hard to tell which one affects the other.

For example, students with high grades may spend more time studying. But it could also be true that students who understand the material better are more confident and choose to study more. The relationship may not be simple.

How do we spot correlation in data?

In 9th Grade maths, we often use a scatter plot to look for correlation. Each point on the graph represents one pair of data values.

• If the points rise from left to right, there is a positive correlation.
• If the points fall from left to right, there is a negative correlation.
• If the points are scattered with no pattern, there is little or no correlation.

A line of best fit or regression line can help show the general trend, but even a strong trend still does not prove causation.

Worked Example 1: Identifying correlation

A class records the number of hours students studied and their test scores.

• Student A: 1 hour, 62
• Student B: 2 hours, 68
• Student C: 3 hours, 74
• Student D: 4 hours, 81
• Student E: 5 hours, 88

As study hours increase, test scores also increase. This shows a positive correlation.

Does this prove causation? It suggests that studying may help improve scores, but the data alone does not prove it completely. Other factors may matter too, such as sleep, class attendance, or prior knowledge.

Worked Example 2: Correlation without causation

Suppose a town records monthly ice cream sales and the number of visits to the beach. Both go up in summer and down in winter.

These variables have a positive correlation.

However, we should not say:

“Buying ice cream causes people to visit the beach.”

A better explanation is that season and temperature affect both variables.

So this is a case where there is correlation, but the likely cause is a third variable.

Worked Example 3: Negative correlation

A student tracks the number of absences and their course grade.

• 1 absence, 92
• 2 absences, 87
• 4 absences, 79
• 6 absences, 70
• 8 absences, 61

As absences increase, grades decrease. This is a negative correlation.

Could there be causation? Possibly. Missing class may cause students to learn less, which can lower grades.

But again, we should be careful. Another factor, such as illness or family problems, might affect both attendance and grades.

Worked Example 4: Decide whether the statement is reasonable

A survey finds that students who spend more time reading tend to have larger vocabularies.

Question: Is it reasonable to say reading causes a larger vocabulary?

Step 1: Identify the relationship.
More reading time is connected with larger vocabulary. This is a positive correlation.

Step 2: Think about causation.
It is reasonable that reading more can help students learn new words.

Step 3: Stay cautious.
Even though causation is possible here, the data alone does not prove it fully. Students who already enjoy language may both read more and know more words.

Conclusion: The statement may be reasonable, but we should not claim proof from correlation alone.

When is causation more believable?

Causation is more believable when:

• there is a clear reason one variable would affect the other,
• the relationship happens again and again,
• other possible causes have been carefully considered.

In maths classes, your job is often to describe the relationship correctly. If a scatter plot shows a trend, you can say there is positive or negative correlation. But unless there is strong evidence, do not say one variable causes the other.

Common mistakes to avoid

• Saying “related” always means “caused by.”
• Ignoring third variables.
• Assuming a strong scatter plot trend proves cause.
• Forgetting that coincidence can happen.

Quick comparison

• Correlation: two variables change together.
• Causation: one variable directly makes the other change.

You can think of it like this:

$$\text{If two variables move together, ask: Why?}$$

The answer might be:

• one causes the other,
• a third variable affects both,
• or the pattern is only a coincidence.

Summary

Correlation describes a pattern between two variables. Causation means one variable directly changes the other. A scatter plot or regression line can show correlation, but they do not prove causation by themselves.

Always be careful when interpreting data. If two variables are connected, look for other explanations before deciding that one causes the other.

Two-Way Frequency Tables help us organize and study data about two categorical variables at the same time.

A categorical variable places data into groups or labels, such as favorite sport, grade level, or whether a student rides the bus.

For example, we might want to study:

• whether students play a sport and whether they have a pet, or
• whether students prefer online or in-person learning and whether they are in 9th or 10th grade.

A two-way frequency table helps us answer questions like:

• How many students are in each combination of categories?
• How many students are in each row or column total?
• What fraction or percent of the data falls into each part of the table?

In this lesson, you will learn how to find and interpret joint frequencies, marginal frequencies, and conditional relative frequencies.

1. What is a two-way frequency table?

A two-way frequency table shows counts for two different categories. One variable is usually listed in the rows, and the other is listed in the columns.

Here is a simple example. A class is asked whether they prefer cats or dogs and whether they prefer reading or sports.

This table tells us there are 30 students total.

It also tells us that:

• 8 students prefer cats and reading,
• 6 students prefer cats and sports,
• 5 students prefer dogs and reading,
• 11 students prefer dogs and sports.

2. Joint frequency

A joint frequency is the number in an inside cell of the table. It shows how many data values belong to both categories at once.

In the table above:

• The joint frequency for cats and reading is 8.
• The joint frequency for dogs and sports is 11.

Think of joint as meaning together. A joint frequency describes the overlap of one row category and one column category.

3. Marginal frequency

A marginal frequency is a row total or column total. These numbers are placed on the margins, or edges, of the table.

In the same table:

• The total who prefer cats is 14.
• The total who prefer dogs is 16.
• The total who prefer reading is 13.
• The total who prefer sports is 17.
• The grand total is 30.

Marginal frequencies help summarize one variable without focusing on the other variable.

4. Relative frequency

A relative frequency compares a count to the total. It is often written as a fraction, decimal, or percent.

The basic idea is:

If 8 out of 30 students prefer cats and reading, then the relative frequency is:

5. Joint relative frequency

A joint relative frequency is the frequency in one inside cell divided by the grand total.

Using the table above:

The joint relative frequency for dogs and sports is:

This means about 36.7% of all students prefer dogs and sports.

6. Marginal relative frequency

A marginal relative frequency is a row total or column total divided by the grand total.

For example, the marginal relative frequency for students who prefer reading is:

This means about 43.3% of all students prefer reading.

7. Conditional relative frequency

A conditional relative frequency describes a part of the table compared to a row total or column total, instead of the grand total.

This means we are asking a question under a condition, such as:

• Among students who prefer cats, what fraction prefer reading?
• Among students who prefer sports, what fraction prefer dogs?

The word among is a clue that you should use a row total or column total, not the grand total.

Example: Among students who prefer cats, what fraction prefer reading?

We only look at the cats row. There are 14 students in that row, and 8 of them prefer reading.

So, among students who prefer cats, about 57.1% prefer reading.

Example: Among students who prefer sports, what fraction prefer dogs?

We only look at the sports column. There are 17 students in that column, and 11 of them prefer dogs.

So, among students who prefer sports, about 64.7% prefer dogs.

Important idea:

• Joint relative frequency uses the grand total.
• Marginal relative frequency uses the grand total.
• Conditional relative frequency uses a row total or column total.

8. How to read the question carefully

When working with two-way tables, the hardest part is often deciding what the whole should be.

Ask yourself:

• Is the question about all data? Use the grand total.
• Is the question about one row? Use that row total.
• Is the question about one column? Use that column total.

Words that often mean you need a conditional relative frequency include:

• among
• out of those who
• given that

Worked Example 1: Finding joint and marginal frequencies

A survey asks 40 students whether they play an instrument and whether they play a sport.

Question A: What is the joint frequency for students who play an instrument and play a sport?

Look at the cell where the Instrument row and Sport column meet.

The answer is 12.

Question B: What is the marginal frequency for students who play no sport?

Look at the total for the No Sport column.

The answer is 18.

Question C: What is the grand total?

The answer is 40.

Worked Example 2: Finding joint and marginal relative frequencies

Use the same table of 40 students.

Question A: What is the joint relative frequency for students who play no instrument and no sport?

The joint frequency is 10, and the grand total is 40.

So, the joint relative frequency is 0.25, or 25%.

Question B: What is the marginal relative frequency for students who play a sport?

The sport column total is 22, and the grand total is 40.

So, the marginal relative frequency is 0.55, or 55%.

Worked Example 3: Finding conditional relative frequencies

Now let us answer questions that use a condition.

Question A: Among students who play an instrument, what fraction also play a sport?

We focus only on the Instrument row. There are 20 students who play an instrument, and 12 of them play a sport.

So, among students who play an instrument, 60% play a sport.

Question B: Among students who play no sport, what fraction play no instrument?

We focus only on the No Sport column. There are 18 students in that column, and 10 of them play no instrument.

So, among students who play no sport, about 55.6% play no instrument.

Worked Example 4: Building and analyzing a table

A teacher surveys 50 students about whether they prefer studying in the morning or evening, and whether they usually do homework at home or somewhere else.

The results are:

• 18 students prefer morning and do homework at home.
• 7 students prefer morning and do homework somewhere else.
• 15 students prefer evening and do homework at home.
• 10 students prefer evening and do homework somewhere else.

First, organize the data into a two-way table.

Question A: What is the joint relative frequency for students who prefer evening and do homework at home?

Question B: What is the marginal relative frequency for students who do homework somewhere else?

Question C: Among students who prefer morning, what fraction do homework at home?

Use the Morning row total of 25.

Question D: Among students who do homework at home, what fraction prefer evening?

Use the Home column total of 33.

This last question is different from Question C because the condition changed. In Question C, we used a row total. In Question D, we used a column total.

9. Common mistakes to avoid

• Using the wrong total. Always decide whether the whole is the grand total, a row total, or a column total.
• Mixing up joint and marginal frequencies. Joint frequencies are inside cells. Marginal frequencies are totals on the edges.
• Forgetting to convert to a percent. If the question asks for a percent, multiply the decimal by 100.
• Reading the condition backward. “Among students who prefer morning” is different from “Among students who do homework at home.”

10. Quick strategy for solving problems

1. Find the correct cell in the table.
2. Decide what the whole should be.
3. Write the fraction: $$\frac{\text{part}}{\text{whole}}$$
4. Simplify or convert to a decimal or percent if needed.
5. Write your answer in words so it is clear what it means.

11. Why two-way tables are useful

Two-way frequency tables help us compare two categories and look for patterns in data.

For example, if two conditional relative frequencies are very different, that may suggest a relationship between the two variables. If they are very similar, the variables may not show much connection in the data.

You do not need advanced math to notice these patterns. You just need to compare the percentages carefully.

Summary

A two-way frequency table organizes data for two categorical variables. The inside cells show joint frequencies, and the row and column totals show marginal frequencies.

To find relative frequencies, divide by a total. Joint and marginal relative frequencies use the grand total. Conditional relative frequencies use a row total or column total, depending on the condition in the question.

If you remember to ask, “What is the whole?”, you will be much more successful with two-way tables.