Bivariate Data and Statistics

Constructing Scatter Plots

A scatter plot is a graph that shows pairs of data on a coordinate plane. It helps us see whether two variables might be related.

For example, you might compare:

• hours studied and test score,
• height and arm span, or
• outside temperature and ice cream sold.

Each piece of data comes as an ordered pair, written like \((x, y)\). The first value goes on the horizontal axis, called the x-axis. The second value goes on the vertical axis, called the y-axis.

When we make a scatter plot, we do not connect the points with lines. Each point stands for one pair of data values.

Why scatter plots are useful

Scatter plots help us:

• organize paired data,
• see patterns,
• notice trends, and
• compare two variables at the same time.

Even before doing any calculations, a scatter plot can show whether values seem to increase together, decrease, or have no clear pattern.

The parts of a scatter plot

• Title: tells what the graph is about.
• x-axis: shows the first variable.
• y-axis: shows the second variable.
• Scale: the counting pattern on each axis, such as by 1s, 2s, 5s, or 10s.
• Points: each point represents one ordered pair.

How to construct a scatter plot

1. Read the data carefully. Make sure you know what each variable means.
2. Choose which variable goes on each axis. Usually the first variable or input goes on the x-axis, and the second variable or result goes on the y-axis.
3. Draw and label the axes. Write the variable names and units if there are any.
4. Choose a scale. The scale should fit all the data values and be easy to read.
5. Plot each ordered pair. Move across to the x-value, then up to the y-value, and place a point.
6. Check your work. Make sure every data pair is plotted once and in the correct location.

Choosing a good scale

A good scale makes the graph easy to read. If the x-values go from 1 to 12, counting by 1s or 2s may work well. If the y-values go from 50 to 100, counting by 5s or 10s may be better.

The scale on the x-axis does not have to match the scale on the y-axis. Each axis should be chosen based on the values it needs to show.

Important reminder about ordered pairs

In the ordered pair \((x, y)\):

• the x-value tells how far to move left or right,
• the y-value tells how far to move up or down.

So the point \((4, 7)\) means move 4 units to the right and 7 units up.

Worked Example 1: Plotting simple data

A student records the number of hours studied and the score earned on a quiz.

Data:

• \((1, 65)\)
• \((2, 70)\)
• \((3, 78)\)
• \((4, 85)\)
• \((5, 90)\)

Step 1: Label the axes.

• x-axis: Hours Studied
• y-axis: Quiz Score

Step 2: Choose scales.

• For hours studied, count by 1s from 0 to 5.
• For quiz score, count by 5s from 60 to 95.

Step 3: Plot the points.

Place points at \((1,65)\), \((2,70)\), \((3,78)\), \((4,85)\), and \((5,90)\).

What do we notice?

As hours studied increase, quiz scores also increase. The points would rise from left to right.

Worked Example 2: Using a table to make a scatter plot

The table shows the temperature outside and the number of cold drinks sold at a snack stand.

Temperature (°F)	Drinks Sold
60	15
65	18
70	22
75	27
80	30

Step 1: Write the ordered pairs.

The data becomes:

• \((60,15)\)
• \((65,18)\)
• \((70,22)\)
• \((75,27)\)
• \((80,30)\)

Step 2: Label the axes.

• x-axis: Temperature (°F)
• y-axis: Drinks Sold

Step 3: Choose scales.

• x-axis: count by 5s from 60 to 80
• y-axis: count by 5s from 0 to 30

Step 4: Plot the points.

Plot all five ordered pairs on the coordinate plane.

What do we notice?

As temperature rises, more drinks are sold. The scatter plot would show an upward pattern.

Worked Example 3: Choosing a scale carefully

A class records the number of books read and the number of minutes spent reading each week.

Data:

• \((1, 40)\)
• \((2, 75)\)
• \((3, 90)\)
• \((4, 130)\)
• \((5, 160)\)

Question: What scale should be used?

Solution:

• The x-values go from 1 to 5, so counting by 1s is a good choice.
• The y-values go up to 160, so counting by 10s, 20s, or 25s could work.

A good graph might use:

• x-axis: 0 to 5 by 1s
• y-axis: 0 to 160 by 20s

Then plot the points:

• \((1,40)\)
• \((2,75)\)
• \((3,90)\)
• \((4,130)\)
• \((5,160)\)

Why this matters:

If you chose a poor scale, such as counting the y-axis by 1s, the graph would be crowded and hard to read. Good scales make patterns easier to see.

Worked Example 4: Avoiding common mistakes

A student wants to plot this data about time spent practicing piano and number of songs learned:

• \((2, 1)\)
• \((4, 2)\)
• \((6, 3)\)
• \((8, 4)\)

The student accidentally plots \((2,1)\) as \((1,2)\).

What went wrong?

The order of the numbers matters. The point \((2,1)\) means 2 units across and 1 unit up. The point \((1,2)\) is in a different location.

Lesson: Always plot the x-value first and the y-value second.

Common mistakes to avoid

• Switching x- and y-values: Remember, \((x,y)\) means across first, then up.
• Forgetting to label axes: A graph should show what each axis means.
• Using a bad scale: Make sure all data fits and the numbers are easy to read.
• Connecting the points: In a scatter plot, points usually stay separate.
• Missing points: Check that every ordered pair is plotted.

How to read a scatter plot after constructing it

Once the scatter plot is made, ask questions like:

• Do the points rise from left to right?
• Do the points fall from left to right?
• Are the points spread out with no clear pattern?

You do not need to make a final conclusion right away. First, your job is to build the graph correctly so the data can be seen clearly.

Quick checklist for constructing scatter plots

• I know what the two variables are.
• I put the first variable on the x-axis.
• I put the second variable on the y-axis.
• I labeled both axes.
• I chose scales that fit the data.
• I plotted each ordered pair correctly.
• I did not connect the points.

Summary

A scatter plot shows paired data as points on a coordinate plane. To construct one, label the axes, choose reasonable scales, and plot each ordered pair carefully.

Remember that the first number in \((x, y)\) goes on the x-axis and the second number goes on the y-axis. A well-made scatter plot helps you see possible relationships between two variables.

Analyzing Associations means looking at two variables and describing how they are related. In 8th grade, we often use scatter plots to study the relationship between two quantitative variables, such as hours studied and quiz score, or temperature and ice cream sales.

When we analyze a scatter plot, we are not just looking at individual points. We are looking for the overall pattern. We want to answer questions like: Do the variables rise together? Does one go up while the other goes down? Do the points follow a straight-line pattern, a curve, or no clear pattern at all?

This lesson will help you recognize and describe different kinds of associations, including positive association, negative association, linear association, nonlinear association, and clustering.

First, remember what a scatter plot shows. A scatter plot has two number lines:

• The x-axis shows one variable.
• The y-axis shows the other variable.
• Each point represents one pair of data values.

For example, a point at \((3, 70)\) could mean a student studied for 3 hours and earned a score of 70.

1. Positive Association

A scatter plot shows a positive association when the points tend to go upward from left to right. This means that as one variable increases, the other variable also tends to increase.

Examples of positive association include:

• More hours studied and higher test scores
• More time spent practicing and more baskets made
• Greater height and greater arm span

If you imagine drawing a line through the middle of the points, that line would slant upward.

2. Negative Association

A scatter plot shows a negative association when the points tend to go downward from left to right. This means that as one variable increases, the other variable tends to decrease.

Examples of negative association include:

• Higher prices and fewer items sold
• More time since a hot drink was poured and lower temperature
• Greater speed and less time needed to travel a fixed distance

If you imagine drawing a line through the middle of the points, that line would slant downward.

3. No Clear Association

Sometimes the points do not show a clear upward or downward trend. In that case, there may be no clear association. The variables do not seem to be related in a useful way based on the scatter plot.

For example, a student’s shoe size and the number of books they checked out last month might not show any clear pattern.

4. Linear Association

An association is linear when the points follow a pattern that is close to a straight line. The line does not have to go through every point. It just needs to match the general trend.

A linear association can be:

• Positive linear: points rise in a straight-line pattern
• Negative linear: points fall in a straight-line pattern

If the points are scattered around an imaginary straight line, the relationship is likely linear.

5. Nonlinear Association

An association is nonlinear when the points follow a curve or some shape that is not a straight line.

For example, a plant may grow quickly at first and then more slowly over time. The points might curve instead of making a straight path.

When describing a nonlinear association, you can still say whether it generally increases or decreases, but you should also mention that the pattern is not linear.

6. Clustering

A cluster is a group of points that are close together. Scatter plots sometimes have one cluster, two clusters, or several clusters.

Clusters can tell us that the data may contain different groups. For example:

• One cluster might represent younger students.
• Another cluster might represent older students.

If points appear in separate groups, it is important to mention that in your description.

7. How to Describe a Scatter Plot

When you are asked to analyze an association, use a step-by-step approach:

1. Look at the overall direction: positive, negative, or no clear association.
2. Decide whether the pattern is linear or nonlinear.
3. Notice whether the points are tightly grouped or widely spread out.
4. Check for clusters or unusual points.

A strong answer often sounds like this: “The scatter plot shows a positive linear association with a cluster near the middle.”

Worked Example 1: Positive Linear Association

Suppose a scatter plot compares hours practiced on the piano and number of songs played correctly. The points are:

• \((1, 2)\)
• \((2, 4)\)
• \((3, 5)\)
• \((4, 7)\)
• \((5, 8)\)

As the number of practice hours increases, the number of songs played correctly also increases. The points move upward from left to right.

The pattern is also close to a straight line, so the association is positive linear.

Answer: The scatter plot shows a positive linear association.

Worked Example 2: Negative Linear Association

A scatter plot compares outside temperature and number of cups of hot chocolate sold. The points are:

• \((30, 40)\)
• \((40, 35)\)
• \((50, 28)\)
• \((60, 20)\)
• \((70, 12)\)

As temperature increases, the number of hot chocolates sold decreases. The points move downward from left to right.

The pattern is close to a straight line, so this is a negative linear association.

Answer: The scatter plot shows a negative linear association.

Worked Example 3: Nonlinear Association

A scatter plot shows time after a ball is thrown and the ball’s height. The points rise at first and then fall:

• \((1, 3)\)
• \((2, 7)\)
• \((3, 9)\)
• \((4, 7)\)
• \((5, 3)\)

The points do not form a straight-line pattern. Instead, they make a curved shape.

This means the association is nonlinear. It increases at first, then decreases.

Answer: The scatter plot shows a nonlinear association.

Worked Example 4: Clustering

A scatter plot compares age and number of push-ups completed. The points form two groups:

• Group 1: \((8, 5), (8, 6), (9, 7), (9, 6)\)
• Group 2: \((13, 15), (13, 16), (14, 18), (14, 17)\)

These points are not spread evenly. They appear in two clusters. One cluster is for younger students with fewer push-ups, and the other is for older students with more push-ups.

Overall, as age increases, the number of push-ups tends to increase, so there is a positive association. The important extra detail is that the data form two clusters.

Answer: The scatter plot shows a positive association with two clusters.

Important Note About Association

Association means there is a pattern or relationship in the data. It does not always mean one variable causes the other to change.

For example, if ice cream sales and swimming pool visits both increase in summer, that does not mean ice cream sales cause people to go swimming. A third factor, like hot weather, may affect both.

Helpful Sentence Starters

• The scatter plot shows a positive association because...
• The scatter plot shows a negative association because...
• The association appears linear because the points are close to a straight line.
• The association appears nonlinear because the points form a curve.
• The data show clustering because the points form separate groups.
• There is no clear association because the points do not show a clear trend.

Common Mistakes to Avoid

• Looking at only one or two points: Always describe the overall pattern.
• Confusing positive with negative: Positive goes up left to right; negative goes down left to right.
• Calling every trend linear: Check whether the points make a straight-line pattern or a curve.
• Ignoring clusters: If the points form groups, mention it.
• Assuming cause: Association does not always mean one thing causes the other.

Quick Check

Ask yourself these questions when you see a scatter plot:

1. Do the points go up, go down, curve, or show no pattern?
2. Do they look close to a straight line?
3. Are there groups of points close together?
4. What is the best complete description of the association?

Summary

To analyze associations in a scatter plot, look at the direction, shape, and grouping of the data. A positive association rises from left to right, and a negative association falls from left to right. A linear association follows a straight-line pattern, while a nonlinear association follows a curve. Clusters are groups of points close together, and they are important to mention when describing the data.

Identifying Outliers

When we look at a scatter plot, we usually try to see the overall pattern in the data. Many points may cluster together or follow a trend, such as going upward from left to right or downward from left to right.

Sometimes, one point is far away from the rest of the data or does not fit the pattern. That point is called an outlier.

Learning to spot outliers is important because an outlier can change how we describe the data. It can affect how strong a pattern looks and may make predictions less accurate.

What is an outlier?

An outlier is a data point that is much higher, lower, or farther away than the other points in a scatter plot. It does not follow the same pattern as most of the data.

In a scatter plot, each point represents a pair of values, written like \,((x,y))\,. An outlier might:

• be far above the other points,
• be far below the other points, or
• be far to the left or right compared with the rest.

How to identify an outlier

To decide whether a point is an outlier, follow these steps:

1. Look for the overall trend. Do the points seem to rise, fall, or cluster in one area?
2. Check for points far from the group. Is one point separated from the others?
3. Ask whether the point fits the pattern. Even if it is not extremely far away, does it still seem different from the trend?
4. Compare carefully. An unusual point is only an outlier if it clearly stands apart from most of the data.

Important idea: An outlier is not just any point near the edge of the graph. It must be noticeably different from the rest of the points.

Outliers and scatter plot patterns

Scatter plots often show one of these patterns:

• Positive association: as \,x\, increases, \,y\, tends to increase.
• Negative association: as \,x\, increases, \,y\, tends to decrease.
• No clear association: the points do not follow a clear trend.

An outlier is easier to notice when most points follow a clear pattern. If almost all the points rise together, but one point is far below them, that point may be an outlier.

Why outliers happen

Outliers can happen for different reasons:

• a measurement or recording mistake,
• an unusual event,
• or a value that is real but uncommon.

For example, if a class usually studies between 20 and 50 minutes and one student studies 180 minutes, that point may be an outlier. It could be a mistake, or it could be a student who studied much longer than everyone else.

Worked Example 1: Finding an obvious outlier

A teacher makes a scatter plot of hours studied and quiz score. Most of the points are near these pairs:

• \((1,62)\)
• \((2,68)\)
• \((3,74)\)
• \((4,81)\)
• \((5,86)\)

There is also one point at \((4,30)\).

Step 1: Look for the pattern. As study hours increase, quiz scores increase. This is a positive association.

Step 2: Compare the point \((4,30)\) to the others. When \,x=4\,, the other nearby scores are around the 80s, not 30.

Conclusion: \((4,30)\) is an outlier because it is far below the overall pattern.

Worked Example 2: A point near the edge is not always an outlier

Suppose a scatter plot has these points:

• \((2,5)\)
• \((3,7)\)
• \((4,9)\)
• \((5,11)\)
• \((6,13)\)
• \((7,15)\)

The point \((7,15)\) is the farthest right and near the top.

Does that make it an outlier? No.

All of the points follow the same rising pattern. Even though \((7,15)\) is at the edge, it still fits the trend. So it is not an outlier.

Lesson: Being at the end of a pattern is different from being outside the pattern.

Worked Example 3: Identifying an outlier from a description of a scatter plot

A scatter plot shows the relationship between temperature and hot chocolate sales. Most points are close to a downward trend. As temperature increases, hot chocolate sales decrease.

One point shows a day with high temperature and very high hot chocolate sales.

Step 1: Identify the trend. This is a negative association.

Step 2: Check whether the unusual point fits. A hot day would normally have lower hot chocolate sales, not very high sales.

Conclusion: That point is an outlier because it does not match the overall downward pattern.

Worked Example 4: Deciding which point is the outlier

A scatter plot includes these data pairs:

• \((1,3)\)
• \((2,5)\)
• \((3,7)\)
• \((4,8)\)
• \((5,11)\)
• \((6,20)\)

Most of these points follow a pattern close to \,y\, increasing by about 2 when \,x\, increases by 1.

Let us compare the last point. If the pattern continues, when \,x=6\,, we would expect \,y\, to be close to 13, not 20.

We can see this by the nearby pattern:

$$ 3,\ 5,\ 7,\ 8,\ 11,\ ? $$

A value of 20 is much larger than expected.

Conclusion: \((6,20)\) is the outlier.

How outliers affect data

An outlier can change how a scatter plot looks. It can make a relationship seem weaker or stronger than it really is.

For example, if most points form a tight upward pattern and one point is far away, that one point can make the graph look less consistent.

Outliers can also affect predictions. If you use the data to guess a new value, an outlier may pull your thinking away from the main pattern.

Tips for students

• Do not choose an outlier just because a point is highest or lowest.
• Look at the whole pattern first.
• Ask, “Does this point fit with most of the others?”
• If one point is clearly separated from the rest, it may be an outlier.
• If all points follow the same trend, there may be no outlier.

Quick check questions

1. If most points on a scatter plot rise from left to right, and one point is far below the others, what is that point likely to be?
2. If a point is on the far right side of the graph but still follows the same trend as the other points, is it an outlier?
3. Why should you look at the overall pattern before deciding?

Answers

1. It is likely an outlier.
2. No. If it fits the trend, it is not an outlier.
3. Because an outlier is a point that does not fit the pattern of most of the data.

Summary

An outlier is a point on a scatter plot that stands far away from the rest of the data or does not follow the overall pattern.

To identify an outlier, first look for the trend in the graph. Then find any point that is clearly separated from the group or does not match the pattern.

Remember: a point can be at the edge of a graph and still not be an outlier if it fits the trend. The key question is always, Does this point belong with the pattern, or does it stand apart?

Line of Best Fit (Trend Line)

When we collect bivariate data, we are looking at two variables for the same group of people or objects. For example, we might compare hours studied and test score, or shoe size and height.

One useful way to display bivariate numerical data is with a scatter plot. A scatter plot shows pairs of values as points on a graph.

Sometimes the points on a scatter plot seem to follow a pattern. When the pattern looks roughly like a straight line, we can draw a line of best fit, also called a trend line.

A line of best fit is a line drawn through a scatter plot that shows the overall direction of the data. It does not need to pass through every point. Instead, it should be placed so that it models the data as closely as possible.

In 8th Grade, you usually draw the line informally. That means you use the graph and your judgment to place the line so that:

• it follows the general trend of the points,
• there are about as many points above the line as below it,
• and the points are not all crowded far away from the line.

Why do we use a line of best fit?

• It helps us see the relationship between two variables.
• It helps us make predictions.
• It shows whether the data has a positive trend, negative trend, or little to no linear trend.

Understanding trends in scatter plots

Before drawing a line of best fit, first look at the trend of the data.

• Positive trend: As one variable increases, the other tends to increase. The line goes upward from left to right.
• Negative trend: As one variable increases, the other tends to decrease. The line goes downward from left to right.
• No clear linear trend: The points do not follow a straight-line pattern, so a line of best fit may not be useful.

For example, if students who study more hours usually get higher scores, the scatter plot would likely show a positive trend.

If a car gets fewer miles per gallon as its weight increases, the scatter plot might show a negative trend.

How to draw a line of best fit

1. Look at the scatter plot and decide whether the pattern is roughly linear.
2. Notice whether the trend is positive or negative.
3. Place a straight line through the middle of the points.
4. Make sure the line has a balance of points above and below it.
5. Try to keep the points as close to the line as possible overall.

Important: The line of best fit does not have to go through the origin \\(0,0\\) unless the data suggests that it should.

Also, the line does not need to go through any actual data point. Its job is to represent the overall pattern.

Using a line of best fit to make predictions

Once a line of best fit is drawn, you can use it to estimate values.

If you know an \\(x\\)-value, move up or down to the line, then across to estimate the \\(y\\)-value.

If you know a \\(y\\)-value, move across to the line, then down or up to estimate the \\(x\\)-value.

These predictions are called estimates because the line is a model, not an exact rule for every point.

Worked Example 1: Identifying the trend

A scatter plot shows the number of hours a student reads each week and the number of vocabulary words they know. The points rise from left to right.

Question: What kind of trend does the scatter plot show?

Solution:

Because the points go upward from left to right, as reading hours increase, vocabulary words also tend to increase.

So the scatter plot shows a positive trend.

A line of best fit for this data would slope upward.

Worked Example 2: Choosing a reasonable trend line

Suppose a scatter plot has these points:

\\((1,2), (2,3), (3,4), (4,4), (5,6)\\)

Question: Would a line going upward from left to right make sense as a line of best fit?

Solution:

Let us look at the data. As the \\(x\\)-values increase from 1 to 5, the \\(y\\)-values generally increase from 2 to 6.

That means the data has a positive linear pattern.

A reasonable line of best fit would go upward from left to right and pass through the middle of the points.

It should not be too steep or too flat. It should be placed so that some points are above it and some are below it.

So yes, an upward-sloping line makes sense as a line of best fit.

Worked Example 3: Estimating with a trend line

A teacher draws a line of best fit on a scatter plot relating study time to quiz score. The line passes near the points \\((2,70)\\) and \\((6,90)\\).

Question: Use the line to estimate the quiz score for a student who studies 4 hours.

Solution:

The line goes through about \\((2,70)\\) and \\((6,90)\\).

From 2 hours to 6 hours is an increase of \\(4\\) hours.

From 70 to 90 is an increase of \\(20\\) points.

That means the score increases about

$$\frac{20}{4}=5$$

points per hour.

Going from 2 hours to 4 hours is 2 more hours, so add about

$$2\cdot 5=10$$

points to 70.

$$70+10=80$$

The estimated quiz score is about 80.

Worked Example 4: Deciding if a prediction is reasonable

A scatter plot compares plant height and number of days growing. A line of best fit shows that at 5 days, a plant is about 8 cm tall, and at 10 days, it is about 13 cm tall.

Question: About how tall would the line predict the plant to be at 8 days?

Solution:

From 5 days to 10 days, the height increases from 8 cm to 13 cm.

That is an increase of

$$13-8=5$$

cm over

$$10-5=5$$

days.

So the plant grows about

$$\frac{5}{5}=1$$

cm per day.

From 5 days to 8 days is 3 days, so add 3 cm to 8 cm.

$$8+3=11$$

The predicted height is about 11 cm.

Things to watch out for

• Do not connect the dots. A line of best fit is not a zigzag line through every point.
• Do not force the line through all points. Real data usually has some scatter.
• Do not use a line if there is no linear pattern. If the points are random or curved, a straight trend line may not fit well.
• Remember that predictions are estimates. They may be close, but not exact.

How line of best fit connects to real life

People use trend lines to study patterns in many situations:

• hours practiced and sports performance,
• temperature and ice cream sales,
• age of a car and its value,
• time spent exercising and heart rate.

In each case, the line of best fit helps show the general relationship between the two variables.

Quick checklist for students

• Is the scatter plot roughly linear?
• Is the trend positive or negative?
• Did I draw the line through the middle of the data?
• Are there about the same number of points above and below the line?
• Am I using the line only to make an estimate?

Summary

A line of best fit, or trend line, is a straight line that models the overall pattern in a scatter plot. It is used when the data shows a roughly linear relationship.

To draw one, place the line through the middle of the data so that the points are balanced above and below it. Then use the line to make reasonable estimates and describe whether the relationship is positive, negative, or has no clear linear trend.

Equation of a Trend Line

When we collect data about two variables, we often want to know whether the variables are related. A scatter plot helps us see that relationship.

If the points on a scatter plot seem to follow a general pattern, we can draw a trend line (also called a line of best fit). This line shows the overall direction of the data.

In this lesson, you will learn how to write the equation of a trend line. To do that, you will choose two points on the trend line, find the slope, and then find the y-intercept. Finally, you will write the equation in the form:

$$y = mx + b$$

Here, m is the slope and b is the y-intercept.

1. What a trend line tells us

A trend line is a straight line that fits the pattern of the points in a scatter plot as closely as possible. It does not have to go through every point. Instead, it should pass through the middle of the data so that about the same number of points are above the line as below it.

If the line goes upward from left to right, the data show a positive association. If the line goes downward from left to right, the data show a negative association.

• Positive slope: as one variable increases, the other tends to increase.
• Negative slope: as one variable increases, the other tends to decrease.
• Slope near 0: little or no linear trend.

2. Choosing points from the trend line

To find the equation, you should choose two points on the trend line, not just two original data points from the scatter plot unless they lie on the line you drew.

Try to choose points that:

• are easy to read from the graph,
• have coordinates close to whole numbers,
• are far apart on the line, which helps make your estimate more accurate.

Suppose the two points you choose are \\((x_1, y_1)\\) and \\((x_2, y_2)\\).

3. Finding the slope

The slope tells how much \\()y\\) changes when \\()x\\) increases by 1. Use the slope formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

This means:

• rise = change in \\()y\\)
• run = change in \\()x\\)
• slope = \\()\frac{\text{rise}}{\text{run}}\\)

If the slope is positive, the line rises. If the slope is negative, the line falls.

4. Finding the y-intercept

The y-intercept is the value of \\()y\\) when \\()x = 0\\). It is where the line crosses the y-axis.

Sometimes you can read the y-intercept directly from the graph. If not, use a point on the line and the slope in the equation \\()y = mx + b\\).

Substitute the coordinates of one point and the slope, then solve for \\()b\\).

For example, if the slope is 2 and a point on the line is \\((3, 11)\\), then:

$$11 = 2(3) + b$$

$$11 = 6 + b$$

$$b = 5$$

So the equation is:

$$y = 2x + 5$$

5. Steps for writing the equation of a trend line

1. Draw or look at the trend line on the scatter plot.
2. Choose two clear points on the trend line.
3. Find the slope using \\()m = \frac{y_2-y_1}{x_2-x_1}\\).
4. Use one point and the slope to find \\()b\\).
5. Write the equation in the form \\()y = mx + b\\).
6. Check whether the equation matches the graph reasonably well.

Worked Example 1: Positive trend line

A scatter plot shows a positive trend. You draw a trend line and choose the points \\((2, 4)\\) and \\((6, 10)\\) on the line.

Step 1: Find the slope.

$$m = \frac{10 - 4}{6 - 2} = \frac{6}{4} = \frac{3}{2}$$

So the slope is \\()\frac{3}{2}\\), or 1.5.

Step 2: Find the y-intercept.

Use \\((2,4)\\) in \\()y = mx + b\\):

$$4 = \frac{3}{2}(2) + b$$

$$4 = 3 + b$$

$$b = 1$$

Step 3: Write the equation.

$$y = \frac{3}{2}x + 1$$

This equation models the trend in the data.

Worked Example 2: Negative trend line

A trend line slopes downward. You choose the points \\((1, 9)\\) and \\((5, 3)\\).

Step 1: Find the slope.

$$m = \frac{3 - 9}{5 - 1} = \frac{-6}{4} = -\frac{3}{2}$$

The slope is negative, so the line goes down from left to right.

Step 2: Find the y-intercept.

Use \\((1,9)\\):

$$9 = -\frac{3}{2}(1) + b$$

$$9 = -\frac{3}{2} + b$$

$$b = 9 + \frac{3}{2} = \frac{18}{2} + \frac{3}{2} = \frac{21}{2}$$

$$b = 10.5$$

Step 3: Write the equation.

$$y = -\frac{3}{2}x + 10.5$$

Worked Example 3: Using a graph estimate

You draw a trend line on a scatter plot about study time and quiz score. Two easy-to-read points on your line are about \\((0, 60)\\) and \\((4, 80)\\).

Step 1: Find the slope.

$$m = \frac{80 - 60}{4 - 0} = \frac{20}{4} = 5$$

This means the quiz score increases about 5 points for each extra hour of study.

Step 2: Find the y-intercept.

Since one point is \\((0,60)\\), the y-intercept is 60.

Step 3: Write the equation.

$$y = 5x + 60$$

Step 4: Interpret the model.

In this situation:

• \\()x\\) = study time in hours
• \\()y\\) = quiz score
• slope 5 means each extra hour of study is associated with about 5 more points
• y-intercept 60 means the model predicts a score of 60 when study time is 0 hours

Worked Example 4: Writing a model from two chosen points

A student draws a trend line for a scatter plot of the number of days after planting and plant height. The student chooses the points \\((2, 7)\\) and \\((8, 19)\\).

Step 1: Find the slope.

$$m = \frac{19 - 7}{8 - 2} = \frac{12}{6} = 2$$

Step 2: Find the y-intercept.

Use \\((2,7)\\):

$$7 = 2(2) + b$$

$$7 = 4 + b$$

$$b = 3$$

Step 3: Write the equation.

$$y = 2x + 3$$

Step 4: Interpret the model.

• The slope 2 means the plant grows about 2 units each day.
• The y-intercept 3 means the model predicts the plant was about 3 units tall at day 0.

6. How to check if your equation makes sense

After writing your equation, ask yourself these questions:

• Does the sign of the slope match the graph? If the graph rises, the slope should be positive. If it falls, the slope should be negative.
• Does the y-intercept fit where the line crosses the y-axis?
• If you substitute one of your chosen points into the equation, does it work?
• Does the equation seem to match the overall pattern of the data?

7. Common mistakes to avoid

• Using data points instead of points on the trend line: You want the equation of the line you drew, so use points on that line.
• Mixing up the slope formula: Make sure you subtract in the same order in the top and bottom.
• Forgetting negative signs: A downward trend should have a negative slope.
• Stopping after finding the slope: You also need the y-intercept to write the full equation.
• Writing only numbers without meaning: In a real situation, think about what the slope and intercept mean.

8. Quick review

To write the equation of a trend line, use two points on the line. First find the slope, then find the y-intercept, and finally write the equation in slope-intercept form.

Remember:

$$m = \frac{y_2-y_1}{x_2-x_1}$$

$$y = mx + b$$

A trend line is a model, so it gives an estimate of the relationship between the two variables. It may not match every data point exactly, but it should describe the overall pattern well.

Interpreting Model Parameters means understanding what the numbers in an equation tell us about a real situation.

When we study bivariate data, we look at how two variables are related. For example, we might compare hours studied and test score, or number of weeks and plant height.

A trend line is a line that shows the general pattern in a scatter plot. It helps us describe how one variable changes when the other variable changes.

Many trend lines can be written in this form:

$$y=mx+b$$

In 8th grade, the two most important model parameters to interpret are:

• Slope, which is the rate of change
• y-intercept, which is the starting value

Let’s learn what each one means in context.

1. Slope: the rate of change

In the equation $$y=mx+b$$, the slope is the number represented by .

The slope tells us how much the y-value changes when the x-value increases by 1.

If the slope is positive, the trend line goes upward from left to right. This means as x increases, y tends to increase.

If the slope is negative, the trend line goes downward from left to right. This means as x increases, y tends to decrease.

So when you interpret slope, always ask:

• What does x represent?
• What does y represent?
• What does “for every 1 unit of x” mean in this situation?

For example, if $$y=3x+5$$ and x is hours and y is dollars, then the slope 3 means:

For every 1 more hour, the amount of money increases by 3 dollars.

2. y-intercept: the initial value

In the equation $$y=mx+b$$, the y-intercept is the number represented by b.

The y-intercept tells us the value of y when $$x=0$$.

This is often the starting amount or initial value.

For example, in $$y=3x+5$$, the y-intercept is 5. That means when $$x=0$$, $$y=5$$.

If x is hours and y is dollars, then this means:

At 0 hours, the amount is 5 dollars.

This might mean someone started with 5 dollars before earning 3 dollars per hour.

Important: The y-intercept only makes sense if $$x=0$$ is reasonable in the situation. If it does not fit the real context, you should be careful about how you interpret it.

How to interpret model parameters step by step

1. Write or look at the equation of the trend line.
2. Identify the slope and y-intercept.
3. Decide what x and y mean.
4. Interpret the slope as “for every 1 unit increase in x, y changes by ...”
5. Interpret the y-intercept as “when x is 0, y is ...”
6. Check whether your interpretation makes sense in the real situation.

Worked Example 1: Saving money

A student records how much money is in a savings jar after several weeks. A trend line is modeled by:

$$y=4x+10$$

where x is the number of weeks and y is the amount of money in dollars.

Step 1: Find the slope.

The slope is 4.

Interpretation: For every 1 more week, the amount of money increases by 4 dollars.

Step 2: Find the y-intercept.

The y-intercept is 10.

Interpretation: At week 0, there were 10 dollars in the jar.

Full context statement: The student started with 10 dollars and then saved about 4 dollars each week.

Worked Example 2: Temperature dropping

A model for the temperature of a drink is:

$$y=-2x+18$$

where x is time in minutes and y is temperature in degrees.

Slope: The slope is 2.

This means for every 1 more minute, the temperature decreases by 2 degrees.

y-intercept: The y-intercept is 18.

This means at 0 minutes, the temperature was 18 degrees.

Full context statement: The drink started at 18 degrees and cooled by about 2 degrees each minute.

Notice that because the slope is negative, the trend is decreasing.

Worked Example 3: Reading score model

A class studies the relationship between number of books read and reading score. A trend line is:

$$y=5x+60$$

where x is the number of books read and y is the reading score.

Slope: 5

Interpretation: For each additional book read, the reading score tends to increase by about 5 points.

y-intercept: 60

Interpretation: If 0 books are read, the model predicts a reading score of 60.

This does not mean every student who reads 0 books will definitely score 60. It means the trend line predicts that starting value based on the data pattern.

This is important because a model gives an estimate, not a guarantee.

Worked Example 4: Taxi fare

A taxi company models the total cost of a ride with:

$$y=2x+7$$

where x is miles traveled and y is cost in dollars.

Slope: 2

This means the cost increases by 2 dollars for every 1 mile traveled.

y-intercept: 7

This means the cost is 7 dollars when 0 miles are traveled.

In context, that 7 dollars could be the starting fee just for getting in the taxi.

Full context statement: The taxi charges a starting fee of 7 dollars and then 2 dollars per mile.

How to tell if a student interpretation is correct

A correct interpretation should include the variables and units.

For example, saying only “the slope is 4” is not enough. A better answer is:

For every 1 week, the amount saved increases by 4 dollars.

A correct interpretation of the y-intercept should mention what happens when $$x=0$$.

For example:

At 0 weeks, the student had 10 dollars.

Common mistakes to avoid

• Mixing up slope and y-intercept
  Remember: slope is the rate of change, and y-intercept is the starting value.
• Forgetting the units
  Always say what is changing and by how much. Use words like dollars, miles, weeks, or points.
• Ignoring the sign of the slope
  A negative slope means the y-value decreases as x increases.
• Giving a math answer but not a context answer
  You should explain what the numbers mean in the situation, not just name them.
• Assuming the model is exact
  A trend line shows a pattern or estimate. Real data points may not all lie on the line.

Quick practice thinking

If a trend line is $$y=6x+2$$, and x is days and y is pages read:

• The slope 6 means: pages read increase by 6 each day.
• The y-intercept 2 means: at day 0, 2 pages had already been read.

If a trend line is $$y=-3x+25$$, and x is hours and y is battery percent:

• The slope 3 means: battery percent drops by 3 percentage points each hour.
• The y-intercept 25 means: at 0 hours, the battery was at 25%.

Summary

When interpreting model parameters in a trend line, the slope tells the rate of change, and the y-intercept tells the initial value.

In a line written as $$y=mx+b$$:

•  tells how much y changes for every 1-unit increase in x.
• b tells the value of y when $$x=0$$.

Always explain these numbers using the meaning of the variables and the units from the data.

Interpolation and Extrapolation are two ways to use data to make predictions.

When we look at bivariate data, we are studying the relationship between two variables. For example, we might compare hours studied and test score, or age of a tree and its height. If the points on a scatter plot follow a pattern, we can use that pattern to make an estimate.

In this lesson, you will learn what interpolation and extrapolation mean, how to use a linear model to make predictions, and why some predictions are more trustworthy than others.

Key idea: A linear model is a line that shows the general trend in the data. We can use that line to estimate values that are not already listed in the data.

1. What is interpolation?

Interpolation means making a prediction between data values that you already have.

Suppose a scatter plot shows data for study times from 1 hour to 5 hours. If you want to predict the test score for 3.5 hours of study, that is interpolation because 3.5 is inside the data range.

Interpolation is usually more reliable because the prediction is based on values that are close to the actual data.

2. What is extrapolation?

Extrapolation means making a prediction outside the data values that you already have.

If the data only goes from 1 hour to 5 hours of study, and you try to predict the score for 8 hours of study, that is extrapolation because 8 is beyond the data range.

Extrapolation is usually less reliable because the pattern may not continue the same way outside the data you collected.

3. Using a linear model

If a scatter plot shows a roughly straight-line pattern, we can model it with a line. Sometimes the line is given. Sometimes you may use a rule in the form

$$y = mx + b$$

In this equation:

• x is the input variable
• y is the predicted output variable
• m is the slope, which tells how much y changes when x increases by 1
• b is the starting value when \(x=0\)

To make a prediction, substitute the given value of \(x\) into the equation and solve for \(y\).

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

1. Look at the smallest and largest \(x\)-values in the data.
2. Compare the new \(x\)-value to that range.
3. If the new value is inside the range, it is interpolation.
4. If the new value is outside the range, it is extrapolation.

5. Why extrapolation can be risky

Real-life data does not always keep following the same pattern forever. A straight-line model may work well for the data you collected, but outside that range, the situation may change.

For example:

• A plant may grow steadily for a while, but later its growth may slow down.
• Studying more may help test scores, but scores cannot rise forever past 100%.
• The cost of an item may increase in a pattern for a short time, then change later.

So, predictions made by extrapolation should be treated carefully, especially when they are far beyond the data range.

Worked Example 1: Identifying interpolation or extrapolation

A data set compares the number of books read \((x)\) and reading score \((y)\) for students. The values of \(x\) range from 2 books to 10 books.

Decide whether each prediction is interpolation or extrapolation:

• Predicting the reading score for 6 books
• Predicting the reading score for 12 books

Solution:

The data range is from \(2\) to \(10\).

• \(6\) is between \(2\) and \(10\), so this is interpolation.
• \(12\) is greater than \(10\), so this is extrapolation.

Worked Example 2: Using a linear model for interpolation

A teacher finds that the relationship between hours studied and quiz score can be modeled by

$$y = 5x + 60$$

Here, \(x\) is the number of hours studied, and \(y\) is the predicted quiz score.

If the data was collected for study times between \(1\) and \(5\) hours, predict the score for a student who studies \(3\) hours.

Step 1: Substitute \(x=3\) into the equation.

$$y = 5(3) + 60$$

Step 2: Multiply.

$$y = 15 + 60$$

Step 3: Add.

$$y = 75$$

The predicted quiz score is 75.

Because \(3\) is inside the data range from \(1\) to \(5\), this prediction is interpolation.

Worked Example 3: Using a linear model for extrapolation

Use the same model:

$$y = 5x + 60$$

Predict the score for a student who studies \(7\) hours.

Step 1: Substitute \(x=7\).

$$y = 5(7) + 60$$

Step 2: Multiply and add.

$$y = 35 + 60 = 95$$

The predicted score is 95.

But this prediction is extrapolation because \(7\) is outside the data range of \(1\) to \(5\).

This answer may be less reliable. The pattern might not continue in exactly the same way for students who study much longer.

Worked Example 4: Reading a situation carefully

A car rental company records the total cost \(y\) for renting a car for \(x\) days. A linear model is

$$y = 40x + 25$$

The company collected data for rentals from \(1\) day to \(6\) days.

Part A: Predict the cost for \(4\) days.

$$y = 40(4) + 25 = 160 + 25 = 185$$

The predicted cost is $185.

Since \(4\) is between \(1\) and \(6\), this is interpolation.

Part B: Predict the cost for \(10\) days.

$$y = 40(10) + 25 = 400 + 25 = 425$$

The predicted cost is $425.

Since \(10\) is outside the data range, this is extrapolation.

This prediction may not be accurate because the company could give discounts for longer rentals, or extra fees might apply.

6. Interpolation and extrapolation on a scatter plot

On a scatter plot, interpolation happens when you estimate a point between the existing data points along the trend line.

Extrapolation happens when you continue the trend line beyond the plotted data to estimate a value.

If the points make a clear straight pattern, a linear model can be useful. If the points are very spread out or do not follow a line well, predictions are less dependable.

7. Tips for answering questions

• Always check the data range before deciding if a prediction is interpolation or extrapolation.
• Use the given equation carefully by substituting the correct value for \(x\).
• If the prediction is extrapolation, remember to mention that it may be less reliable.
• The farther outside the data range you go, the less trustworthy the prediction usually is.

8. Common mistakes to avoid

• Mixing up interpolation and extrapolation
• Forgetting to compare the new value to the data range
• Assuming a model is always accurate far beyond the data
• Making arithmetic mistakes when substituting into the equation

Summary

Interpolation is predicting a value within the range of known data, and extrapolation is predicting a value outside the range of known data.

We often use a linear model, such as \(y = mx + b\), to make these predictions. Interpolation is usually more reliable because it stays within the actual data, while extrapolation is riskier because the pattern may change outside the observed range.

Two-Way Frequency Tables help us organize information about two different categories at the same time.

For example, we might want to know whether students prefer soccer or basketball and whether they are in 7th or 8th grade. A two-way table lets us sort that data clearly so we can compare groups.

This lesson will show you how to read, make, and use two-way frequency tables. You will also learn how to find row totals, column totals, and the grand total.

1. What is a two-way frequency table?

A frequency tells how many times something happens. A two-way frequency table shows the number of data values that fit into each combination of two categories.

One category is usually written across the top in columns, and the other category is written down the left side in rows.

Here is the basic structure:

$$ \begin{array}{c|c|c|c} & \text{Category B1} & \text{Category B2} & \text{Total} \\ \hline \text{Category A1} & & & \\ \text{Category A2} & & & \\ \hline \text{Total} & & & \end{array} $$

Each box inside the table shows how many data values belong to both categories.

• Rows go across.
• Columns go up and down.
• Row total is the sum across a row.
• Column total is the sum down a column.
• Grand total is the total number of all data values.

2. Why do we use two-way tables?

Two-way tables help us:

• organize bivariate categorical data,
• compare two groups,
• see patterns,
• answer questions about totals and parts of the data.

The word bivariate means the data involves two variables. In this lesson, both variables are categorical, which means they are labels or groups, not measurements.

Examples of categorical variables include:

• favorite subject,
• grade level,
• pet type,
• yes or no answers.

3. How to read a two-way frequency table

Suppose a class survey asked students whether they prefer reading fiction or nonfiction.

$$ \begin{array}{c|c|c|c} & \text{Fiction} & \text{Nonfiction} & \text{Total} \\ \hline \text{Boys} & 12 & 8 & 20 \\ \text{Girls} & 15 & 5 & 20 \\ \hline \text{Total} & 27 & 13 & 40 \end{array} $$

Here is how to read the table:

• 12 boys prefer fiction.
• 8 boys prefer nonfiction.
• 15 girls prefer fiction.
• 5 girls prefer nonfiction.
• 20 boys were surveyed in all.
• 20 girls were surveyed in all.
• 27 students prefer fiction altogether.
• 40 students were surveyed altogether.

Notice that each inside box belongs to one row and one column. For example, the number 12 is in the row for boys and the column for fiction.

4. How to find row totals, column totals, and the grand total

To find a row total, add the numbers across the row.

To find a column total, add the numbers down the column.

To find the grand total, add all the values in the table, or add the row totals, or add the column totals.

For example, if one row has 9 and 11, then the row total is:

$$9+11=20$$

If one column has 9 and 6, then the column total is:

$$9+6=15$$

If all four inside values are 9, 11, 6, and 14, then the grand total is:

$$9+11+6+14=40$$

5. Worked Example 1: Reading a completed table

A survey asked students whether they walk to school or ride the bus.

$$ \begin{array}{c|c|c|c} & \text{Walk} & \text{Bus} & \text{Total} \\ \hline \text{6th Grade} & 18 & 12 & 30 \\ \text{7th Grade} & 10 & 20 & 30 \\ \hline \text{Total} & 28 & 32 & 60 \end{array} $$

Question 1: How many 7th grade students ride the bus?

Look at the row for 7th Grade and the column for Bus. The answer is 20.

Question 2: How many students walk to school?

Look at the total at the bottom of the Walk column. The answer is 28.

Question 3: How many students were surveyed in all?

Look at the bottom-right corner, which is the grand total. The answer is 60.

Question 4: Which is greater: 6th graders who walk or 7th graders who walk?

6th graders who walk: 18

7th graders who walk: 10

Since \(18>10\), more 6th graders walk.

6. Worked Example 2: Filling in missing totals

A club asked students whether they like art or music better.

$$ \begin{array}{c|c|c|c} & \text{Art} & \text{Music} & \text{Total} \\ \hline \text{Boys} & 14 & 9 & \\ \text{Girls} & 11 & 16 & \\ \hline \text{Total} & & & \end{array} $$

We will fill in the missing totals step by step.

Step 1: Find each row total.

Boys total:

$$14+9=23$$

Girls total:

$$11+16=27$$

Step 2: Find each column total.

Art total:

$$14+11=25$$

Music total:

$$9+16=25$$

Step 3: Find the grand total.

$$23+27=50$$

Now the completed table is:

$$ \begin{array}{c|c|c|c} & \text{Art} & \text{Music} & \text{Total} \\ \hline \text{Boys} & 14 & 9 & 23 \\ \text{Girls} & 11 & 16 & 27 \\ \hline \text{Total} & 25 & 25 & 50 \end{array} $$

Always check your work. The row totals and column totals should agree with the grand total.

Here,

• row totals: \(23+27=50\)
• column totals: \(25+25=50\)

Both give the same grand total, so the table is correct.

7. Worked Example 3: Making a table from data

A teacher asked 12 students two questions:

• Do you prefer cats or dogs?
• Are you a student athlete or not?

The results are listed below:

• Athlete, Cat
• Athlete, Dog
• Not Athlete, Dog
• Athlete, Dog
• Not Athlete, Cat
• Not Athlete, Dog
• Athlete, Cat
• Not Athlete, Cat
• Athlete, Dog
• Not Athlete, Dog
• Athlete, Cat
• Not Athlete, Cat

Step 1: Choose rows and columns.

Let the rows be Athlete and Not Athlete.

Let the columns be Cat and Dog.

Step 2: Count each combination.

• Athlete and Cat: 3
• Athlete and Dog: 3
• Not Athlete and Cat: 3
• Not Athlete and Dog: 3

Step 3: Put the counts into the table.

$$ \begin{array}{c|c|c|c} & \text{Cat} & \text{Dog} & \text{Total} \\ \hline \text{Athlete} & 3 & 3 & 6 \\ \text{Not Athlete} & 3 & 3 & 6 \\ \hline \text{Total} & 6 & 6 & 12 \end{array} $$

This table shows that the preferences are evenly split in this group.

8. Worked Example 4: Solving with a missing value

Suppose this table shows favorite school lunch choice.

$$ \begin{array}{c|c|c|c} & \text{Pizza} & \text{Tacos} & \text{Total} \\ \hline \text{8th Grade} & 17 & x & 25 \\ \text{7th Grade} & 12 & 9 & 21 \\ \hline \text{Total} & 29 & & 46 \end{array} $$

We need to find \(x\).

In the 8th Grade row, the total is 25. One part is 17, so the other part must be:

$$17+x=25$$

Subtract 17 from both sides:

$$x=25-17=8$$

So the missing value is 8.

Now we can also find the Tacos column total:

$$8+9=17$$

The completed table is:

$$ \begin{array}{c|c|c|c} & \text{Pizza} & \text{Tacos} & \text{Total} \\ \hline \text{8th Grade} & 17 & 8 & 25 \\ \text{7th Grade} & 12 & 9 & 21 \\ \hline \text{Total} & 29 & 17 & 46 \end{array} $$

9. Tips for success

• Read the row and column labels carefully before answering a question.
• Make sure each number matches both categories.
• Add across for row totals.
• Add down for column totals.
• Check that the row totals and column totals give the same grand total.

10. Common mistakes to avoid

• Mixing up rows and columns: Trace with your finger if needed.
• Using the wrong total: A row total is different from a column total.
• Forgetting the grand total: The bottom-right corner is important.
• Counting data twice: Each person or item should fit in only one box in the table.

11. Quick practice questions

Use this table:

$$ \begin{array}{c|c|c|c} & \text{Yes} & \text{No} & \text{Total} \\ \hline \text{Has Pet} & 13 & 7 & 20 \\ \text{No Pet} & 9 & 11 & 20 \\ \hline \text{Total} & 22 & 18 & 40 \end{array} $$

1. How many students have a pet and answered yes?
2. How many students do not have a pet?
3. How many students answered no?
4. How many students were surveyed in all?

Answers:

1. 13
2. 20
3. 18
4. 40

12. Summary

A two-way frequency table organizes data for two categorical variables.

The inside boxes show how many data values fit each pair of categories. The numbers at the ends of rows are row totals, the numbers at the bottoms of columns are column totals, and the bottom-right number is the grand total.

If you read the labels carefully and check your totals, two-way tables become a simple and powerful way to understand data.

Relative Frequencies help us compare data in a two-way table by turning counts into fractions, decimals, or percents. This makes it easier to see patterns and possible relationships between two categorical variables.

For example, a table might show whether students prefer cats or dogs and whether they are in 7th or 8th grade. The counts tell how many students are in each group, but relative frequencies tell what part or percent of a group each count represents.

This is useful because raw counts can sometimes be misleading. One group may be much larger than another, so comparing percentages is often more fair than comparing just the number of people.

Key idea: A relative frequency compares a part to a total.

The basic formula is:

$$ \text{Relative Frequency} = \frac{\text{Frequency Count}}{\text{Total}} $$

You can leave the answer as a fraction, change it to a decimal, or write it as a percent.

To turn a decimal into a percent, multiply by 100.

$$ \text{Percent} = \text{Decimal} \times 100 $$

In two-way tables, there are three common kinds of relative frequency:

• Joint relative frequency: compares one cell to the grand total.
• Row relative frequency: compares one cell to its row total.
• Column relative frequency: compares one cell to its column total.

Let’s look at what each one means.

1. Joint relative frequency

This tells what fraction of the entire table is in one specific cell.

Formula:

$$ \text{Joint Relative Frequency} = \frac{\text{Cell Count}}{\text{Grand Total}} $$

2. Row relative frequency

This tells what fraction of a row total is in a specific cell. Use this when you want to compare parts within the same row.

Formula:

$$ \text{Row Relative Frequency} = \frac{\text{Cell Count}}{\text{Row Total}} $$

3. Column relative frequency

This tells what fraction of a column total is in a specific cell. Use this when you want to compare parts within the same column.

Formula:

$$ \text{Column Relative Frequency} = \frac{\text{Cell Count}}{\text{Column Total}} $$

When row or column relative frequencies are very different across categories, that can suggest a possible association between the two variables.

If the relative frequencies are very similar, that suggests there may be little or no association.

Worked Example 1: Finding a simple relative frequency

A class surveyed 20 students. Out of those 20 students, 8 said math is their favorite subject.

Find the relative frequency that chose math.

Use the formula:

$$ \frac{8}{20} = 0.4 $$

Convert to a percent:

$$ 0.4 \times 100 = 40\% $$

Answer: The relative frequency is 0.4, or 40%.

This means 40% of the students chose math.

Worked Example 2: Joint relative frequency in a two-way table

A survey asked 50 students whether they prefer reading fiction or nonfiction.

	Fiction	Nonfiction	Total
Girls	18	7	25
Boys	10	15	25
Total	28	22	50

Find the joint relative frequency for girls who prefer fiction.

The cell count is 18, and the grand total is 50.

$$ \frac{18}{50} = 0.36 $$

As a percent:

$$ 0.36 \times 100 = 36\% $$

Answer: The joint relative frequency is 0.36, or 36%.

This means 36% of all students surveyed were girls who prefer fiction.

Worked Example 3: Row relative frequencies

Use the same table.

Find the row relative frequencies for the Girls row.

The row total for girls is 25.

• Girls who prefer fiction: $$\frac{18}{25} = 0.72 = 72\%$$
• Girls who prefer nonfiction: $$\frac{7}{25} = 0.28 = 28\%$$

So in the girls row:

• 72% prefer fiction
• 28% prefer nonfiction

Now find the row relative frequencies for the Boys row. The row total for boys is also 25.

• Boys who prefer fiction: $$\frac{10}{25} = 0.40 = 40\%$$
• Boys who prefer nonfiction: $$\frac{15}{25} = 0.60 = 60\%$$

Now compare the rows:

• Among girls, 72% prefer fiction.
• Among boys, 40% prefer fiction.

These percentages are quite different, so there may be an association between gender and reading preference.

Worked Example 4: Column relative frequencies and interpreting association

A school surveyed students about whether they play a sport and whether they ride the bus.

	Ride Bus	Do Not Ride Bus	Total
Play Sport	24	16	40
Do Not Play Sport	6	14	20
Total	30	30	60

Find the column relative frequencies.

Ride Bus column total = 30

• Play Sport and Ride Bus: $$\frac{24}{30} = 0.80 = 80\%$$
• Do Not Play Sport and Ride Bus: $$\frac{6}{30} = 0.20 = 20\%$$

Do Not Ride Bus column total = 30

• Play Sport and Do Not Ride Bus: $$\frac{16}{30} \approx 0.53 \approx 53\%$$
• Do Not Play Sport and Do Not Ride Bus: $$\frac{14}{30} \approx 0.47 \approx 47\%$$

Now compare the columns:

• Of students who ride the bus, 80% play a sport.
• Of students who do not ride the bus, about 53% play a sport.

Because these percentages are not close, there appears to be a possible association between riding the bus and playing a sport.

How to decide which total to use

This is one of the most important parts of relative frequency.

• If the question asks about the whole table, use the grand total.
• If the question asks about values within a row, use the row total.
• If the question asks about values within a column, use the column total.

Look for clue words:

• "Out of all students" means use the grand total.
• "Of the girls" means use the girls row total.
• "Of the students who ride the bus" means use the ride bus column total.

Common mistakes to avoid

• Using the wrong total in the denominator.
• Forgetting to convert a decimal to a percent.
• Comparing counts instead of relative frequencies when groups have different sizes.
• Thinking that a possible association proves one variable causes the other. Relative frequencies only help show patterns.

Quick check

Suppose a table shows that 12 out of 30 students in a row chose pizza.

• Relative frequency as a decimal: $$\frac{12}{30} = 0.4$$
• Relative frequency as a percent: $$40\%$$

If another row has 24 out of 60 students choosing pizza, then:

$$ \frac{24}{60} = 0.4 = 40\% $$

Even though the counts are different, the relative frequencies are the same. That means the two groups chose pizza at the same rate.

Summary

Relative frequencies turn counts into fractions, decimals, or percents so data is easier to compare.

In a two-way table:

• Joint relative frequency uses the grand total.
• Row relative frequency uses a row total.
• Column relative frequency uses a column total.

By comparing row or column relative frequencies, you can look for a possible association between two categorical variables.

Always ask yourself: What total should I compare this count to? Once you choose the correct total, finding the relative frequency is straightforward.