5. Bivariate Quantitative Data
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Chapter 4
5.
Bivariate Quantitative Data
Bivariate quantitative data involves two sets of numerical values and their relationship. Visual representations, such as scatter plots, help in spotting patterns or trends within these data sets. A line of fit, often drawn on scatter plots, provides a generalized trajectory, suggesting how one set of data might predict or influence the other. For instance, in real-world applications, bivariate data can be used to compare the ages and incomes of individuals, temperatures and ice cream sales, or even the years of experience and job performance ratings. Through these methods, analysts, researchers, and students can draw meaningful conclusions from complex sets of data.
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| | 9 Theory slides |
| | 7 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
This lesson will explore how two quantities are related and how to make predictions by finding a line of fit for scatter plots.
Catch-Up and Review
The recommended reading is information that is helpful or necessary to understand before beginning the lesson.
- These are some topics that help in data analysis.
- These are some important topics about linear equations.
Explore
A Set of Points on a Coordinate Plane
The following applet shows three graphs — each with a set of points distributed on a coordinate plane.
Pay close attention to the behavior of the dependent variable as the independent variable increases. Does y increase, decrease, or can it not be distinguished?
Challenge
Attendance at an Aquatic Park
Magdalena is fascinated by her local aquatic park and is eager to analyze how temperatures influence attendance. The following graph represents the data she collected — the average number of people that attend the park at specific temperatures.
Discussion
Analyzing Scatter Plots and Identifying Relationships Between Data Sets
Valuable conclusions and predictions are made about a situation based on collected data. Before such statements can be made, the data is analyzed by using tools such as graphs. A scatter plot, for example, is used to identify the correlation between a pair of data sets.
Concept
Scatter Plot
A scatter plot is a graph that shows each observation of a bivariate data set as an ordered pair in a coordinate plane. Consider the following example, where a scatter plot illustrates the results gathered at a local ice cream parlor. This study records the number of ice creams sold and the corresponding air temperature.
Concept
Correlation
A correlation is a relation between two data sets. For example, consider two data sets, one consisting of temperatures and the other consisting of the number of coats sold. A decrease in the temperature may imply an increase in the number of coats sold. Based on the trend of the bivariate data, three types of correlations are possible which can be described using scatter plots.
Knowing the type of correlation helps analyze trends and make predictions based on data. Furthermore, the shape of the patterns formed by positive and negative correlations can be thought to have a positive and negative slope, respectively. The applet below shows how a data set transforms from a random pattern to a positive or a negative correlation.
Pop Quiz
Identifying Correlations from Scatter Plots
The following applet shows different scatter plots. Select the type of correlation that matches the scatter plot shown.
Discussion
Lines of Fit for Scatter Plots
Once the scatter plot of a data set is drawn and the type of correlation is identified, predictions can be made about the trend of the data by using lines of fit.
Concept
Line of Fit
When data sets have a positive or negative correlation, the trend of the data can be modeled using a line of fit, also called a trend line. This line is drawn on a scatter plot near most of the data points, which appear evenly distributed above and below the line.
The scatter plot above shows the mean weights of kittens from the same litter in relation to their age. In this case, a line of fit could be drawn quite seamlessly. When drawing such a line of fit, the following characteristics should be considered.
- The data needs to have either a positive or negative correlation.
- While a line of fit is not unique and does not create an exact distribution, ideally, about half of the points should be above the line and about half below the line.
- An equation of the line can be found using two of its points. These points do not necessarily belong to the bivariate data set.
Example
Analyzing Daily Situations Using Scatter Plots and Lines of Fit
At an aquatic park, a student-volunteer named Tadeo noticed a dedicated person who swims long distances in the lazy lagoon every Saturday morning.
Tadeo is amazed and wants to analyze how many calories the swimmer burns compared to the distance swam. He observes and records the swimmer diligently.
| Distance (km) | Calories Burned |
|---|---|
| 16 | 980 |
| 15 | 880 |
| 14 | 860 |
| 13 | 740 |
| 12 | 720 |
| 11 | 680 |
| 10 | 595 |
| 9 | 560 |
| 8 | 490 |
| 7 | 400 |
| 6 | 380 |
a Make a scatter plot of the data.
b What type of correlation does the data have? Justify the answer.
c Draw a line of fit for the scatter plot.
d Find an equation for the line of fit.
Answer
a Example Answer:
b Positive, see solution.
c Example Answer:
d Example Equation: y=50x+110
Solution
a To draw the scatter plot, let x be the distance in kilometers and y the calories burned. With this in mind, the information from the table can be shown on a scatter plot.
b From the scatter plot previously drawn, it can be seen that as the distance increases, the number of calories burned also increases. Therefore, the bivariate data has a positive correlation.
c Since the data has a positive correlation, it can be modeled with a line of fit. The line of fit is not unique. However, ideally, the number of points below and above the line is expected to be similar.
d Because the equation of a line can be found using any two points on the line, two points whose coordinates can be easily identified will be marked on the graph of the line of fit.
y−y1y−y1=m(x−x1)⇓=50(x−x1)
To complete the equation, any of the two points can be substituted above. For simplicity, (4,310) will be used.
Example
Making Predictions Using Lines of Fit
Zosia and Vincenzo are poster designers at the aquatic park. Right now, they are promoting a 3D movie about the life of dolphins called Above and Below the Line.
They recorded the number of tickets sold each week with the purpose of using the data to determine whether they should continue to advertise the movie on a billboard. The scatter plot shows the collected data.
a Draw a line of fit for the scatter plot.
b Find an equation for the line of fit in slope-intercept form.
c If the expected number of tickets sold on week 12 is more than 150000, Vincenzo and Zosia will keep the movie for at least two more weeks on the billboard. Use the line of fit to predict if the movie stays on the billboard.
Answer
a Example Line:
b Example Equation: y=-25x+525
c Example Answer: Because the expected number of tickets sold on week 12 is about 237000, they may decide to keep the movie.
Hint
a What type of correlation does the scatter plot have?
b Use two points on the line of fit to find the slope and the y-intercept of the line.
c Evaluate the equation found in Part B for x=12.
Solution
a The scatter plot shows that the number of tickets sold decreases as time passes. Therefore, the data has a negative correlation and can be modeled by a line of fit. Recall that the line of fit is close to most of the data points, while the points are ideally half above and half below the line of fit.
b The equation of a line in slope-intercept form has the following form.
y=mx+b
In this equation, m is the slope and b the y-intercept of the line. The slope of the line of fit can be found by using the Slope Formula.
m=x2−x1y2−y1
The points (1,500) and (7,350) are on the line of fit. This means that they can be substituted in the above formula.
m=x2−x1y2−y1
SubstitutePoints
Substitute (1,500) & (7,350)
m=7−1350−500
▼
Evaluate right-hand side
SubTerms
Subtract terms
m=6-150
MoveNegNumToFrac
Put minus sign in front of fraction
m=-6150
CalcQuot
Calculate quotient
m=-25
y=mx+b substitute y=-25x+b
Now, the y-intercept can be found by substituting any of the points into the partial equation. In this case, (1,500) will be used. y=-25x+b
SubstituteII
x=1, y=500
500=-25(1)+b
▼
Solve for b
IdPropMult
Identity Property of Multiplication
500=-25+b
AddEqn
LHS+25=RHS+25
525=b
RearrangeEqn
Rearrange equation
b=525
y=-25x+b substitute y=-25x+525
c The line of fit describes the trend of the data. This means that it can be used to predict how many tickets would be sold on the 12th week. Therefore, by evaluating the equation of the line of fit for x=12, the expected number of tickets sold can be found.
Closure
Applying Lines of Fit to Solve Real-Life Situations
In this lesson, it was taught how to analyze bivariate data using scatter plots and lines of fit. These mathematical concepts can now be used to solve the Challenge. It is now recognizable that Magdalena created a scatter plot to show the aquatic park visitors in relation to the temperatures.
Answer
Yes, see solution.
Solution
The scatter plot shows that the number of attendants increases as the temperature increases, which means the data has a positive correlation. Therefore, it can be modeled with a line of fit.
y=mx+b
In this equation, m is the slope and b the y-intercept of the line. The slope can be calculated by using the Slope Formula.
m=x2−x1y2−y1
Because the points (70,7000) and (80,8000) are on the line of fit, they will be used to find the slope.
Now, the slope m=100 can be substituted to obtain a partial equation of the line of fit. y=mx+b substitute y=100x+b
By substituting one of the points on the line of fit, the y-intercept can be found. In this case, (70,7000) will be used.
Therefore, the y-intercept b is 0. With this information, the equation of the line of fit can be written.
y=100x+b substitute yy=100x+0=100x
Finally, using this equation, the number of people that would attend the park if the temperature is about 110∘F can be predicted. To do so, the equation needs to be evaluated for x=110.
It is expected that more than 10000 people will attend the park next week. What a brilliant way to make a prediction. Magdalena has really helped the aquatic park prepare for the influx of visitors surely to come.
The scatter plot shows the average number of attendants per year at the London Football Club's games.
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A linear equation in point-slope form with slope m and y-intercept b has the following format. y= mx+ b To write an equation for the line of fit, we first need to determine the slope of the line. To do so we will substitute the given points into the Slope Formula.
We can substitute the slope m= 10 to obtain a partial equation of the line of fit. y= mx+b [ m= 10]Substitute y= 10x+b Next, we wil substitute one of the points on the line of fit and solve the equation for b to find the y-intercept. Let's use ( 2, 25)
We can now complete the equation for the line of fit. y=10x+ b [ b= 5]Substitute y=10x+ 5
Consider the equation of the line of fit found in the previous part. y=10x+5 We will use this equation to calculate the expected attendance number in 2023. Since the year 2023 is 23 years after 2000, we will evaluate the equation when x= 23 to predict attendance in the year 2023.
Since the number of attendance is given in thousands, we can predict that the average attendance in 2023 will be about 235 000 people.
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Examine each of the given situations and decide which one does not belong to the group.
A.B.C.D. Hours worked and amount of money earned Height of a student and favorite food Amount of ice cream sold and temperature Time spent running and calories burned
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We will analyze each situation separately to determine which one is different from the others.
Situation A
The first situation talks about the hours worked and the amount of money earned. Usually, the amount of money someone earns depends on the number of hours worked. This means that this situation indicates a positive correlation.
Situation B
The second situation is about the height of a student and his favorite food. Note that favorite foods depend on a personal preference, not genetics. This means there is no correlation between the variables in this situation.
Situation C
We will now analyze the third situation. Generally, people tend to seek a cool treat like ice cream when it is hot. This means that there is a positive correlation in this situation. As the temperature increases, the amount of ice cream sold increases.
Situation D
When doing exercise like running, we burn calories. Furthermore, while more time is spent on exercise, more calories are expected to burn. This indicates a positive correlation between the variables in this situation.
Conclusion
We have analyzed each situation. Let's summarize this information in a table.
| Situation | Description | Correlation |
|---|---|---|
| A | Hours worked and amount of money earned. | Positive Correlation |
| B | Height of a student and favorite food. | No Correlation |
| C | Number of ice creams sold and temperature. | Positive Correlation |
| D | Time spent running and calories burned. | Positive Correlation |
We can see that the variables given for each case are positively correlated, except for B. Therefore, B does not belong to the group.
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Bivariate Quantitative Data
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