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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Lesson Settings & Tools
| | 9 Theory slides |
| | 7 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Bivariate Quantitative Data
Slide of 9
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 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.
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 150 000, 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 237 000, 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.
m = y_2-y_1/x_2-x_1
SubstitutePoints
Substitute ( 1,500) & ( 7,350)
m=350- 500/7- 1
▼
Evaluate right-hand side
SubTerms
Subtract terms
m=-150/6
MoveNegNumToFrac
Put minus sign in front of fraction
m=-150/6
CalcQuot
Calculate quotient
m=-25
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
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 12^\text{th} 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.
Bivariate Quantitative Data
Exercise 1.1
Consider the following table of values.
| x | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| y | -20 | -12 | -11 | -8 | 0 | 3 | 5 | 13 | 13 | 21 | 25 |
Which scatter plot represents the data?
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{"type":"choice","form":{"alts":["Positive Correlation","Negative Correlation","No Correlation"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
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We will draw a scatter plot to determine which option represents the data. We will do it by treating the information in the table as a set of points.
We can see that this scatter plot corresponds to option A.
Consider the three general outcomes for the correlation of data.
| Observation | Type of Correlation |
|---|---|
| As x increases, y increases. | Positive |
| As x increases, y decreases. | Negative |
| There is no visible pattern. | No |
Using this information, we can identify the correlation type of the given data. We can see that as the x-values increase, the y-values also increase.
To visualize it, let's draw 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 the same.
We can conclude that the data has a positive correlation.
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Determine if x and y show a positive correlation, a negative correlation, or no correlation.
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We will analyze the scatter plot to determine the type of correlation the data shows. Let's first review the three general outcomes for the correlation of data.
| Observation | Type of Correlation |
|---|---|
| As x increases, y increases. | Positive |
| As x increases, y decreases. | Negative |
| There is no visible pattern. | No Correlation |
We can now consider the scatter plot.
It appears that there is some kind of correlation. We will draw a line of fit to help us identify the type of correlation. Recall that the line of fit is not unique. However, the number of points below and above the line is expected to be similar.
Note that as the x-values increase, the y-values decrease. This means that there is a negative correlation between x and y.
Consider the given scatter plot.
Observing the graph, we see that the data does not follow any pattern. Therefore, there is no correlation between x and y.
Let's look at the given graph.
It seems that the data follows some pattern. We will draw a line of fit to identify the type of correlation the data follows.
We can see that as the x-values increase, the y-values also increase. This indicates that there is a positive correlation between x and y.
We will begin by looking at the given scatter plot.
Again, it appears that the data follows some pattern. A line of fit can help us to identify the type of correlation. Let's draw it!
We can note that as the x-values increase, the y-values decrease. This means there is a negative correlation between x and y.
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Ali was reading a real estate magazine that he found in the garage of his house. He found out that the price of a house (in thousands of dollars) is connected to the number of square meters of the house. Two places called Ali's attention: Comfyville and Dreamville. The prices of the houses of each place are predicted by the following equations.
| Price of Houses | |
|---|---|
| Comfyville | Dreamville |
| p=159+0.05A | p=299+0.2A |
{"type":"choice","form":{"alts":["Comfyville","Dreamville"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}
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We are given two equations, and each can be used to predict the price p in thousands of dollars for a house in different cities given the area of the house in square feet, A.
| Price of Houses | |
|---|---|
| Comfyville | Dreamville |
| p=159+0.05 A | p=299+0.2 A |
The house Ali is looking at in the advertisement costs 933 thousands of dollars and has an area of 3200 square meters. Therefore, we need to evaluate both equations when A= 3200 and see which of the equations gives roughly 933.
Comftyville
Let's evaluate the equation for Comfyville.
Dreamville
Now, let's evaluate the equation for Dreamville.
Conclusion
We have evaluated both equations when A=3200. We can also note that none of the equations gave 933. However, because 939 is the closest to 933, the house shown in the advertisement is probably located in Dreamville.
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The scatter plot shows the average number of people attending a beach club at specific temperatures.
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To determine the average number of people attending the beach club when the temperature is 70^(∘)F, we will draw a horizontal segment to the y-axis to find its y-coordinate.
We can see the segment intercepts the y-axis at 100. This means that when it is 70^(∘)F, the average number of people attending the beach club is 100.
We now want to determine the average temperature given that the beach club had 350 visitors. In this case, we need to draw a vertical segment to the x-axis to determine the x-coordinate corresponding to the number of visitors.
Note that the segment intercepts the x-axis at 110. This means that the temperature last Saturday was 110^(∘)F.
To find which sentence best describes the scatter plot, let's first draw a line of fit through the points on the scatter plot.
Observing the line of fit, we can see that as the temperature increases the number of people attending the beach club also increases. This corresponds to option B.
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Bivariate Quantitative Data
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