Unit 6 Associations In Data — Unit Plan

Consider the data collected from pulling back a toy car and then letting it go forward. In the first table, the data may not seem to have an obvious pattern. The second table has the same data and shows that both values are increasing together.

A scatter plot of the data makes the pattern clear enough that we can estimate how far the car will travel when it is pulled back 5 in.

Patterns in data can sometimes become more obvious when reorganized in a table or when represented in scatter plots or other diagrams. If a pattern is observed, it can sometimes be used to make predictions. This is a scatter plot for this scenario:

A scatterplot. Horizontal, from 0 to 12, by 2’s, labeled distance pulled back, inches. Vertical, from 0 to 40, by 10’s, labeled distance traveled, inches. 6 data points.  Trend linearly upward and right.

Histograms show us how measurements of a single attribute are distributed. For example, a veterinarian saw 25 dogs in her clinic one week. She measured the height and weight of each dog.

This histogram shows how the weights of the dogs are distributed.

This histogram shows how the heights of the dogs are distributed.

These histograms tell us how the weights of the dogs and how the heights of dogs were distributed. But, they do not give any evidence of a connection between a dog’s height and its weight.

Scatter plots allow us to investigate possible connections between two attributes. In this example, each plotted point corresponds to 1 of the 25 dogs, and its coordinates tell us the height and weight of that dog. Examination of the scatter plot allows us to see a connection between height and weight for the dogs. 

A scatterplot. Horizontal, from 6 to 30, by 3’s, labeled dog height, inches. Vertical, from 0 to 112, by 16’s, labeled dog weight, pounds. 24 data points.  Trend upward and to right.

Scatter plots show two measurements for each individual from a group. For example, this scatter plot shows the weight and height for each dog from a group of 25 dogs. 

A scatterplot. Horizontal, from 6 to 30, by 3’s, labeled dog height, inches. Vertical, from 0 to 112, by 16’s, labeled dog weight, pounds. 24 data points.  Trend upward and to right.

We can see that the tallest dogs are 27 inches, and that one of those tallest dogs weighs about 75 pounds while the other weighs about 110 pounds. This shows us that dog weight is not a function of dog height because there would be two different outputs for the same input. But we can see a general trend: taller dogs tend to weigh more than shorter dogs. There are exceptions. For example, there is a dog that is 18 inches tall and weighs over 50 pounds, and there is another dog that is 21 inches tall but weighs less than 30 pounds.

When we collect data by measuring attributes like height, weight, area, or volume, we call the data numerical data (or measurement data), and we say that height, weight, area, or volume is a numerical variable.

Sometimes, we can use a linear function as a model of the relationship between two variables. For example, here is a scatter plot that shows heights and weights of 25 dogs together with the graph of a linear function which is a model for the relationship between a dog’s height and its weight.

For some dogs, we can see that the model does a good job of predicting the weight given the height. These correspond to points on or near the line. The model doesn’t do a very good job of predicting the weight given the height for the dogs whose points are far from the line.

For example, there is a dog that is about 20 inches tall and weighs a little more than 16 pounds. The model predicts that the weight would be about 48 pounds. We say that the model overpredicts the weight of this dog. There is also a dog that is 27 inches tall and weighs about 110 pounds. The model predicts that its weight will be a little less than 80 pounds. We say the model underpredicts the weight of this dog. For most of the dogs in this data set, though, the model does a good job of predicting the weight from the height.

Sometimes a data point is far away from the other points or doesn’t fit a trend that all the other points fit. We call these outliers. 

When a linear function fits data well, we say there is a “linear association” between the variables. For example, the relationship between height and weight for 25 dogs with the linear function whose graph is shown in the scatter plot.

We say there is a positive association between dog height and dog weight because knowledge about one variable helps predict the other variable, and when one variable increases, the other tends to increase as well.

What do you think the association between the weight of a car and its fuel efficiency is?

We say that there is a negative association between fuel efficiency and weight of a car because knowledge about one variable helps predict the other variable, and when one variable increases, the other tends to decrease.

Here is a scatter plot that we have seen before. As noted earlier, we can see from the scatter plot that taller dogs tend to weigh more than shorter dogs.

Another way to say it is that weight tends to increase as height increases.

When we have a positive association between two variables, an increase in one means there tends to be an increase in the other.

A scatterplot. Horizontal, from 6 to 30, by 3’s, labeled dog height, inches. Vertical, from 0 to 112, by 16’s, labeled dog weight, pounds. 24 data points.  Trend upward and to right.

We can quantify this tendency by fitting a line to the data and finding its slope.

For example, the equation of the fitted line is $w = 4.27h -37$, where $h$ is the height of the dog and $w$ is the predicted weight of the dog.

The slope is 4.27, which tells us that for every 1-inch increase in dog height, the weight is predicted to increase by 4.27 pounds.

In our example of the fuel efficiency and weight of a car, the slope of the fitted line shown is -0.01.

This tells us that for every 1-kilogram increase in the weight of the car, the fuel efficiency is predicted to decrease by 0.01 mile per gallon (or, after multiplying both values by 100, every 100-kilogram increase corresponds to a predicted decrease of 1 mpg). 

When we have a negative association between two variables, an increase in one means there tends to be a decrease in the other.

Sometimes a scatter plot shows an association that is not linear:

In this scatter plot, the data initially shows a negative trend then later a positive trend. Because the variables appear to be associated, but not in a linear way, we call this a non-linear association. In later grades, you will study functions that can be models for non-linear associations.

Sometimes in a scatter plot we can see separate groups of points.

A scatterplot with two groups of points.  The first begins near the origin and trends up and to the right toward 8 comma 13. Second group begins near 3 comma 25 and trends up and right toward 9 comma 45.

A scatterplot with two groups of points.  The first begins near the origin and trends up and to the right toward 8 comma 13. Second group begins near 3 comma 25 and trends up and right toward 9 comma 45. Each group is circled. 

We call these groups “clusters.” Clusters often appear when multiple patterns are present within the data. There may be subgroups within the overall data set that affect the variables.

People often collect data in two variables to investigate possible associations between two numerical variables and use the connections that they find to predict more values of the variables.

Data analysis usually follows these steps:

1. Collect data
2. Organize and represent the data, then look for an association
3. Identify any outliers and try to explain why these data points are exceptions to the trend that describes the association
4. Find a function that fits the data well

Although computational systems can help with data analysis by graphing the data, finding a function that might fit the data, and using that function to make predictions, it is important to understand the process and think about what is happening. A computational system may find a function that does not make sense or use a line when the situation suggests that a different model would be more appropriate.