6.2.5: Observing More Patterns in Scatter Plots

Lesson

Let's look for other patterns in data.

Exercise \(\PageIndex{1}\): Notice and Wonder: Nonlinear Scatter Plot

What do you notice? What do you wonder?

Figure \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\): Scatter Plot City

Your teacher will give you a set of cards. Each card shows a scatter plot.

1. Sort the cards into categories and describe each category.
2. Explain the reasoning behind your categories to your partner. Listen to your partner’s reasoning for their categories.
3. Sort the cards into two categories: positive associations and negative associations. Compare your sorting with your partner’s and discuss any disagreements.
4. Sort the cards into two categories: linear associations and non-linear associations. Compare your sorting with your partner’s and discuss any disagreements.

Exercise \(\PageIndex{3}\): Clustering

How are these scatter plots alike? How are they different?

Figure \(\PageIndex{2}\): Four scatterplots. Plot A, points start at 0 comma 40 and trend down and to the right. Plot B has 2 sets of points, 1 starts near the origin and trends up and to the right. The second starts near 3 comma 25 and trends up and to the right. Plot C has 2 sets of points, 1 starts near the origin and trends up and to the right. The second starts near the origin and trends down and to the right. Plot D points starts near 0 comma 40 and trends down and to the right toward 24 negative 20, but there are no points between x = 6 and x = 12.

Summary

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

Figure \(\PageIndex{3}\)

We call such an association 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.

Figure \(\PageIndex{4}\): 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. The second image is the same scatterplot, with each group circled.

We call these groups clusters.

Glossary Entries

Definition: Negative Association

A negative association is a relationship between two quantities where one tends to decrease as the other increases. In a scatter plot, the data points tend to cluster around a line with negative slope.

Different stores across the country sell a book for different prices.

The scatter plot shows that there is a negative association between the the price of the book in dollars and the number of books sold at that price.

Figure \(\PageIndex{5}\)

Definition: Outlier

An outlier is a data value that is far from the other values in the data set.

Here is a scatter plot that shows lengths and widths of 20 different left feet. The foot whose length is 24.5 cm and width is 7.8 cm is an outlier.

Figure \(\PageIndex{6}\)

Definition: Positive Association

A positive association is a relationship between two quantities where one tends to increase as the other increases. In a scatter plot, the data points tend to cluster around a line with positive slope.

The relationship between height and weight for 25 dogs is shown in the scatter plot. There is a positive association between dog height and dog weight.

Figure \(\PageIndex{7}\)

Practice

Exercise \(\PageIndex{4}\)

Literacy rate and population for the 12 countries with more than 100 million people are shown in the scatter plot. Circle any clusters in the data.

Figure \(\PageIndex{8}\)

Exercise \(\PageIndex{5}\)

Here is a scatter plot:

Figure \(\PageIndex{9}\)

Select all the following that describe the association in the scatter plot:

1. Linear association
2. Non-linear association
3. Positive association
4. Negative association
5. No association

Exercise \(\PageIndex{6}\)

For the same data, two different models are graphed. Which model more closely matches the data? Explain your reasoning.

Figure \(\PageIndex{10}\)

Figure \(\PageIndex{11}\)

(From Unit 6.2.3)

Exercise \(\PageIndex{7}\)

Here is a scatter plot of data for some of the tallest mountains on Earth.

The heights in meters and year of first recorded ascent is shown. Mount Everest is the tallest mountain in this set of data.

1. Estimate the height of Mount Everest.
2. Estimate the year of the first recorded ascent of Mount Everest.

Figure \(\PageIndex{12}\)

(From Unit 6.2.1)

Exercise \(\PageIndex{8}\)

A cone has a volume \(V\), radius \(r\), and a height of 12 cm.

1. A cone has the same height and \(\frac{1}{3}\) of the radius of the original cone. Write an expression for its volume.
2. A cone has the same height and 3 times the radius of the original cone. Write an expression for its volume.

(From Unit 5.5.2)