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Scatter Plots and Linear Correlation

Plot points and estimate the line that best represents them

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Use a Scatterplot to Interpret Data
License: CC BY-NC 3.0

Fillmore Factories is expanding their chain of furniture stores. They have compiled information of floor area and weekly profits from various stores that already exist.

Floor Area (m2) 22 25 29 34 40 41 47 52 55 56
Weekly Profits ($) 128 235 217 244 316 308 362 374 439 474

Create a scatterplot to display the relationship between Floor Space and Weekly Profit and draw a trend line. Is there a positive correlation between the two variables?

In this concept, you will learn to use a scatterplot to interpret data.

Scatterplots

A scatterplot is a type of graph where corresponding values from a set of data are placed as points on a coordinate plane. A relationship between the points is sometimes shown to be positive, negative, strong, or weak. Sometimes a scatterplot shows that there is no relationship at all. Aside from finding relationships, scatterplots are useful in predicting values based on the relationship that was revealed.

Take a look at this scatterplot.

License: CC BY-NC 3.0

You can see that there is a relationship between the independent and dependent values of the chart. The relationship is moving up to the right and therefore is a positive correlation.

Let’s look at how you can examine these relationships.

What happens to people’s heating bills as the temperature outside goes up?

You might imagine that as the temperature outside goes up, people’s heating bills go down because they use their heaters less. As one variable goes up, the other goes down. This is a negative relationship or a negative correlation.

What happens to the gasoline consumption in a vehicle as the miles traveled goes up?

You might imagine that as the miles traveled in a car go up, the amount of gasoline consumed also goes up. So, as one variable goes up, the other goes up, too. This is a positive relationship or a positive correlation.

What happens to the number of accidents as the number of blue cars increases on the road?

You can imagine that there would be no relationship. While one variable goes up, the other may go up, down, or stay the same; the number of accidents is independent of the number of blue cars. This oftentimes occurs, too. This is an example of no relationship or no correlation.

These three trends, positive, negative, and no relationship are evident on scatterplots. This is what they look like:

License: CC BY-NC 3.0

This is a positive relationship. As the x-values increase, the y-values increase. Some points may not follow an exact pattern but the overall trend, the general tendency or movement, is clearly from the lower left to the upper right of the plot.

License: CC BY-NC 3.0

This is a negative relationship. In this case, as the x-values increase, the y-values decrease. You may argue that the slope is not as steep which is true. However, the general tendency is evident. This graph moves from the upper left to the lower right.

License: CC BY-NC 3.0

At times, like the one shown above, there is no relationship between variables. The scatterplots of these situations will show no trend. In other words, there seems to be no definite pattern with the points; you cannot see any particular direction that they take.

Scatterplots are as useful for finding a relationship between variables as they are for making predictions. Here, you will make a trend line, or a line that best describes the data on a scatterplot, in order to estimate unknown outputs for given inputs.

A trend line is a straight line that best represents the points on a scatterplot. The trend line may go through some points but need not go through them all. The trend line is used to show the pattern of the data. This trend line may show a positive trend or a negative trend. However, if there is no relationship, then no trend line can be adequately drawn.

Your trend line is your best approximation of the pattern of the data.

License: CC BY-NC 3.0

The line on this graph is the trend line; it is the line that best describes the data. About half of the points should be on either side of the line. You may notice that outliers are practically ignored when a trend line is drawn. This trend line goes from the lower left to the upper right and shows a positive relationship.

License: CC BY-NC 3.0

Notice that this trend goes down and indicates a negative correlation or relationship. You can also see that it goes off of the chart. Therefore, you could use a chart like this one to predict the trend. It is likely that the trend will continue to go down.

Examples

Example 1

Earlier, you were given a problem about expanding furniture store. The furniture company has collected the following data:

Floor Area (m2) 22 25 29 34 40 41 47 52 55 56
Weekly Profits ($) 128 235 217 244 316 308 362 374 439 474

You need to create a scatterplot to display the relationship between Floor Space and Weekly Profit. Draw a trend line. Finally you need to see if there is a positive correlation between the two variables?

First, draw the scatterplot.

License: CC BY-NC 3.0

Next, draw the trend line.

License: CC BY-NC 3.0

Then, look at the scatterplot to see if there is a pattern.

There is a definite pattern in the positive direction. This data shows a positive correlation.

Example 2

What kind of relationship is shown by the data?

License: CC BY-NC 3.0

As one variable increases, the other variable increases as well. This scatterplot shows a positive correlation in the data.

Example 3

What kind of correlation will describe this scatterplot?

License: CC BY-NC 3.0

The answer is that this scatterplot shows no correlation.

Example 4

What kind of correlation will describe this scatterplot?

License: CC BY-NC 3.0

The answer is that this scatterplot shows a positive correlation.

Example 5

If the distance of a car increases as its speed increases, what kind of correlation will the data have?

The answer is that this would show a positive correlation.

Review

What type of relationship is shown in the following scatterplots?

1. If the data decreases as one variable increases, what type of relationship is shown?

2. 

License: CC BY-NC 3.0

3. Use the following table to make a scatter plot.

x 3 6 8 14 18 23 29 32 37
y 55 50 46 40 37 18 26 20 18

4. Draw a trend line.

5. Identify the type of relationship.

A zoologist studied the relationship between the kilometers from a lake and number of felines per 100 square kilometers. She found the following data:

Distance from Lake 3 1 4 3 4.5 5 5 2 2.5 3.5 8 6 5
# of Felines 5 10 2 8 6 5 8 8 6 6 0 2 4

6. Make a scatterplot that illustrates this data.

7. Draw a trend line.

8. What is the correlation?

9. Estimate the number of felines 1.5 kilometers from a lake.

Define the following terms.

10. Input value

11. Output value

12. Positive correlation

13. Negative correlation

14. No correlation

15. Data set

Review (Answers)

 To see the Review answers, open this PDF file and look for section 10.8. 

Resources

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Vocabulary

bivariate

Bivariate data has two variables

correlation

Correlation is a statistical method used to determine if there is a connection or a relationship between two sets of data.

curvilinear relationships

Non-linear relationships are called curvilinear relationships.

direct relationship

If the line on a line graph rises to the right, it indicates a direct relationship.

homogeneity

When a group is homogeneous, or possesses similar characteristics, the range of scores on either or both of the variables is restricted.

indirect relationship

If the line on a line graph falls to the right, it indicates an indirect relationship.

linear relationship

A linear relationship appears as a straight line either rising or falling as the independent variable values increase.

negative correlation

A negative correlation appears as a recognizable line with a negative slope .

non-linear relationship

A non-linear relationship may take the form of any number of curved lines but is not a straight line.

positive correlation

A positive correlation appears as a recognizable line with a positive slope               .

scatter plot

A scatter plot is a plot of the dependent variable versus the independent variable and is used to investigate whether or not there is a relationship or connection between 2 sets of data.

Slope

Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the y over the change in the x.” The symbol for slope is m

strong correlation

Two variables with a strong correlation will appear as a number of points occurring in a clear and recognizable linear pattern.

trends

Trends in data sets or samples are indicators found by reviewing the data from a general or overall standpoint

weak correlation

Two variables with a weak correlation will appear as a much more scattered field of points, with only a little indication of points falling into a line of any sort.

Image Attributions

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