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This module provides an overview of Linear Regression and Correlation: Scatter Plots as a part of Collaborative Statistics collection (col10522) by Barbara Illowsky and Susan Dean.

Before we take up the discussion of linear regression and correlation, we need to examine a way to display the relation between two variables x and y . The most common and easiest way is a scatter plot . The following example illustrates a scatter plot.

From an article in the Wall Street Journal : In Europe and Asia, m-commerce is popular. M-commerce users have special mobilephones that work like electronic wallets as well as provide phone and Internet services. Users can do everything from paying for parking to buying a TV set or soda from amachine to banking to checking sports scores on the Internet. For the years 2000 through 2004, was there a relationship between the year and the number of m-commerce users?Construct a scatter plot. Let x = the year and let y = the number of m-commerce users, in millions.

x (year) y (# of users)
2000 0.5
2002 20.0
2003 33.0
2004 47.0
Table showing the number of m-commerce users (in millions) by year.
A scatter plot with the x-axis representing the year and the y-axis representing the number of m-commerce users in millions.  There are four points plotted, at (2000, 0.5), (2002, 20.0), (2003, 33.0), (2004, 47.0).
Scatter plot showing the number of m-commerce users (in millions) by year.

A scatter plot shows the direction and strength of a relationship between the variables. A clear direction happens when there is either:

  • High values of one variable occurring with high values of the other variable or low values of one variable occurring with low values of the other variable.
  • High values of one variable occurring with low values of the other variable.

You can determine the strength of the relationship by looking at the scatter plot and seeing how close the points are to a line, a power function, an exponential function,or to some other type of function.

When you look at a scatterplot, you want to notice the overall pattern and any deviations from the pattern. The following scatterplot examples illustrate these concepts.

Positive linear pattern (strong)

Scatterplot of 6 points in a straight ascending line from lower left to upper right.

Linear pattern w/ one deviation

Scatterplot of 6 points in a straight ascending line from lower left to upper right with one additional point in the upper left corner.

Negative linear pattern (strong)

Scatterplot of 6 points in a straight descending line from upper left to lower right.

Negative linear pattern (weak)

Scatterplot of 8 points in a wobbly descending line from upper left to lower right.

Exponential growth pattern

Scatterplot of 7 points in a exponential curve from along the x-axis on the left to slowly ascending up the graph in the upper right.

No pattern

Scatterplot of many points scattered everywhere.

In this chapter, we are interested in scatter plots that show a linear pattern. Linear patterns are quite common. The linear relationship is strong if the points are close to a straight line.If we think that the points show a linear relationship, we would like to draw a line on the scatter plot. This line can be calculated through a process called linear regression . However, we only calculate a regression line if one of the variables helps to explain orpredict the other variable. If x is the independent variable and y the dependent variable, then we can use a regression line to predict y for a given value of x .

Before we calculate the regression equation there are some condition we should think about first; quantitative data condition, straight enough condition, and outlier condition. The quantitative data condition is making sure that you are working with two numerical variables. Remember that sometimes we use number to represent categories. The straight enough condition is making sure the data is linear. Look at the scatterplot, do the data look linear? Lastly, the outlier condition is looking to see if you have data points on the scatterplot that do not seem to fit the overall trend in the data. If you identify points that you think are outliers you may want to remove them from the dataset and calculate the regression equation again without them in the dataset.

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Source:  OpenStax, Collaborative statistics using spreadsheets. OpenStax CNX. Jan 05, 2016 Download for free at http://legacy.cnx.org/content/col11521/1.23
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