A **scatterplot **is a plot or graph of two variables that shows how the score for an individual on one variable associates with his or her score on the other variable. If the correlation is *high positive,* the plotted points will be close to a straight line (the **linear regression line**) from the lower left corner of the plot to the upper right. The linear regression line will slope downward from the upper left to the lower right if the correlation is *high negative*. For correlations *near zero*, the regression line will be flat with many points far from the line, and the points form a pattern more like a circle or random blob than a line or oval.

Doing a scatterplot with this program is somewhat cumbersome, as you will see, but it provides a visual picture of the correlation. Each dot or circle on the plot represents a particular individual’s score on the two variables, with one variable being represented on the X axis and the other on the Y axis. The plot also allows you to see if there are bivariate outliers (circles/dots that are far from the regression line, indicating that the way that person’s score on one variable relates to his/her score on the other is different from the way the two variables are related for most of the other participants), and it may show that a better fitting line would be a curve rather than a straight line. In this case the assumption of a linear relationship is violated and a Pearson correlation would not be the best choice.

- What are the scatterplots and linear regression line for (a)
*math achievement*and*grades in h.s.*and for (b)*math achievement*and*mosaic pattern score*?

To develop a scatterplot of *math achievement* and *grades*, follow these commands:

**Graphs → Legacy Dialogs → Scatter/Dot**. This will give you Fig. 8.1.- Click on
**Simple Scatter.**

** Fig. 8.1. Scatterplot.**

- Click on
**Define**, which will bring you to Fig. 8.2. - Now, move
*math achievement*to the**Y Axis**and*grades in h.s.*to the**X Axis**.__Note: the presumed outcome or dependent variable goes on the Y axis__. However, for the correlation itself there is no distinction between the independent and dependent variable.

** Fig. 8.2. Simple scatterplot**

- Next, click on
**Titles**(in Fig. 8.2). Type**Correlation of math achievement with high school grades**(see Fig. 8.3). Note we put the title on two lines. - Click on
**Continue**, then on You will get Output 8.1a, the scatterplot. You will not print this now because we want to add the regression line first in order to get a better sense of the relationship and how much scatter or deviation there is.

** Fig. 8.3. Titles.**

**Output 8.1a: Scatterplot Without Regression Line**

GRAPH

/SCATTERPLOT(BIVAR)=grades WITH mathach

/MISSING=LISTWISE

/TITLE= ‘Correlation of math achievement with’ ‘high school grades’.

__Double click on the scatterplot in Output 8.1a__. The Chart Editor (Fig. 8.4) will appear.- Click on a circle in the scatterplot in the
**Chart Editor;**__all the circles will be highlighted in____yellow.__

- Click on the button circled in Fig. 8.4 to create a
**Fit Line**. The**Properties**window (see Fig. 8.5) will appear as well as a blue fit line in the Chart Editor.

- Be sure that
**Linear**is checked (see Fig. 8.5). - Click on
**Close**in the**Properties**window and click**File ^ Close**to close the**Chart Editor**in order to return to the Output window (Output 8.1b).

** Fig. 8.5. Properties.**

- Now
__add a new scatterplot__to Output 8.1b by doing the same steps that you used for Problem 8.1a__for a new pair of variables__:*math achievement*(**Y-Axis**) with*mosaic*(**X- Axis**). - Don’t forget to click on
**Titles**and change the second line before you run the scatterplot so that the title reads: Correlation of math achievement with (1^{st}line) mosaic pattern score (2^{nd }line). - Then add the linear regression line as you did earlier using Figs. 8.4 and 8.5.
- Now,
__double click once more on the chart you just created__. We want to add a__quadratic regression line__. The Chart Editor (similar to Fig. 8.4) should appear. - Again, click on
__the button circled in Fig. 8.4__to bring up the**Properties** - In the
**Properties**window (Fig. 8.5),__click on__instead of**Quadratic** - Click
**Apply**and then**Close**. You will see that a curved line was__added__to the second scatterplot in Output 8.1b below.

Do your scatterplots look like the ones in Output 8.1b?

**Output 8.1b: Three Scatterplots With Regression Lines**

GRAPH

/SCATTERPLOT(BIVAR)=grades WITH mathach

/MISSING=LISTWISE

/TITLE= ‘Correlation of math achievement with’ ‘high school grades’.

**Interpretation of Output 8.1b**

Both scatterplots shown in Output 8.1b show the best fit of a straight or linear regression line (i.e., it minimizes the squared differences between the points and the line). Note that for the first scatterplot (grades in h.s.), the points fit the line pretty well; *r*^{2} = .25 and thus *r* is .50. The second scatterplot shows that mosaic and math achievement are only weakly correlated; the points do not fit the line very well, *r*2 = .05, and *r* is .21. Note that in the second scatterplot we asked the program to fit a quadratic (one bend) curve as well as a linear line. It seems to fit the points better; *r*2 = .10. If so, the linear assumption would be violated and a Pearson correlation may not be the most appropriate statistic.

Source: Morgan George A, Leech Nancy L., Gloeckner Gene W., Barrett Karen C.

(2012), *IBM SPSS for Introductory Statistics: Use and Interpretation*, Routledge; 5th edition; download Datasets and Materials.

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