Heiman G. Basic Statistics for the Behavioral Sciences

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In the example

Step 6 Determine the degrees of freedom.

A3subs

5 147.33 2 63.33 2 11.33 5 72.67

388 APPENDIX A / Additional Statistical Formulas

is the number of levels of factor A. (In the example, , so is 2.)df

5 3k

is the number of levels of factor A, and is the number of participants. In the

example with three levels of factor A and five subjects, .

Compute and to check the above . The . The

, where is the total number of scores in the experiment. The is also

equal to the sum of all other .

Step 7 Place the sum of squares and the in the summary table. For the example,

Summary Table of One-Way, Within-Subjects ANOVA

Source Sum of Squares df Mean Square F

Subjects 11.33 4

Factor A (dress) 63.33 2 MS

Interaction

(A ⫻ subjects) 72.67 8 MS

A⫻subs

Total 147.33 14

Because there is only one factor of interest here (type of dress), we will find the

only for factor A.

Step 8 Find the mean square for factor A and the interaction.

obt

dfs

tot

Ndf

tot

5 N 2 1

subs

5 k

subs

2 1dfdf

tot

subs

A3subs

5 1221425 8

subs

The degrees of freedom between groups for factor A is

5 k

2 1

The degrees of freedom for the interaction is

A3subs

5 1k

2 121k

subs

2 12

The mean square for factor A is

In our example,

63.33

5 31.67

A.3 The One-Way, Within-Subjects Analysis of Variance 389

In the example,

Step 9 Find .F

obt

A3subs

72.67

5 9.08

In the example,

The finished summary table is

Source Sum of Squares df Mean Square F

Subjects 11.33 4

Factor A (dress) 63.33 2 31.67 3.49

Interaction

(A ⫻ subjects) 72.67 8 9.08

Total 147.33 14

Step 10 Find the critical value of in Table 5 of Appendix C. Use as the degrees

of freedom between groups and as the degrees of freedom within groups. In

the example for , , and , the is 4.46.F

crit

A3subs

5 8df

5 2␣ 5 .05

A3subs

obt

A3subs

31.67

9.08

5 3.49

The mean square for the interaction between factor A and

subjects is

A3subs

The within-subjects F-ratio is

obt

A3subs

ADDITIONAL STATISTICAL FORMULAS

Interpreting the Within-Subjects F

Interpret the above the same way that you would a between-subjects . Because

in the above example is not larger than , it is not significant. Thus, we do not

have evidence that the means from at least two levels of type of dress represent differ-

ent populations of scores. Had been significant, it would indicate that at least two

of the level means differ significantly. Then, for post hoc comparisons, graphing, eta

squared, and confidence intervals, follow the procedures discussed in Chapter 13. How-

ever, in any of those formulas, in place of the term use .

Note: The related-samples t-test in Chapter 12 is more powerful than the independent-

samples -test because the variability in the scores is less. For the same reason, a within-

subjects ANOVA is more powerful than a between-subjects ANOVA for the same data.

Application Questions

(Answers for odd-numbered questions are in Appendix D.)

1. You read in a research report that the repeated-measures factor for a person’s

weight gain led to a decrease in his or her mood. (a) What does this tell you about

the design? (b) What does it tell you about the results?

2. Which of these relationships suggest using a repeated-measures design? (a) Exam-

ining the improvement in language ability as children grow older (b) Measuring

participants’ reaction when the experimenter surprises them by unexpectedly

shouting, under three levels of volume of shouting (c) Comparing the dating strate-

gies of males and females (d) Comparing memory ability under the conditions of

participants’ consuming different amounts of alcoholic beverages.

3. We study the influence of practice on performing a task requiring eye–hand

coordination. We test people with no practice, after 1 hour of practice, and after

2 hours of practice. In the following data, higher scores indicate better

performance. (a) What are and ? (b) Complete the ANOVA summary table.

post hoc comparisons. (e) What is the effect size in this study? (f) What should you

conclude about this relationship?

Amount of Practice

Subjects None 1 Hour 2 Hours

1 43 6

2 35 5

3 14 3

4 34 6

5 15 6

6 26 7

7 24 5

8 13 8

obt

␣ 5 .05

A3subs

obt

crit

obt

390 APPENDIX A / Additional Statistical Formulas

4. You measure 21 students’ degree of positive attitude toward statistics at four

equally spaced intervals during the semester. The mean score for each level is time

1, 62.50; time 2, 64.68; time 3, 69.32; time 4, 72.00. You obtain the following

sums of squares:

Source Sum of Squares df Mean Square F

Subjects 402.79

Factor A 189.30

A ⫻ subjects 688.32

Total 1280.41

(a) What are and (b) Complete the ANOVA summary table (c) With

, what do you conclude about ? (d) Perform the appropriate post

hoc comparisons. (e) What is the effect size in this study? (f) What should you

conclude about this relationship?

obt

␣ 5 .05

A.3 The One-Way, Within-Subjects Analysis of Variance 391

ADDITIONAL STATISTICAL FORMULAS

392

Using SPSS

This appendix describes how to use the computer program called SPSS (or PASW ) to

compute the statistics described in this textbook. These instructions are appropriate for

most recent versions of SPSS, although copies of “screens” were made from Student

Version 17.0 and different versions may have minor differences.

It is best if you perform the following sections in order. For each, type in the exam-

ple data and perform the other steps as they are described. We’ll refer to example

problems from earlier chapters so that you can compare your answer when using a

formula to the answer when using the computer. (Slight differences may arise

because of rounding.)

B.1 ENTERING DATA

Install and start SPSS using the instructions included with the program. The opening

window is shown in Screen B.1.

The window in the foreground gives several options. Usually we will input data, so

using the left mouse button click at Type in data and then click the OK button. The fore-

ground window disappears, leaving the window in the background, containing the grid

of rectangles. This is the “Data Editor” and in it we enter our data. Use the horizontal

scroll bar at the bottom of the Data Editor to see columns to the right.

Across the top of the Data Editor is the typical “Menu Bar,” with buttons for File,

Edit, and so on. Some useful information is available by clicking Help: Click Topics for

an index, Tutorial for instructions and examples, and Statistics Coach for help in select-

ing a procedure.

The first step in any data analysis is to name the variables and enter the scores.

B.1 Entering Data

B.2 Frequency Distributions and Percentile

B.3 Central Tendency, Variability, and z-Scores

B.4 Correlation Coefficients and the Linear Regression Equation

B.5 The One-Sample t-Test and Significance Testing of Correlation

Coefficients

B.6 Two-Sample t-Tests

B.7 The One-Way, Between-Subjects ANOVA

B.8 The Two-Way, Between-Subjects ANOVA

B.9 Chi Square Procedures

B.10 Nonparametric Tests for Ranked Scores

B.11 The One-Way, Within-Subjects ANOVA

Naming Variables

The program’s default will name your variables var00001, var00002, and so on, but

give them a more meaningful name. For example, say that we want to input the follow-

ing scores, which measure creativity and intelligence:

Participant Creativity Intelligence

1 50 110

2 47 116

3 65 125

4 66 127

5 48 100

Begin with a blank Data Editor: To clear old data, on the Menu Bar click File,

point to New, and click Data.

Name the variable(s): At the bottom left of the Data Editor, click on Variable View.

In the left column under “Name” click on the first rectangle and type the variable’s

name. (For example, Creativity.) Press Enter on the keyboard.

The information that appears next to a variable’s name are the SPSS defaults, which

will work for our statistics, assuming that you want scores to have no more than two

decimal places. (Type in a three decimal number, and it will be rounded.) Click any rec-

tangle in a row to change the default value. In the “Label” column, you can add infor-

mation about the variable that you want to remember.

B.1 Entering Data 393

USING SPSS

SCREEN B.1

Opening Windows of SPSS

In the “Name” column, click the second rectangle and enter the second variable’s

name (Intelligence).

Entering Data

Click on Data View at the bottom left of the Data Editor. Input the scores under the cor-

responding variable. When participants’ are measured on both variables, each row in

the Data Editor holds the scores from the same participant. Thus, participant 1 scored

50 on creativity, so in the Creativity column, click the rectangle next to “1” and type

50. Next you have three choices: (1) Press the Tab key on the keyboard, and the rectan-

gle to the right of 50 is highlighted, ready for participant 1’s intelligence score;

(2) press the Enter key and the rectangle below 50 is highlighted, ready for participant 2’s

creativity score; (3) click any rectangle in either column to highlight it and type in the

score that belongs there. Enter the remaining scores.

You can enter data from several variables and then only analyze some of it at one

time. (We can analyze only creativity, only intelligence, or both.)

To correct a mistake: Click on the score or rectangle and retype. Always check that

the Data Editor contains the correct scores. (Here the columns should look like our

original columns of data above.)

Saving Data

To save data: The first time that you save, on the Menu Bar click File, click Save as,

and name the file. For saves after that, click File and Save.

To open a saved file: Open a file to add more data or to analyze it. Below the Menu

Bar, click the file folder icon. Click your file’s name and click Open.

Saving or Printing the Output

Later we’ll see that SPSS shows the results of calculations by opening a new window

called the “Output.” Although you’ll have many options with this window, the most

useful are to select the appropriate icon to save or print the results.

Close SPSS by clicking the X in the upper right-hand corner.

For Practice

Enter the following scores. Your Data Editor should look like that shown in Screen B.2.

Then save and then retrieve the file.

394 APPENDIX B / Using SPSS

SCREEN B.2

Practice Data Editor

Participant Test 1 Test 2

11911

2177

3155

4167

51711

61912

7 19 3.431

B.2 FREQUENCY DISTRIBUTIONS AND PERCENTILE

SPSS creates frequency tables and graphs, and it computes percentiles as discussed in

Chapter 3. (Graphs can also be exported into a report that you are writing.) For exam-

ple, say that we have these 15 scores:

645679634658577

Creating a Frequency Table with Percentiles

Enter the data: Name one variable (for example, Scores) and enter the data as above.

Select a frequency table: We’ll often do the type of move shown in Screen B.3. On

the Menu Bar, click Analyze, move the cursor to Descriptive Statistics, and then click

Frequencies. A “box” appears that is labeled “Frequencies,” as shown in Screen B.4

B.2 Frequency Distributions and Percentile 395

USING SPSS

SCREEN B.3

Creating Frequency

Tables

SCREEN B.4

Frequencies Box

Select the variable(s): Click and the highlighted variable will move to under

“Variable(s).” Or, you may “drag and drop” the highlighted variable into the space

under “Variable(s).”

To remove a variable: Highlight the variable under “Variable(s)” and click the

reversed-arrow button.

The output: Click OK. The frequency table in Screen B.5 appears. In the left-hand

column are the original scores, but arranged in increasing order. In the next column

is each score’s simple frequency. Under “Percent” is the percent each score’s is of

(its times 100). Under “Cumulative Percent” is percentile. (Ignore “Valid

Percent.”)

rel. fN

396 APPENDIX B / Using SPSS

SCREEN B.5

Frequency Table Output

Determining the Score at a Particular Percentile

Select a frequency table: On the Menu Bar, again click Analyze, Descriptive Statis-

tics, and Frequencies. Then move each variable to “Variables(s).”

Select a percentile: Click Statistics. A “Frequencies: Statistics” box appears. Click

to place a check mark at Percentile(s) and in the space next to it, type the percentile that

you seek. Say that we seek the score at the 10th percentile, so type 10. Click Add and

10 appears in the larger white box. Add other percentiles in the same way. Or click

Quartiles to produce the 25th, 50th, and 75th percentile. Or, use Cut points for the

score at the 10th percentile, the 20th, and so on. (To remove a percentile from the white

box, click it and click Remove.) Click Continue.

The output: Back at the “Frequencies” box, click OK. The percentile(s) will be listed

in the “Statistics” output table shown below. (The score at the 10th percentile is 3.6.)

Plotting Bar Graphs and Histograms

Select a frequency table: Repeat the original steps for a frequency table: On the Menu

Bar, click Analyze, Descriptive Statistics, and Frequencies. In the “Frequencies” box,

move your variable to “Variables(s).”

Select Charts: Click Charts. In the “Frequencies: Charts” box, click Bar charts or

Histograms. Click Continue, and click OK. (You may need to scroll down in the output

window to see the graph.)

Plotting Frequency Polygons

On the Menu Bar, click Graphs and click Chart Builder: Click OK on the Chart Builder

box that appears. Under Choose from, click Histogram. Place the cursor over the Fre-

quency Polygon icon and drag it into the Chart preview area. Close the Element Prop-

erties box. Drag your variable from the Variables column to the area under the X axis.

Click OK. The polygon will not show zero frequency when a score did not occur—the

line will connect the frequencies of the scores above and below it. (For older versions

of SPSS, on the Menu Bar click Graphs, then Line then Simple. Click Define and move

the variable to Category Axis. Click of cases and click OK.)

For Practice

1. Create a frequency table for the data in Application Question 21 in Chapter 3.

2. For the above data, create a bar graph and a frequency polygon.

Answers

1. Your output should give the same answers as in Appendix D.

B.3 Central Tendency, Variability, and z-Scores 397

USING SPSS

46.00 48.00 50.00 52.00 54.00

0.0

1.0

2.0

3.0

Frequency

Scores

4.0

5.0

B.3 CENTRAL TENDENCY, VARIABILITY, AND z-SCORES

This section describes how to (1) compute measures of central tendency (as in Chapter 4)

and variability (as in Chapter 5) for a sample of scores and (2) transform a sample of raw

scores into z-scores (as in Chapter 6).

Summarizing a Sample of Scores

SPSS often computes the mean and standard deviation of scores as part of other proce-

dures, including computing them for each condition when analyzing an experiment. The

following section describes how, for one sample of scores, we can also simultaneously