Spss Tutorial Guide Complete
Spss Tutorial Guide Complete
Spss Tutorial Guide Complete
Our written step-by-step tutorials are organized into sections. Users can work through the tutorials in order or skip through to
topics of interest.
Section 1: Intro to the SPSS Environment is intended for new users of SPSS. In this section, you'll learn how to:
Navigate the SPSS interface using the drop-down menus or syntax.
Create a new dataset or import data from a file.
Section 2: Working with Data covers data manipulation and cleaning of all kinds. In this section, you'll learn how to:
Create, modify, or compute new variables.
Manipulate a dataset by splitting, merging, or transposing techniques.
Section 3: Exploring the Data is intended for users who have mastered the skills in Section 1 and are ready to begin data
analysis. In this section, you'll learn how to:
Generate descriptive statistics for numeric variables.
Create frequency tables and cross-tabulations of categorical variables.
Graph the distributions or relationships of variables.
Interpret these measures.
Section 4: Analyzing the Data is intended for users who are ready to begin data analysis that incorporates hypothesis testing.
In this section, you'll learn how to conduct and interpret various analyses, including:
Associations (Chi-Square, Pearson's Correlation)
Comparing means (One Sample t Test, Independent Samples t Test, Paired Samples t Test, One-Way ANOVA)
Predictive models (Multiple Regression, Logistic Regression, Ordinal Regression)
EXPLORING DATA
Introduction
Before doing any kind of statistical testing or model building, you should always examine your data using summary statistics
and graphs. This process is called exploratory data analysis, and it's a crucial part of every research project. Exploratory data
analysis is about "getting to know" your data: which values are typical, which values are unusual; where is it centered, how
spread out is it; what are its extremes. More importantly, it's an opportunity to identify and correct any problems in your data
that would affect the conclusions you draw from your analysis.
How do we "get to know" our data? The answer is different depending on whether our variables are numeric or categorical. In
this section, we'll demonstrate which statistics and SPSS procedures to use for both types of data.
Part 1: Descriptive Statistics for Continuous Variables
When summarizing a quantitative (continuous/interval/ratio) variable, we are typically interested in things like:
How many observations were there? How many cases had missing values? (N valid; N missing)
Where is the "center" of the data? (Mean, median)
Where are the "benchmarks" of the data? (Quartiles, percentiles)
How spread out is the data? (Standard deviation/variance)
What are the extremes of the data? (Minimum, maximum; Outliers)
What is the "shape" of the distribution? Is it symmetric or asymmetric? Are the values mostly clustered about the
mean, or are there many values in the "tails" of the distribution? (Skewness, kurtosis)
In Part 1, we discuss how to explore quantitative (continuous/interval/ratio scale) data using the Descriptives, Compare
Means, Explore, and Frequencies procedures. Each of these procedures offers different strengths for summarizing continuous
variables. The Descriptives and Frequencies commands provide summary statistics for an entire sample, while the Explore and
Compare Means commands can produce descriptive statistics for subsets of the sample.
Descriptives
Descriptives (Analyze > Descriptive Statistics > Descriptives) is best to obtain quick summaries of numeric variables, or
to compare several numeric variables side-by-side.
Compare Means
Compare Means (Analyze > Descriptive Statistics > Descriptives) is best used when you want to summarize several
numeric variables across the categories of a nominal or ordinal variable. It is especially useful for summarizing numeric
variables simultaneously across multiple factors.
Explore
Explore (Analyze > Descriptive Statistics > Explore) is best used to deeply investigate a single numeric variable, with or
without a categorical grouping variable. It can produce a large number of descriptive statistics, as well as confidence
intervals, normality tests, and plots.
Frequencies Part I (Continuous Variables)
Frequencies (Analyze > Descriptive Statistics > Frequencies) is typically used to analyze categorical variables, but can
also be used to obtain percentile statistics that aren't otherwise included in the Descriptives, Compare Means, or
Explore procedures.
Part 2: Descriptive Statistics for Categorical Variables
When summarizing qualitative (nominal or ordinal) variables, we are typically interested in things like:
How many cases were in each category? (Counts)
What proportion of the cases were in each category? (Percentage, valid percent, cumulative percent)
What was the most frequently occurring category (i.e., the category with the most observations)? (Mode)
In Part 2, we describe how to obtain descriptive statistics for categorical variables using
the Frequencies and Crosstabs procedures.
Frequencies Part II (Categorical Variables)
Frequencies (Analyze > Descriptive Statistics > Frequencies) is primarily used to create frequency tables, bar charts,
and pie charts for a single categorical variable.
Crosstabs
The Crosstabs procedure (Analyze > Descriptive Statistics > Crosstabs) is used to create contingency tables, which
describe the interaction between two categorical variables. This tutorial covers the descriptive statistics aspects of the
Crosstabs procedure, including and row, column, and total percents.
Multiple Response Sets / Working with "Check All That Apply" Survey Data
Check-all-that-apply questions on surveys are recorded as a set of binary indicator variables for each checkbox option.
Frequency tables and crosstabs alone don't capture the dependent nature of this data -- and that's where Multiple
Response Sets come in.
ANALYZING DATA
Part 1: Inferential Statistics for Association
In Part I, we cover common inferential statistics for testing the relationship or association between variables.
Pearson Correlation
Pearson correlation (Analyze > Correlate > Bivariate) is used to assess the strength of a linear relationship between
two continuous numeric variables.
Chi-square Test of Independence
The Chi-Square Test of Independence is used to test if two categorical variables are independent of each other.
Part 2: Inferential Statistics for Comparing Means
In Part 2, we cover common inferential statistics for testing and comparing means.
One Sample t Test
One sample t tests (Analyze > Compare Means > One Sample T Test) are used to test if the mean of a continuous
numeric variable is equal to a hypothesized value of the population mean.
Paired-Samples T Test
Paired t tests (Analyze > Compare Means > Paired-Samples T Test) are used to test if the means of two paired
measurements, such as pretest/posttest scores, are significantly different.
Independent Samples T Test
Independent samples t tests (Analyze > Compare Means > Independent-Samples T Test) are used to test if the means
of two independent groups are significantly different.
One-Way ANOVA
One-Way ANOVA (Analyze > Compare Means > One-Way ANOVA) is used to test if the means of two or more groups
are significantly different.
PEARSON CORRELATION
The bivariate Pearson Correlation produces a sample correlation coefficient, r, which measures the strength and
direction of linear relationships between pairs of continuous variable. By extension, the Pearson Correlation evaluates whether
there is statistical evidence for a linear relationship among the same pairs of variables in the population correlation coefficient,
p a(“rho”). The Pearson Correlation is a parametric measure.
This measure is also known as:
1. Pearson’s Correlation
2. Pearson Product-Moment Correlation (PPMC)
COMMON USES
The bivariate Pearson Correlation is commonly used to measure the following:
1. Whether a statistically significant linear relationship exist between two continuous variables.
2. The strength of a linear relationship (i.e., how close the relationship is to being a perfectly straight line)
3. The direction of a linear relationship (increasing or decreasing)
Note: The bivariate Pearson Correlation cannot address non-linear relationship or relationship among categorical variables. If
you wish to understand relationship that involve categorical variables and/or non-linear relationships, you will need to choose
another measure of association.
Note: The bivariate Pearson Correlation only reveals associations among continuous variables. The bivariate Pearson
Correlation does not provide any inferences about causation, no matter how large the correlation coefficient is.
DATA REQUIREMENTS
To use Pearson Correlation, your data must meet the following requirements:
1. Two or more continuous variables (i.e., interval or ration level)
2. Cases must have non-missing values on both variables.
3. Linear relationship between variables.
4. Independent cases (i.e., independence of observations)
There is no relationship between the values of variables between cases.
This means that:
a. The values for all variables across cases are related.
b. For any case, the value for any variable cannot influence the value of any variable for other cases.
c. No case can influence another case on any variable.
The bivariate Pearson Correlation coefficient and corresponding significance test are not robust when
independence is violated.
5. Bivariate normality
Each pair of variables is bivariately normally distributed.
Each pair of variables is bivariately normally distributed at all levels of the other variable(s).
This assumption ensures that the variables are linearly related; violations of this assumption may indicate
that non-linear relationships among variables exist. Linearity can be assessed visually using a scatter plot
of the data.
6. Random sample of data from the population.
7. No outliers.
HYPOTHESES
The null hypothesis (Ho) and the alternative hypothesis (H1) of the significance test for correlation can be expressed in
the following ways, depending on whether a one-tailed or two-tailed test is requested:
Test Statistics
The sample correlation coefficient between two variables x and y is denoted r or rxy, and can be computed as :
cov ( x , y )
r xy =
√ var ( x ) √ var ( y )
Where cov(x,y) is the sample covariance of x and y, var (x) is the sample variance of x, and var (y) is the sample variance of
y.
Correlation can take on any value in the range [1,-1]. The sign of the correlation coefficient indicates the direction of the
relationship, while the magnitude of the correlation (how close it is to -1 or +1) indicates the strength of the relationship.
-1: perfectly negative linear relationship
0: no relationship
+1: perfectly positive linear relationship
Note: The direction and strength of a correlation are two distinct properties. The scatterplots below show correlations that are
r = +0.90, r = 0.00, and r = -0.90, respectively. The strength of the non-zero correlations are the same; 0.90. But the
direction of the correlations is different: a negative correlation corresponds to a decreasing relationship, while and a positive
correlation corresponds to an increasing relationship.
Figure 1
Note that the r = 0.00 correlation has no discernable increasing or decreasing linear pattern in this particular graph. However,
keep in mind that Pearson Correlation is only capable of detecting linear associations, so it possible to have a pair of variables
with a strong nonlinear relationship and a small Pearson Correlation coefficient. It is good practice to create scatterplots of
your variables to corroborate your correlation coefficients.
DATA SET UP
Your dataset should include two or more continuous numeric variables, each defined as scale, which will be used in the
analysis.
Each row in the dataset should represent one unique subject, person, or unit. All of the measurements taken on that
person or unit should appear on multiple rows—for example, if you have measurements from different time points on separate
rows—you should reshape your data to “wide” format before you compute the correlations.
Figure 2
The Bivariate Correlations window opens, where you will specify the variables to be used in the analysis. All of the variables in
your dataset appear in the list on the left side. To select variables for the analysis, select the variables in the list on the left and
click the blue arrow button to move them to the right, in the Variables field.
Figure 3
A Variables: The variables to be used in the bivariate Pearson Correlation. You must select at least two continuous variables,
but may select more than two. The test will produce correlation coefficients for each pair of variables in the list.
B Correlation Coefficients: There are multiple types of correlation coefficients. By default, Pearson is selected. Selecting
Pearson will produce the test statistics for a bivariate Pearson Correlation.
C Test of Significance: Click Two-tailed or One-tailed, depending on your desired significance test. SPSS uses a two tailed
test by default.
D Flag significant correlations: Checking this option will include asterisks (**) next to statistically significant correlations in
the output. By default, SPSS marks statistically significance at the alpha = 0.05 and alpha = 0.01 levels, but not at the alpha =
0.001 level (which is treated as alpha = 0.01)
E Options: Clicking Options will open a window where you can specify which statistics to include (i.e., Means and
standard deviations, Cross-product deviations and covariances) and how to address Missing Values (i.e., Exclude
cases pairwise or exclude cases listwise). Note that the pairwise/listwise setting does not affect your computations if you
are only entering two variable, but can make a very large difference if you are entering three or more variables into the
correlation procedure.
Figure 4
PROBLEM STATEMENT
Perhaps you would like to test whether there is a statistically significant linear relationship between two continuous
variables, weight and height (and by extension, infer whether the association is significant in the population). You can use a
bivariate Pearson Correlation to test whether there is a statistically significant linear relationship between height and weight,
and to determine the strength and direction of the association.
Figure 5
To add a linear fit like one depicted, double click on the plot in the Output Viewer to open the Chart Editor. Click
Elements>Fit Line at Total. In the Properties window, make sure the Fit Method is set to Linear, then click Apply. (Notice
that adding the linear regression trend line will also add the R-squared value in the margin of the plot. If we take the square
root of this number, it should match the value of the Pearson correlation we obtain.
From the scatterplot, we can see that as the Height increases, Weight also tends to increase. There does appear to be
some linear relationship.
OUTPUT
Tables
The results will display the correlations in a table, labeled Correlations.
Figure 6
A Correlation of Height with itself (r =1), and the number of nonmissing observations for height (n=408).
B Correlation of height and weight (r =0.513), based on n=354 observations with pairwise nonmissinfg values.
C Correlation of height and weight (r =0.513), based on n=354 observations with pairwise nonmissinfg values.
D Correlation of Weight with itself (r =1), and the number of nonmissing observations for weight (n=376).
The important cells we want to look at either B or C. (Cells B and C are identical, because they include information
about the same pair of variables.) Cells B and C contain the correlation coefficient for the correlation between height and
weight, its p-value, and the number of complete pairwise observations that the calculation was based on.
The correlations in the main diagonal (cells A and D) are equal to 1. This is because a variable is always perfectly
correlated with itself. Notice, however, that the sample sizes are different in cell A (n=408) versus cell D (n=376). This is
because of missing data—there are more missing observations for variable weight than there are for variable height.
If you have opted to flag significant correlations, SPSS will mark a 0.05 significance level with one asterisks (*) and a
0.01 significance level with two asterisks (**). In cell B (repeated in cell C), we can see that the Pearson Correlation coefficient
for height and weight is 0.513, which is significant (p˂0.001 for a two-tailed test), based on 354 complete observations (i.e.,
cases with nonmissing values for both the height and weight).
Cases represent subjects, and each subject appears once in the dataset. That is, each row represents an observation
from a unique subject.
The dataset contains at least two nominal categorical variables (string or numeric). The categorical variables used in
the test must have two or more categories.
IF YOU HAVE FREQUENCIES (EACH ROW IS A COMBINATION OF FACTORS):
An example of using the chi-square test for this type of data can be found in the Weighting Cases tutorial.
B Column(s): One or more variables to use in the columns of the crosstab(s). You must enter at least one Column variable.
Also note that if you specify one row variable and two or more column variables, SPSS will print crosstabs for each pairing of
the row variable with the column variables. The same is true if you have one column variable and two or more row variables,
or if you have multiple row and column variables. A chi-square test will be produced for each table. Additionally, if you include
a layer variable, chi-square tests will be run for each pair of row and column variables within each level of the layer variable.
C Layer: An optional "stratification" variable. If you have turned on the chi-square test results and have specified a layer
variable, SPSS will subset the data with respect to the categories of the layer variable, then run chi-square tests between the
row and column variables. (This is not equivalent to testing for a three-way association, or testing for an association between
the row and column variable after controlling for the layer variable.)
D Statistics: Opens the Crosstabs: Statistics window, which contains fifteen different inferential statistics for comparing
categorical variables. To run the Chi-Square Test of Independence, make sure that the Chi-square box is checked off.
E Cells: Opens the Crosstabs: Cell Display window, which controls which output is displayed in each cell of the crosstab.
(Note: in a crosstab, the cells are the inner sections of the table. They show the number of observations for a given
combination of the row and column categories.) There are three options in this window that are useful (but optional) when
performing a Chi-Square Test of Independence:
1 Observed: The actual number of observations for a given cell. This option is enabled by default.
2 Expected: The expected number of observations for that cell (see the test statistic formula).
F Format: Opens the Crosstabs: Table Format window, which specifies how the rows of the table are sorted.
CROSSTABS
/TABLES=Smoking BY Gender
/FORMAT=AVALUE TABLES
/STATISTICS=CHISQ
/CELLS=COUNT
/COUNT ROUND CELL
/BARCHART.
OUTPUT
Tables
The first table is the Case Processing summary, which tells us the number of valid cases used for analysis. Only cases with
non-missing values for both smoking behavior and gender can be used in the test.
The next tables are the cross-tabulation and chi-square test results.
The key result in the Chi-Square Tests table is the Pearson Chi-Square.
The value of the test statistic is 3.171.
The footnote for this statistic pertains to the expected cell count assumption (i.e., expected cell counts are all greater
than 5): no cells had an expected count less than 5, so this assumption was met.
Because the test statistic is based on a 3x2 cross-tabulation table, the degrees of freedom (df) for the test statistic is
df=(R−1)∗(C−1)=(3−1)∗(2−1)=2∗1=2df=(R−1)∗(C−1)=(3−1)∗(2−1)=2∗1=2
.
The corresponding p-value of the test statistic is p = 0.205.
DECISION AND CONCLUSIONS
Since the p-value is greater than our chosen significance level ( α = 0.05), we do not reject the null hypothesis. Rather, we
conclude that there is not enough evidence to suggest an association between gender and smoking.
Based on the results, we can state the following:
No association was found between gender and smoking behavior (Χ2(2)> = 3.171, p = 0.205).
Example: Chi-square Test for 2x2 Table
PROBLEM STATEMENT
Let's continue the row and column percentage example from the Crosstabs tutorial, which described the relationship between
the variables RankUpperUnder (upperclassman/underclassman) and LivesOnCampus (lives on campus/lives off-campus).
Recall that the column percentages of the crosstab appeared to indicate that upperclassmen were less likely than
underclassmen to live on campus:
The proportion of underclassmen who live off campus is 34.8%, or 79/227.
The proportion of underclassmen who live on campus is 65.2%, or 148/227.
The proportion of upperclassmen who live off campus is 94.4%, or 152/161.
The proportion of upperclassmen who live on campus is 5.6%, or 9/161.
Suppose that we want to test the association between class rank and living on campus using a Chi-Square Test of
Independence (using α = 0.05).
BEFORE THE TEST
The clustered bar chart from the Crosstabs procedure can act as a complement to the column percentages above. Let's look at
the chart produced by the Crosstabs procedure for this example:
The height of each bar represents the total number of observations in that particular combination of categories. The "clusters"
are formed by the row variable (in this case, class rank). This type of chart emphasizes the differences within the
underclassmen and upperclassmen groups. Here, the differences in number of students living on campus versus living off-
campus is much starker within the class rank groups.
RUNNING THE TEST
1. Open the Crosstabs dialog (Analyze > Descriptive Statistics > Crosstabs).
2. Select RankUpperUnder as the row variable, and LiveOnCampus as the column variable.
3. Click Statistics. Check Chi-square, then click Continue.
4. (Optional) Click Cells. Under Counts, check the boxes for Observed and Expected, and under Residuals,
click Unstandardized. Then click Continue.
5. (Optional) Check the box for Display clustered bar charts.
6. Click OK.
OUTPUT
Syntax
CROSSTABS
/TABLES=RankUpperUnder BY LiveOnCampus
/FORMAT=AVALUE TABLES
/STATISTICS=CHISQ
/CELLS=COUNT EXPECTED RESID
/COUNT ROUND CELL
/BARCHART.
Tables
The first table is the Case Processing summary, which tells us the number of valid cases used for analysis. Only cases with
nonmissing values for both class rank and living on campus can be used in the test.
The next table is the cross-tabulation. If you elected to check off the boxes for Observed Count, Expected Count, and
Unstandardized Residuals, you should see the following table:
With the Expected Count values shown, we can confirm that all cells have an expected value greater than 5.
Computation of the expected cell counts and residuals (observed minus expected) for the cross-tabulation of class rank by living on
campus.
Off-Campus On-Campus
COMMON USES
The One Sample t -Test is common used to test the following:
1. Statistical Difference between a mean and a known or hypothesized value of the mean in the population.
2. Statistical Difference between a change score and zero.
This approach involved creating a change score from two variables, and then comparing the mean
change score to zero, which will indicate whether any change occurred between the two time points for the
original measures. If the mean change score is not the significantly different from zero, no significant change
occurred.
Note: The One Sample t -Test can only compare a simple mean to a specified constant. In cannot compare sample
means between two or more groups. If you wish to compare the means of multiple groups to each other, you will likely
want to run an Independent Samples t -Test (to compare the means of two groups) or a One-Way ANOVA (to compare
the means of two or more groups).
DATA REQUIREMENTS
Your data must meet the following requirements:
1. Test variable that is continuous (i.e. interval or ratio)
2. Scores on the test variable are independent (i.e. independence of observations)
There is no relationship between scores on the teat variable
Violation of this assumption will yield an inaccurate p value
3. Random sample of data from the population.
4. Normal Distribution (approximately) of the sample and population on the test variable.
Non-normal population distributions, especially those that are thick-tailed or heavily skewed,
considerably reduce the power of the test.
Among moderate or large samples, a violation of normality may still yield accurate p values.
5. Homogeneity of variances (i.e. variances approximately equal in both the sample and population)
6. No outliers
HYPOTHESES
The Null hypothesis (Ho) and (two-tailed) Alternative Hypothesis (H1) of the one sample t -Test can be expressed as:
Ho: µ1 = µ2 ( The population mean is equal to the proposed population mean)
H1: µ1 ≠ µ2 ( The population mean is not equal to the proposed population mean)
Where µ is the true population mean and µo is the proposed value of the population mean.
TEST STATISTIC
The test statistic for a One Sample t -Test is denoted by t, which is calculated using the following formula:
x−µ o s
t= where sx=
sx √n
Where µo= the test value – the proposed constant for the population mean
x = sample mean
n = sample size (i.e. number of observations)
s = sample standard deviation
s
s x = estimated standard error of the mean =
√n
The calculated t value is then compared to the critical t-values from the t distributions table with the degrees of
freedom df= n-1 and chosen confidence level. If the calculated t-value ¿critical t-value, then we reject the null hypothesis.
DATA SET-UP
Your data should include one continuous, numeric variable (represented in a column) that will be used in the analysis.
The variable’s measurement level should be defined as scales in the variable view window.
RUN A ONE SAMPLE t-Test
To run a One sample t-Test in SPSS, click ANALYZE¿COMPARE MEANS¿ONE SAMPLE t-Test
One Sample t-Test window opens where you will specify the variables to be used in the analysis. All the variables in your data
set appear in the list on the left side. Move variables to the Test variable(s) area by selecting them in the list and clicking the
arrow button.
Figure 1
A Test variable(s): The variable whose mean will be compared to the hypothesized population mean (i.e. Test value). You may
run multiple One sample t-Test simultaneously by selecting more than one Test variable. Each variable will be compared to the
same Test value.
B Test Value: The hypothesized population mean against which your test variables will be compared.
C Options: clicking Options will open a window where you can specify the Confidence Interval percentage and how the
analysis will address missing values (i.e. Exclude cases analysis by analysis or Exclude cases list wise). Click Continue when
you are finished making specifications.
Figure 2
Click OK to run the One Sample t-Test
EXAMPLE
PROBLEM STATEMENT
According to CDC, the mean height of US adults ages 20 and older is about 66.5 inches. (69.3 inches for males, 63.8
inches for females).
In our sample data, we have a sample of 435 college students from a single college. Let’s test if the mean height of
students at this college is significantly different than 66.5 inches using a One Sample t-Test.
The Null Hypothesis and Alternative Hypothesis of this test will be:
Ho: µheight = 66.5 inches (The mean height is equal to 66.5”)
H1: µheight ≠ 66.5 inches (The mean height is not equal to 66.5”)
Figure 3
To add vertical reference lines at the mean (or another location) double click on the plot to open the chart editor, then
click Options¿ X-Axis Reference Line. In the properties window, you can enter a specific location on the x-axis for the
vertical line, or you can choose to have the reference line at the mean, median of the sample data (using the sample data).
Click Apply to make sure your new line is added to the chart. Here, we have added two reference lines: One at the sample
mean (solid black) and the other at the 66.5 (dashed red)
Figure 4
TABLES
Two section (Boxes) appear in the output: One Sample Statistics and One Sample t-Test. The first section, One Sample
Statistics provides basic information about the selected variable, Height including their valid (non-missing) sample size ( n),
mean, standard deviation, and standard error. In this example, the mean height of the sample is 68.03 inches which is based
on 408 n0n-missing observations.
Figure 5
The second section: One Sample Test, displays the results most relevant to the One Sample t-Test.
Figure 6
A Test Value: The number we entered as the test value in the One-Sample t-Test window.
B t Statistic: The test statistic of the One sample t-Test, denoted by t. In this example, t= 5.810. Note that t is calculated by
dividing the mean difference (E) by the standard error mean (from the One Sample Statistics box)
C df: The degrees of freedom for the test. For a One Sample t-Test, df = n-1, so here, df = 408-1 =407
D Sig. (2 –tailed): The two-tailed p-value corresponding to the test statistic.
E Mean Difference: The difference between the “observed” sample mean (from the One Sample Statistics box) and the
“expected” mean ( the specified test value (A)). The sign of the mean difference corresponds to the sign of the t value (B). The
positive t value in this example indicates that the mean height of the sample is greater than the hypothesized value (66.5).
F Confidence Interval for the difference: The confidence interval for the difference between the specified test value and
the sample mean.
COMMON USES
The Paired Samples t-Test is commonly used to test the following:
1. Statistical difference between two time points.
2. Statistical difference between two conditions.
3. Statistical difference between two measurements.
4. Statistical difference between a matched pair.
Note: The Paired Samples t-Test can only compare the means for two (and only two) related (paired) units on a
continuous outcome that is normally distributed.
Paired Samples t-Test is not appropriate for analyses involving the following:
1. Unpaired data
2. Comparison between more than two units/groups
3. A continuous outcome that is not normally distributed
4. An ordinal/ ranked outcome
To compare unpaired means between two independent groups on a continuous outcome that is normally distributed,
choose Independent Sample t-Test.
To compare unpaired means between more than two groups on a continuous outcome that is normally distributed,
choose ANOVA.
To compare paired means for continuous data that are not normally distributed, choose the Non-parametric Wilcoxon
Signed-Ranks Test.
To compare paired means for ranked data, choose the Non-parametric Wilcoxon Signed-Ranks Test.
DATA REQUIREMENTS
Your data must meet the following requirements:
1. Dependent variable that is continuous. (i.e. interval or ratio level)
Note: The paired measurements must be recorded in two separate variables.
2. Related samples/ groups. (i.e. dependent observations)
The subject in each sample, or group, are the same. This means that the subjects in the first group are also in
the second group.
3. Random sample of data from the population.
4. Normal distribution (approximately) of the difference between the paired values.
5. No outliers in the difference between the two related groups.
Note: When testing assumptions related to normality and outliers, you must use variables that represents the difference
between the paired values. – not the original variables themselves.
Note: When one or more of the assumptions for the Paired Samples t-Test are not met, you may want to run the non-
parametric Wilcoxon Signed-Ranks Test instead.
HYPOTHESES
The hypotheses can be expressed in two different ways that express the idea and are mathematically equivalent.
Ho: µ1=µ2 (The paired population means are equal)
H1: µ1 ≠ µ 2 (The paired population means are not equal)
or
Ho: µ1−µ2=0 (The difference between the paired population means is equal to 0)
H1: µ1−µ2 ≠ 0 (The difference between the paired population means is not 0)
Where
µ1 is the population mean of variable 1 and
µ2 is the population mean of variable 2
TEST STATISTIC
The test statistic for the Paired Samples t-Test, denoted by t, follows the same formula as the One Sample t-Test.
x diff −0 s diff
t= sx=
sx √n
Where
x diff =¿ sample mean of the differences
n = sample size (i.e. number of observations)
sdiff =¿ sample standard deviation of the differences
s
s x =¿ estimated standard of error of the mean ( diff )
√n
The calculated t value is then compared to the critical t value with df = n-1 from the t distribution table for a chosen
confidence level. If calculated t value is greater than the critical t value, then we reject the null hypothesis. (and conclude that
the means are significantly different)
DATA SET-UP
Your data should include two continuous numeric variables (represented in columns) that will be used in the analysis.
The two variables should represent the paired variables for each subject (row). If your data are arranged differently (e.g. cases
represent repeated units/subjects), simply restructure the data to reflect this format.
Figure 1
The Paired Samples t-Test window opens where you will specify the variables to be used in the analysis. All the
variables in your data set appear in the list on the left side. Move variables to the right by selecting them in the list and clicking
the blue arrow buttons. You will specify the paired variables in the Paired Variables area.
Figure 2
A Pair: The “pair” column represent the number of Paired Samples t-Test to run. You may choose to run Multiple Paired
Samples t-Test simultaneously by selecting multiple sets of matched variables. Each new pair will appear on a new line.
B Variable1: The first variable, representing the first group of matched values. Move the variable that represents the first
group to the right where it will be listed beneath the “Variable 1” column.
C Variable 2: The second variable, representing the second group of matched values. Move the variable that represents the
second group to the right where it will be listed beneath the “Variable 2” column.
D Options: Clicking Options will open a window where you can specify the Confidence Interval Percentage and how the
analysis will address Missing Values (i.e. Exclude cases analysis by analysis or Exclude case listwise). Click Continue
when you are finished making specifications.
Figure 3
Setting the confidence interval percentage does not have any impact on the calculation of the p value.
If you are only running one Paired Sample t-Test, the two “missing values” settings will produce the same results.
There will only be differences if you are running two or more Paired Samples t-Test. (This would look like having two
or more rows in the main Paired Samples t-Test dialog window)
EXAMPLE
Problem Statement
The sample dataset has placement test scores (out of 100 points) for four subject areas: English, Reading, Math and
Writing. Students in the sample completed all 4 placement test when they enrolled in the University. Suppose we are
particularly interested in the English and Math sections, and want to determine whether students tended to score higher on
their English or Math test, on average. We could use a Paired t-Test to test if there was a significant difference of the two
tests.
Before the Test
Variable English has a high of 101.95 and a low of 59.83, while Variable Math has a high of 93.78 and a low of 35.32.
(ANALYZE¿DESCRIPTIVE STATISTICS¿DESCRIPTIVES). The mean English score is much higher than the mean Math
score. (82.79 vs 65.47). Additionally, there were 409 cases with non-missing English scores and 422 cases with non-missing
Math scores, but only 398 cases with non-missing observations for both variables. (Recall that the sample data set has 435
cases in all).
Let’s create a comparative box plot of these variables to help visualize those numbers. Click ANALYZE¿
DESCRIPTIVE STATISTICS¿EXPLORE. Add English and Math to the dependent box, then change display Option to Plots.
We’ll also need to tell SPSS to put these two variables on the same chart. Click the Plots button, and in the boxplot area,
change the selection to Dependents Together. You can also uncheck stem and leaf. Click Continue. Then Click OK to run
the procedure.
Figure 4
We can see from the box plot that the center of the English scores is much higher than the center of the Math
scores, and that there is slightly more spread in the Math scores than in the English scores. Both variables appear to be
symmetrically distributed. It’s quite possible that the Paired Samples t-Test could come back significant.
RUNNING THE TEST
1. Click ANALYZE¿COMPARE MEANS¿PAIRED SAMPLES t-TEST
2. Select the variable English and move it to the variable 1 slot in the Paired Variable box. Then select the variable
Math and move it to the Variable 2 slot in the Paired Variable box
3. Click OK
OUTPUT
TABLES
There are three tables: Paired Samples Statistics, Paired Samples Correlations and Paired Samples t-Test. Paired
Samples Statistics gives univariate descriptive statistics (mean, sample size, standard deviation and standard error) for each
variable entered. Notice that the sample size here is 398; this is because paired t-test can only use cases that have non-
missing values for both variables. Paired Samples Correlations shows the bivariate Pearson Correlation Coefficient (with two
tailed test of significance) for each pair of variables entered. Paired Samples t-Test gives the hypothesis test results.
Figure 5
The Paired Samples Statistics output repeats what we examined before we can run the test. The Paired Samples
Correlation table adds the information that English and Math scores are significantly positively correlated (r=0.243)
TIP
Why does SPSS report the correlation between the two variables when you run a Paired t-Test? Although our primary
interest when we run a Paired t-Test is finding out if the means of the two variables are significantly different, it’s also
important to consider how strongly the two variables are associated with one another, especially when the variables being
compared are pretest/posttest measures. For more information about correlation, check out the Pearson Correlation Test.
Figure 6
COMMON USES
The Independent Samples t-Test is commonly used to test the following:
1. Statistical difference between the means of two groups.
2. Statistical difference between the means of two interventions.
3. Statistical difference between the means of two change scores.
Note: The Independent Samples t-Test can only compare the means of two (and only two) groups. It cannot make
comparisons among more than two groups. If you wish to compare the means across more than two group, you will likely
want to run an ANOVA.
DATA REQUIREMENTS
Your data must meet the following requirements:
1. Dependent variable that is continuous. (i.e. interval or ratio)
2. Independent variable that is categorical. (i.e. two or more groups)
3. Cases that have values on both the dependent and independent variables.
4. Independent Samples/groups. (i.e. Independent observations)
There is no relationship between the subject in each sample. This means that:
a. Subjects in the first group cannot also be in the second group.
b. No subject in either group can influence subjects in the other group.
c. No group can influence the other group.
Violation of this assumption will yield an inaccurate p-value.
5. Random sample of data from the population.
6. Normal distribution (approximately) of the dependent variable for each group.
Non-normal population distributions, specially those that are thick-tailed or heavily skewed,
considerably reduce the power of the test.
Among moderate or large samples, a violation of normality may still yield accurate p-values.
7. Homogeneity of variances (i.e. variances approximately equal across groups)
When assumptions is violated and the sample scores for each group differ p-value is not
trustworthy. However, the Independent Samples t-Test output also includes an approximate t
statistic that is not based on assuming equal variances. This alternative statistic, called Welch t Test
statistic may be used when equal variances among population cannot be assumed. The Welch t-Test is
also known as Unequal Variance t-Test or Separate Variances t-Test.
8. No outliers
Note: When one or more of the assumptions for the Independent Samples t-Test are not met, you may want to run the non-
parametric Mann-Whitney U Test instead.
HYPOTHESES
The null hypothesis (Ho) and the alternative hypothesis (H1) of the Independent Samples t-Test can be expressed in
two different but equivalent ways:
Ho: µ1¿ µ2 (the two population means are equal)
H1: µ1≠ µ2 (the two population means are not equal)
or
Ho: µ1−¿ µ2 ¿ 0 (The difference between the two population means is equal to 0)
H1: µ1−¿µ2 ≠ 0 (The difference between the two population means is not equal to 0)
Where µ1 and µ2 are the population means for the group1 and group2, respectively. Notice that the second set of hypotheses
can be derived from the first set by simply subtracting µ 2 from both side of the equation.
TEST STATISTIC
The test statistic for an Independent Sample t-Test is denoted by t. These are actually two forms of the test statistic
for this test, depending on whether or not equal variances are assumed. SPSS produces both forms of the test, so both forms
of the test are described here.
Note that the null and alternative hypothesis are identical for both forms of the test statistic.
x 1−x 2
Where:
t=
sp
√ 1 1
+
n1 n2
with s p=
√ ( n1 −1 ) s12 + ( n 2−1 ) s 22
n 1+ n2−2
( )
2 2 2
s1 s2
+
n1 n2
df =
( ) ( )
2 2 2 2❑
1 s1 1 s2
+
n1−1 n1 n 2−1 n2
and the chosen confidence level. If the calculated t-value ¿critical t-value, then we reject the null hypothesis. Note that this
form of the Independent Samples t-Test statistic does not assume equal variances. This is why both the denominator of the
test statistic and the degrees of freedom of the critical value of t are different from the equal variances form of the test
statistic.
DATA SET UP
Your data should include two variables (represented in columns) that will be used in the analysis. The Independent
variable should be categorical and include exactly two groups. (Note that this SPSS restricts categorical indicators to numeric or
short string values only. The dependent variable should be continuous. (i.e. interval or ratio) SPSS can only make use of cases
that have non-missing values for the Independent and the dependent variables. So if a case has missing value for either
variable, it cannot be included in the test.
The number of rows in the data set should correspond to the number of subjects of the study. Each row of the data
set should represent a unique subject, person or unit and all the measurements taken on that person or units should appear in
that row.
Figure 1
A Test Variable(s): The dependent variable(s). This is the continuous variable whose means will be compare between the
two groups. You may run multiple t test simultaneously by selecting more than one test variable.
B Grouping Variable: The independent variable. The categories (or groups) of the independent variable will define which
samples will be compared in the t test. The grouping variable must have at least two categories (groups); it may have more
than two categories but a t-test can only compare two groups, so you will need to specify which two groups to compare. You
can also use a continuous variable by specifying a cut point to create two groups (i.e. values at or above the cut point and
values below the cut point).
C Define Groups: Click Define Groups to define the category indicators (groups) to use in the t-test. If the button is not
active, make sure that you have already moved your independent variable to the right in the Grouping Variable field. You
must define the categories of your grouping variable before you can run the Independent Samples t-Test procedure.
D Options: The option section is where you can set your desired confidence level for the confidence interval for the mean
difference, and specify how SPSS should handle the missing values.
When finished, Click OK to run the Independent Samples t-Test, or click Paste to have the syntax corresponding to your
specified settings written to an open syntax window (If you do not have syntax window open, a new window will open for you)
DEFINE GROUPS
Clicking the Define Groups button C opens the Define Groups window
Figure 2
1 Use specified values: f your grouping variable is categorical, select Use specified values. Enter the values for
the categories you wish to compare in the Group 1 and Group 2 fields. If your categories are numerically coded, you will
enter the numeric codes. If your group variable is string, you will enter the exact text strings representing the two categories.
If your grouping variable has more than two categories (e.g. takes on values of 1, 2, 3, 4), you can specify two of the
categories to be compared (SPSS will disregard the other categories in this case)
Note that when computing the test statistic, SPSS will subtract the mean of the Group 2 from the mean of Group 1.
Changing the order of the subtraction affects the sign of the result, but does not affect the magnitude of the results.
2 Cut Point: If your grouping variable is numeric and continuous, you can designate a cut point for dichotomizing the
variable. This will separate the cases into two categories based on the cut point. Specifically, for a given cut point x, the new
categories will be:
Group 1: All cases where grouping variable ≥ X
Group 2: All cases where grouping variable ¿ X
Note that this implies that cases where the grouping variable is equal to the cut point itself will be included in the “greater than
or equal to” category. (If you want your cut point to be included in a “less than or equal to” group, then you will need to use
Recode into Different Variables or use DO IF syntax to create this grouping variable yourself.) Also note that while you can use
cut points on any variable that has a numeric type, it may not make practical sense depending on the actual measurement
level of the variable (e.g. nominal categorical variables coded numerically). Additionally, using a dichotomized variable created
via a cut point generally reduces the power of the test compared to using a non-dichotomized variable.
OPTIONS
Clicking the Options button D opens the Options window
Figure 3
The Confidence Interval Percentage box allows you to specify the confidence level for a confidence interval. Note that this
setting does NOT affect the test statistic or p-value or standard error; it only affects the computed upper and lower bounds of
the confidence interval. You can enter any value between 1 and 99 in this box (although in practice, it only makes sense to
enter numbers between 90 and 99).
Missing Values section allows you to choose if cases should be excluded “analysis by analysis” (i.e. pairwise deletion) or
excluded listwise. This setting is not relevant if you have only specified one dependent variable; it only matters if you are
entering more than one dependent (continuous numeric) variable. In that case, excluding “analysis by analysis” will use all
nonmissing values for a given variable. If you exclude “listwise”, it will only use the case with nonmissing values for all of the
variables entered. Depending on the amount of missing data you have, listwise deletion could greatly reduce your sample size.
In the sample data, we will use two variables: Athlete and MileMinDur. The variable Athlete has values of either
“0”(non-athlete) or “1” (athlete). It will function as the independent variable in this T-Test. The varable MileMinDur is a
numeric duration variable (h:mm:ss), and it will function as the dependent variable in SPSS.
The first few rows of the data looks like this:
Figure 4
If the variances were indeed equal, we would expect the total length of the boxplots to be the same for both groups.
However, from this boxplot, it is clear that the spread of observations for non-athletes is much greater than the spread of
observations for athletes. Already, we can estimate that the variances for these two groups are quite different. It should not
come as a surprise if we run the Independent Samples t-Test and see that Levene’s Test is significant.
Additionally, we should also decide on a significance level (typically denoted by using the Greek letter alpha, a). before
we perform our hypothesis tests. The significance level is the threshold we use to decide whether a test result is significant.
For this example, let’s use a = 0.05.
OUTPUT
Tables
Two sections(boxes) appear in the output. Group Statistics and Independent Samples Test. The first section,
Group Statistics, provides basic information about the group comparisons, including the sample size (n), mean, standard
deviation and standard error for mile times by group. In this example, there are 166 athletes, and 226 non-athletes. The mean
mile time for athletes is 6 minutes 51 seconds, and the mean mile time for non-athletes is 9 minute 6 seconds.
Figure 6
The second section, Independent Samples Test displays the result most relevant to the Independent Samples t-
Test. There are two parts that provide different pieces of information: (A) Leven’s Test for Equality of Variances and (B) t-Test
for Equality Means.
Figure 7
A Levene’s Test for Equality of Variances: This section has the test results for Levene’s Test. From left to right:
F is the test statistic of Levene’s test
Sig. is the p-value corresponding to this test statistic.
The p-value of Levene’s test is printed as “.000” (but should be read as p¿ 0.001—i.e. p very small)., so we reject the
null Levene’s test and conclude that the variance in mile time of athletes is significantly different than that of non-
athletes. This tells us that we should look at the “Equal variances not assumed” row for the t-test (and
corresponding confidence interval) result. (If this test result had not been significant – that is , if we had
observed p¿ a -- then we would have used the “Equal variances assumed” output.)
B t-test for Equality of Means provides the results for the actual Independent Samples t-Test. From left to right:
T is the computed test statistic
df is the degrees of freedom
Sig. (2 tailed) is the p-value corresponding to the given test statistic and degrees of freedom.
Mean Difference is the difference between the sample means; it also corresponds to the numerator of
the test statistic.
Standard Error Difference is the standard error; it also corresponds to the denominator of the test
statistic
Note that the mean difference is calculated by subtracting the mean of the second group from the mean of the first group. In
this example the mean mile time for athletes was subtracted from the mean mile time for non-athletes (9:06-6:51=2:14) The
sign of the mean difference corresponds to the sign of the t-value. The positive t-value in this example indicates that the mean
mile time for the first group, non-athlete, is significantly greater than the mean for the second group, athletes.
The associated p-value is printed as “0.000”; double-clicking on the p-value will reveal the un-rounded number. SPSS
Rounds p-values to three decimal places, so any p-value too small to round up to 0.001 will print as 0.000. (In this particular
example, the p-values are on the order of 10−40.)
C Confidence Interval of the Difference: This part of the t-Test output compliments the significance test results.
Typically, if the CI for the mean difference contains 0 within the interval—i.e. if the lower boundary of the CI is negative
number and the upper boundary of the CI is a positive number – the results are not significant at the chosen significance level.
In this example, the 95% CI is [01:57, 02:32], which does not contain zero; this agrees with the small p-value of the
significance test.
One-Way ANOVA
One-Way ANOVA (“Analysis of Variance”) compares the means of two or more independent groups in order to
determine whether there is statistical evidence that the associated population means are significantly different. One-Way
ANOVA is a parametric test.
COMMON USES
The One-Way ANOVA is often used to analyze data from the following types of studies:
Field Studies
Experiments
Quasi-experiments
The One-Way ANOVA is commonly used to test the following:
Statistical differences among the means of two or more groups.
Statistical differences among the means of two or more interventions.
Statistical differences among the means of two or more change scores.
Note: Both the One-Way ANOVA and the Independent Samples t-Test can compare the means for two groups. However, only
the One-Way ANOVA can compare mean across three or more groups.
Note: If the grouping variable has only two groups, then the result of One-Way ANOVA and the Independent Samples t-Test
will be equivalent. In fact, if you run both an Independent Samples t-Test and One-Way ANOVA in this situation, you should be
able to confirm that t2=F.
DATA REQUIREMENTS
Your data must meet the following requirements:
1. Dependent variable that is continuous (i.e. interval or ratio level)
2. Independent variable that is categorical (i.e. two or more groups)
3. Cases that have values on both the dependent and independent variables.
4. Independent samples/groups (i.e. Independence observations)
a. There is no relationship between the subjects in each sample. This means that:
i. Subjects in the first group cannot also be in the second group.
ii. No subject in either group can influence subjects in the other group.
iii. No group can influence the other group.
Note: When the normality, homogeneity of variances, or outliers assumptions for One-Way ANOVA are not met, you may want
to run the non-parametric Kruskal-Wallis test instead.
HYPOTHESES
The Null and Alternative hypotheses of One-Way ANOVA can be expressed as:
Ho: µ1 = µ2 = µ3 =…= µk (“all k population means are equal”)
H1: at least one µi different (“at least one of the k population means is not equal to the others”) .
Where
µi is the population mean of the ith group (i= 1,2,…,k)
Note: The One-Way ANOVA is considered an omnibus (Latin for “all”) test because the F test indicates whether the model is
significant overall—i.e. whether or not there are any significant differences in the means between any of the groups. (Stated
another way, this says that at least one of the means is different from the others.) However, it does not indicate which mean is
different. Determining which specific pairs of means are significantly different requires either contrast or post hoc (Latin for
“after this”) tests.
TEST STATISTIC
Test statistic for One-Way ANOVA is denoted as F. For an independent variable with k groups, the F statistic evaluates
whether the group means are significantly different. Because the computation of the F statistic is slightly more involved than
computing the paired or independent samples t test statistics, it’s extremely common for all of the F statistic components to be
depicted in a table like the following:
Where:
SSR = the regression sum of squares
SSE = the error sum of squares
SST = the total sum of squares (SST=SSR+SSE)
dfr = the model degrees of freedom (equal to dfr = k-1)
dfe = the error degrees of freedom (equal to dfe = n-k-1)
k = the total number of groups (levels of the independent variable)
n = the total number of valid observations
dft = the total degrees of freedom (equal to dft = dfr + dfe = n-1)
MSR = SSR / dfr = the regression mean square
MSE = SSE / dfe = the mean square error
Note: In some texts you may see the notation df1 or v1 for the regression degrees of freedom, and df2 or v2 for the error
degrees of freedom. The latter notation uses the Greek letter nu (v) for the degrees of freedom.
Some texts may use “SSTr” (Tr=”treatment”) instead of SSR (R=”regression”), and may use SSTo (To =”total”)
instead of SST.
The terms Treatment (or Model) and Error are the terms most commonly used in natural sciences and in traditional
experiment design texts. In the social sciences, it is more common to see the terms Between groups instead of “Treatment”,
and Within groups instead of “Error”. The between/within terminology is what SPSS uses in the One-Way ANOVA procedure.
DATA SET UP
Your data should include at least two variables (represented in columns) that will be used in the analysis. The
independent variable should be categorical (nominal or ordinal) an d include at least two groups, and the dependent variable
should be continuous (i.e. interval or ratio). Each row of the data set should represent a unique subject or experimental unit.
Note: SPSS restricts categorical indicators to numeric or short string values only.
Figure 1
The One Way ANOVA window opens, where you will specify the variables to be used in the analysis. All of the variables
in your data set appear in the list on the left side. Move variables to the right by selecting them in the list and clicking the blue
arrow buttons. You can move a variable(s) to either of the two areas: Dependent List or Factor.
Figure 2
A Dependent List: The dependent variable(s). This is the variable whose means will be compared between the samples
(groups). You may run multiple means comparison simultaneously by selecting more than one dependent variable.
B Factor: The independent variable. The categories (or groups) of the independent variable will define which samples will be
compared. The independent variable must have at least two categories (groups), but usually has three or more groups when
used in a One-Way ANOVA.
C Contrasts: (Optional) Specify contrasts, or planned comparisons, to be conducted after the overall ANOVA test.
Figure 3
When initial F test indicates that significant differences exists between group means, contrasts are useful for determining which
specific means are significantly different when you have specific hypotheses that you wish to test.
Contrasts are decided before analyzing the data (i.e. a priori). Contrasts break down the variance into component parts. They
may involve using weights, non-orthogonal comparisons, standard contrasts, and polynomial contrasts (trend analysis).
Many online and print resources detail the distinctions among these options and will help users select appropriate contrasts.
For more information about contrasts, you can open the IBM SPSS help manual from within SPSS by clicking the “Help” button
at the bottom of the One-Way ANOVA dialog window.
D Post Hoc: (Optional) Request post hoc (also known as multiple comparisons) tests. Specific post hoc tests can be selected
by checking the associated boxes.
Figure 4
1 Equal variances assumed: Multiple comparisons options that assume homogeneity of variance (each group has
equal variance). For detailed information about the specific comparison methods, click the Help button in this window.
2 Test: By default, a 2-sided hypothesis test is selected. Alternatively, a directional, one-sided hypothesis test can be
specified if you choose to use a Dunnett post hoc test. Click the box next to Dunnett and then specify whether the Control
Category is the Last or First Group, numerically, of your grouping variable. In the Test area, click either¿<¿ Control or¿
Control. The one-tailed options require that you specify whether you predict that the mean for the specified control group will
be less than ¿ Control) or greater than ( ¿Control) another group.
3 Equal variance not assumed: Multiple comparisons options that do not assume equal variances. For detailed
information about the specific comparison methods, click the Help button in this window.
4 Significance level: The desired cutoff for statistical significance. By default, significance level is set at 0.05.
When the initial F test indicates that significant differences exists between group means, post hoc tests are useful for
determining which specific means are significantly different when you do not have specific hypotheses that you wish to test.
Post hoc test compare each pair of means (like t-tests), but unlike t-tests, they correct the significance estimate to account for
the multiple comparisons.
E Options: Clicking Options will produce a window where you can specify which statistics to include in the output
(Descriptive, Fixed and random effects, homogeneity of variance test, Brown-Forsythe, Welch), whether to include a Means
plot, and how the analysis will address Missing Values (i.e., Exclude cases analysis by analysis or Exclude cases
Listwise). Click Continue when you are finished making specifications.
Figure 5
Example
To introduce One-Way ANOVA, let’s use an example with relatively obvious conclusion. The goal here is to show the
thought process behind a One-Way ANOVA.
PROBLEM STATEMENT
In the sample data set, the variable Sprint is the respondent’s time (in seconds) to sprint a given distance, and
Smoking is an indicator about whether or not the respondent smokes ( 0= nonsmoker, 1= Past smoker, 2 = Current smoker).
Let’s use ANOVA to test if there is a statistically significant difference in sprint time with respect to smoking status. Sprint time
will serve as the dependent variable, and smoking status will act as the independent variable.
The sprint times are a continuous measure of time to sprint a given distance in seconds. From the Descriptive
procedure (Analyze¿Descriptive Statistics¿Descriptives), we see that the times exhibit a range of 4.5 to 9.6 seconds,
with a mean of 6.6 seconds (based on n=374 valid cases). From the Compare Means procedure (Analyze¿Compare Means¿
Means), we see these statistics with respect to the groups of interest.
Notice that, according to the Compare Means procedure, the valid sample size is actually n=353. This is because Compare
Means (and additionally, the one-way ANOVA procedure itself) requires there to be nonmissing values for both the sprint time
and the smoking indicator.
Lastly, we’ll also want to look at a comparative box plot to get an idea of the distribution of the data with respect to the
groups:
Figure 6
From the box plots, we see that there are no outliers; that the distributions are roughly symmetric; and that the center of the
distribution don’t appear to be hugely different. The median sprint time of the past and current smokers.
OUTPUT
The output displays a table entitled ANOVA
Figure 7
Figure 8
The Means plot is a visual representation of what we saw in the Compare Means Output. The points on the chart are the
average of each group. It’s much easier to see from this graph that the current smokers had the slowest mean sprint time,
while the nonsmokers had the fastest mean sprint time.
DISCUSSION AND CONCLUSION
We conclude that the mean sprint time is significantly different for at least one of the smoking groups ( F2,350=9.209, p˂0.001).
Note that the ANOVA alone does not tell us specifically which means were different from one another. To determine that, we
would need to follow up with multiple comparisons (or post hoc) tests.