Chapter Four: Univariate
Statistics
Univariate analysis, looking at single variables, is typically the first
procedure one does when examining first time data. There are a number of
reasons why it is the first procedure, and most of the reasons we will cover at
the end of this chapter, but for now let us just say we are interested in the
"basic" results. If we are examining a survey, we are interested in
how many people said, "Yes" or "No", or how many people
"Agreed" or "Disagreed" with a statement. We aren't really
testing a traditional hypothesis with an independent and dependent variable; we
are just looking at the distribution of responses.
The
SPSS tools for looking at single variables include the following procedures: Frequencies, Descriptives and Explore all located under the Analyze menu.
This
chapter will use the GSS02A file used in earlier chapters, so start SPSS and
bring the file into the Data Editor. ( See Chapter 1
to refresh your memory on how to start SPSS). To begin the process start SPSS,
then open the data file. Under the Analyze menu, choose Descriptive Statistics and the procedure desired: Frequencies, Descriptives, Explore, Crosstabs.
Frequencies
Generally
a frequency is used for looking at detailed information in a nominal (category)
data set that describes the results. Categorical data is for variables such as
gender i.e. males are coded as "1" and females are coded as
"2." Frequencies options include a table
showing counts and percentages, statistics including percentile values, central
tendency, dispersion and distribution, and charts including bar charts and
histograms. The steps for using the frequencies procedure is to click the Analyze menu choose Descriptive Statistics then from the submenu
choose Frequencies and select your variables
for analysis. You can then choose statistics options, choose chart options,
choose format options, and have SPSS calculate your request.
For
this example we are going to check out attitudes on the abortion issue. The 2002
General Social Survey, GSS02A, has the variable ABANY with the label "ABORTION--FOR
ANY REASON." We will look at this variable for our initial investigation.
Choosing Frequencies
Procedure:
From
the Analyze menu, highlight Descriptive Statistics, Figure 4‑1, then
move to the sub menu and click on Frequencies.
A
dialog box, Figure 4‑2, will appear providing a
scrollable list of the variables on the left, a Variable(s) choice box, and buttons for
Statistics, Charts and Format options.[1]
Selecting Variables for
Analysis:
First
select your variable from the main frequencies dialog box, Figure
4‑2, by clicking the variable name once. (Use the scroll bar if you
do not see the variable you want.) In this case ABANY is the first variable and
will be selected (i.e., highlighted). Thus, you need not click on it.
Click
the arrow to the left of the Variable(s): box, Figure
4‑2, to move ABANY into the box. All variables
selected for this box will be included in any procedures you decide to run. We
could click OK to obtain a frequency and percentage distribution of the
variables. In most cases we would continue and choose one or more statistics.
Choosing Statistics for
Variables:
Click
the Statistics button, bottom of Figure 4‑2, and a dialog box of statistical choices
will appear, Figure 4‑3.
This
variable, ABANY is a nominal (category)
variable so click only the Mode box within the central
tendency choices. See Figure 4‑3.
After
clicking the Mode box click the Continue button, top right, and we return to the main Frequencies dialog box, Figure 4‑2.
We
could now click OK and SPSS would calculate
and present the frequency and percent distribution (click OK if you want) but, in the more typical manner, we will continue and
include choices for charts and check out the Options possibilities. If you
clicked OK, just press the Analysis menu then choose Descriptive Statistics and then Frequencies from the sub menu and you
will be back to this point with your variable and statistics chosen.
Choosing Charts for
Variables:
On
the main frequencies window, click the Charts button, Figure
4‑2, and a dialog box of chart choices, Figure 4‑4,
will appear.
Click
Bar Chart, as I have done, since this
is a categorical variable, then click Continue to return to the main Frequencies window box. If you have a continuous
variable choose Histograms and the With Normal Curve option would be available. Choose the With Normal Curve option to have a normal curve drawn over the
distribution so that you can visually see how close the distribution is to normal. Note: Frequencies is automatically chosen for
chart values but if desired you could change that to Percentages.
Now
click OK on the main frequencies
dialog box and SPSS will calculate and present a frequency and percent
distribution with our chosen format, statistics, and chart. (Note: We could
look to see if additional choices should be made by clicking the Format button. In this case we don't need to do this because all the Format defaults are appropriate since we are looking at one variable.)
Looking at Output from
Frequencies:
We
will now take a brief look at our output from the SPSS frequencies procedure.
(Processing time for SPSS to perform the analysis in the steps above will
depend on the size of the data set, the amount of work you are asking SPSS to
do and the CPU speed of your computer). The SPSS
Output Navigator,
left side, and the output, right side, will appear when SPSS has completed its
computations. Either scroll down to the chart in the right window, or click the
Bar Chart icon in the outline pane to
the left of the output as we did in Figure 4‑5.
Interpreting the Chart:
We now see the chart, Figure 4‑6. The graphic is a bar chart with the
categories at the bottom, the X axis, and the frequency scale at the left, the
Y axis. To display the chart, drag the scroll bar on the right of your table.
The variable label ABORTION‑‑ FOR ANY REASON is displayed at the
top of the chart. We see from the frequency distribution that there are more
"no", 18.6%, answers than "yes", 14.0% answers (see Figure 4‑7), when respondents were asked if a woman
should be able to get an abortion for any reason. A much
smaller number, which does not appear on this chart, .8% (see Figure 4‑7), selected "don't know",
"DK." If a chart were the only data presented for this
variable in a report, you should look at the frequency output and report the
total responses and/or percentages of YES, NO and DK answers. Not asked this
question, NA, were 66.6% of the sample. You should also label the chart with
frequencies and/or percentages. There are a lot of possibilities for enhancing
this chart within SPSS (Chapter 9 will discuss presentation).
If
we choose to copy our chart to a word processor program for a report, first
select the chart by clicking the mouse on the bar chart. A box with handles
will appear around the chart. Select Copy from the Edit menu. Start your word
processing document, click the mouse where you want the chart to appear then
choose Paste Special from the Edit menu. Choose Picture in the paste special dialog
box that appears and click OK to paste the chart into
your document.
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Interpreting Frequency
Output:
To
enlarge the frequency distribution, move the scroll bar on the right of our
output window to maximize the size of the table. Another way is to click the Frequencies icon in the outline box to the left of the output window. To view a
large table you may want to click on the Maximize Arrow in the upper right
corner of the SPSS Output Navigator window to enlarge the output
window. Use the scroll bars to display different parts of a large table. The
most relevant part of the frequency distribution for ABANY is in Figure 4‑7.
We
can now see some of the specifics of the SPSS frequencies output for the
variable ABANY. At the top is the variable label ABORTION‑‑
FOR ANY REASON. The major part of the display
shows the value labels (YES, NO, Total), and the missing categories NAP [Not Appropriate],
DK [Don't Know], NA [Not Answered], Total and the Frequency, Percent, Valid Percent,
Cumulative Percent (the cumulative % for values as they increase in size), for
each classification of the variable. The "Total" frequency and
percent is listed at the bottom of the table. When asked if a woman should be
able to have an abortion for any reason 18.6 % responded no. DK, don't know was chosen by .8 % and .0%
were NA [Not Answered]. The 66.6 % "NAP" [Not Appropriate] was that
portion of the sample that were not asked this
question. In a written paper you should state that the "Valid Percent"
excludes the "missing" answers.
Variable Names, Variable
Labels,
Values, Value Labels, Oh My!
Options in Displaying
Variables and Values
It
is important to use these concepts correctly so a review at this point is
appropriate. A Variable name is the
short name you gave to each variable, or
question in a survey. The table below is
designed to help you keep these separate.
|
Variable Name |
Variable Label. |
Value |
Value Label |
|
SEX |
Respondent's gender. |
1 or 2 |
Male, Female |
|
AGE |
Respondent's age at last birthday. |
18, 19, 20, 21… 89, 98, 99 |
None needed |
|
AGED |
Should aged live with their children. |
1, 2, 3, 0, 8, 9 |
A good idea, Depends, A bad idea NAP [Not Appropriate], DK [Don't
Know], NA [Not Answered] |
|
BIBLE |
Feelings about the bible |
1, 2, 3, 4, 0, 8, 9 |
Word of God, Inspired Word, Book of Fables, NAP, DK, NA |
Understanding
these allows you to intelligently customize SPSS for Windows so that it is
easier for you to use. You can set SPSS so that you can see the Variable Names
when you scroll through a listing of variables, or so that you can see the
Variable Labels as you scroll through the listing. You can set SPSS so that you
get only the Values, only the Labels, or both in the output. Below are two
examples of a frequencies dialog box.
Figure 4-8 shows the listing as Variable Labels. This is the default setting when SPSS for
Windows is installed. This example has
the cursor on the variable ABPOOR so
ABORTION-LOW INCOME-CAN'T AFFORD MORE CHILDREN (is displayed). You can change the listing however, so that
you see only variable names as in Figure 4-9. Changing this is a matter of personal
taste. This chapter uses variable names, Figure 4-9.
Changing
the display option for the variable selection dialog box must be done before the data file is opened. If you have SPSS open with a data file click File, New and Data and the data editor will be
cleared.
To
set the display option click Edit then choose Options. The General tab on the options dialog
box will appear, Figure 4-10. Under Variable Lists section, top right quadrant, click your choices
then OK. For
this chapter we choose Display Names and Alphabetical so that variable names will
be displayed alphabetically as in Figure 4-9.
Displaying Values, Value
Labels or Both in Your Output
One
other option you might want to make is in the table format for your SPSS
output. You can choose to have displayed
variable labels, values (e.g. 1, 2, 3, etc), Value Labels (YES, No, DK, etc.)
or both values and labels (1 YES, 2 NO, 3 DK).
To make these choices click the Edit menu and choose Options, then click the Output Labels, click the options dialog
box and make your choices. My choices
are seen in Figure 4-11. The output resulting from my choices is Figure 4-12.
Descriptives
Descriptives (Analysis, Descriptive
Statistics, Descriptives, Figure 4-13) is used to obtain summary information about
the distribution, variability, and central tendency of continuous variables.
Possibilities for Descriptives include mean, sum, standard
deviation, variance, range, minimum, maximum, S.E. mean, kurtosis and skewness.
For this example we are going to look at the distribution of age and education
for the General Social Survey sample. Since both these variables were measured
at interval/ratio level, different statistics from our previous example will be
used.
Choosing Descriptive
Procedure:
First
click the Analyze menu and select Descriptive Statistics, then move across to the sub menu and select Descriptives (see Figure
4-13).
Selecting Variables for
Analysis:
First
click on AGE, the variable name for AGE OF RESPONDENT. Click the select arrow
in the middle and SPSS will place AGE in the Variable(s) box. Follow the same steps
to choose EDUC the variable name for HIGHEST YEAR OF SCHOOL COMPLETED. The dialog box should look like Figure 4-14.
We
could click OK and obtain a frequency and
percentage distribution, but we will click the Options button and decide on
statistics for our output. Click Options and the Descriptives: Options dialog box, Figure 4-15, will open.
Since
these variables are interval/ratio measures, choose: Mean, Std. deviation, Minimum and Maximum. We will leave the defaults
for the Distribution and Display Order.
Next,
click the Continue button to return to the
main Descriptives dialog box, (Figure 4-14). Click OK in the main Descriptives dialog box and SPSS will
calculate and display the output seen in Figure 4-16.
Interpretation of the
Descriptives Output
In
the Interpretation of Figure 4-16, AGE OF RESPONDENT
has a mean of 46.28 and a standard deviation of 17.370. The youngest respondent
was 18 and the oldest was 89. On your SPSS output HIGHEST YEAR OF SCHOOL
COMPLETED, has a mean of 13.36 (a little more then 1 year beyond high school)
and a standard deviation of 2.9. Some respondents indicated no "0"
years of school completed. The most education reported was 20 years.
Explore
Explore
is primarily used to visually examine the central tendency and distributional
characteristics of continuous variables. Explore statistics include M‑estimators,
outliers, and percentiles. Grouped frequency tables and displays, as well as
Stem‑and‑leaf and box‑plots, are available. Explore will aid in checking assumptions with Normality plots and Spread vs.
Level with the Levene test.
Choosing the Explore Procedure:
From
the Analyze menu choose Descriptive Statistics, drag to the sub menu and select Explore.
Selecting Variables:
As
in the other procedures, find and click the variable you want to explore, and
then click the select arrow to include your variable in the Dependent List box. Choose the variable EDUC The dialog box should look
like Figure 4-17.
Selecting Displays:
In
the Display box on the bottom left, you
may choose either Both, Statistics, or Plots. We left the default selection, Both to display statistics and plots.
Selecting Statistics:
Click
the Statistics button, bottom middle of Figure 4-17, and the Explore:
Statistics
dialog box will open, Figure 4-18.
Leave
checked the default box for Confidence
Interval for the Mean 95%, and click the Outliers box so we can look at the
extreme observations for our variable. Click Continue to return to the main
explore dialog window.
Selecting Plots:
Click
the Plots button on the main Explore
Dialog Box, Figure 4-17, and the Explore:
Plots dialog
box, Figure 4-19, will open.
The
default choices in the Boxplots box are good so click Stem‑and‑leaf and Histogram in the Descriptive box. Click on Normality
Plots with Test"
so we can see how close the distribution of this variable is to
normal. Leave the default for Spread vs Level with Levene Test. Click Continue to return to the main
explore dialog box.
Selecting Options:
Click
the Options button in the main explore
dialog box, Figure 4-17, and the Explore:
Options dialog
box, Figure 4-20, will be displayed.
No
changes are needed here since the default of Exclude cases listwise is appropriate. Now click Continue to return to the main Explore dialog box, Figure 4-17. Click OK in the main Explore dialog box and SPSS will
perform the chosen tasks and display the data in the SPSS
Output Navigator.
Interpretation of Explore
Output:
Use
the scroll bar to view any part of the output. The first part of the output is
the Case Processing Summary, Figure 4-21.
We
can see that 2753 (99.6%) of our respondents answered this question. The other 12,
.4% of the sample, was Missing, not answering the question
in this case. The GSS in recent years has had a split sample where not all
respondents in the sample are asked the same questions. This is a question where
all respondents where asked the question, so the total sample size was 2765
(100%).
The
Descriptives statistics output should
look like Figure 4-22.
We
can see all the typical descriptive statistics on this output: mean (13.26),
lower bound (13.16) and upper bound (13.37) for a 95% confidence of the mean
(in polling terminology this says that we are 95% confident that the mean for
the population is between 13.14 and 13.36), median (13.00), variance (8.23),
standard deviation (2.89), minimum (0), maximum (20), range (20), inter
quartile range (4.00), skewness (‑.134), kurtosis (.781). A narrative
explaining the education of first marriage for the
Our
sample from the General Social Survey of 2002,
indicates that the average education for those over 18 in the
The
Extreme Values can be seen in Figure 4-23. This Figure shows the five highest and the
five lowest values for our variable. More then five respondents listed their
years of education as 20. On the low end
there was one with 0 education, two with 2 years of
education and more then five people listed their education as 3 years in our
sample. The Test of Normality is shown next (see Figure
4-24). This shows that this distribution is significantly different from
the expected normal distribution. This is a pretty stringent test,
most researchers would not require the distribution to be this close to
normality.
The
histogram, Figure 4-25, shows a rough bell shaped
distribution. SPSS divided our distribution into nine groups with a width of
2.5 years of education for each group.
The
largest group has a little more then 1100 cases, a visual estimate. The
smallest group has very few cases (we know there are a number of respondents
who reported 3 years of education from our Extreme Values and the 2.5 bar). The
statistics on the histogram tell us that the standard deviation is 2.87 with a
mean of 13.3 for a total N of 2808.
The
Stem‑and‑Leaf is next. Figure 4-26, once
again, shows a close but not quite normal distribution with significant
outliers on the end of the distribution and a high number of observations above
the mode. We saw this in our earlier output.
Interpretation of the Q‑Q
Plot of Age:
Continue
scrolling down the SPSS Output
Navigator to
the Normal Q‑Q Plot of HIGHEST YEAR OF SCHOOL COMPLETED (see Figure
4-27).
A q‑q
plot charts observed values against a known distribution, in this case a normal
distribution. If our distribution is normal, the plot would have observations
distributed closely around the straight line. In Figure
4-27, the expected normal distribution is the straight line and the line of
little boxes is the observed values from our data. Our plot shows the
distribution deviates somewhat from normality at the low end. The high end of
the distribution is pretty much normal.
The
Detrended Normal Q‑Q plot,
shows the differences between the observed and expected values of a normal
distribution. If the distribution is normal, the points should cluster in a
horizontal band around zero with no pattern. Figure 4-28,
of HIGHEST YEAR OF SCHOOL COMPLETED, indicates some deviation from normal
especially at the lower end. Our overall conclusion is that this distribution
is not normal. Many researchers would
see this as close enough to treat as a normal distribution.
Interpretation of the
Boxplot:
In
the SPSS Output Navigator, scroll to the boxplot of HIGHEST YEAR OF SCHOOL
COMPLETED. The boxplot should look like Figure 4-29.
Once
again the major part of our distribution is not normal and there are
significant outliers, the cases beyond the lower line of our boxplot. Our outliers are at the lowest end of the
distribution, people with little or no education. There are also more observations
above than below the mode.
Conclusion
In
performing univariate analysis the level of
measurement and the resulting distribution determine appropriate analysis as
well as further multivariate analysis with the variables studied. The specific
output from SPSS one uses in a report is chosen to clearly display the
distribution and central tendencies of the variables analyzed. Sometimes you
report a particular output to enable comparison with other studies. In any
case, choose the minimal output that best accomplishes this goal. Don't report
every SPSS output you obtained.
Univariate Analysis as Your First Step
in Analysis
Why
do univariate analysis as your first step in data analysis? There are five
reasons:
1. As
discussed at the beginning of this chapter, the frequency distribution may
actually be all you are interested in. You may be doing research for people
with little statistical background and/or they are really only interested in
the percentage or count of people that said "Yes" or "No"
to some question.
2. You can check for "dirty" data. Dirty data is
incorrectly entered data. "Data cleaning" is correcting these errors.
Remember, in Chapter 2 you were instructed to give each case an ID number. One
primary reason for the ID number is to help us clean our data in case there are
data entry errors. One way to do this is by determining when there are codes in
the data outside the range of the question asked and determining which cases,
the ID number, is in error. You can then check all the way back to the original
questionnaire and correct the entry or if that's not possible change the
erroneous code to the "Missing values" code.
An example might be if you had a question in a
questionnaire where responses were coded in the following way:
1 is the
code used for "Strongly Agree"
2 is the
code used for "Agree"
3 is the
code used for "Neutral"
4 is the code used for "Disagree"
5 is the
code used for "Strongly Disagree"
But suppose you run a frequency distribution and
find that two respondents have a code of "6." That wasn't one of the codes! What happened?
Your data entry person, who may have been you, hit the 6 on the keyboard
instead of some other number. We can correct this error. In fact, when we
locate this error, we may find others because often errors occur in streaks.
The data entry person gets something out of order, or they get their fingers on
the wrong keys. These problems can happen to any of us. The trick is to correct
the errors as best possible.
You can have SPSS for Windows select only those
cases that have the code of "6" (see chapter 3) for that variable,
and then tell it to do a Frequencies on the variable ID. This will tell you the
case numbers that have the error and you can correct it. Be sure to double
check the codes, before and after, to make sure they are correct.
3. A third reason for running a Frequencies on your variables as
your first step in analysis is that you can tell if you need to combine
categories, and if so, what codes should be combined. You would know if there
were too few respondents giving "Strongly Agree" or "Strongly
Disagree" and for analysis they should be folded into either
"Agree" or "Disagree". Another common combination of
categories is for age groups. For example you would do this if you wanted to
compare age groups born before and after a significant event (i.e. those born
before
4. Related to number 3 is that you can find if everything that
should be defined as "Missing" is actually defined as missing. For
example, if you find that 8 "Don't Know" is a response that has been
left in your calculations, your analysis will include all of the eight's. Even
your mean statistics will have these "extra" eight's included in the
calculation. You need to go into the definition of the variable and make these
codes "Missing values" or recode these so they are not included, say
as a "System Missing" value (Chapter 3).
5. Finally, you may want to examine the distributions for your
variables. This should help you determine characteristics of your sample, make
some conclusions and decide further steps in your analysis. You might find that
in a 1‑5 agree/disagree question, discussed in step 2 above, almost
everyone disagreed. You may discover you do not have a normal distribution and
may decide that you want to "fix" the distribution using various
transformation techniques to convert the data into a normal distribution. These
and related techniques are often referred to as "exploratory data
analysis" and are beyond the scope of this text.
Chapter Four Exercises
These
exercises are designed to familiarize you with the SPSS univariate procedures.
They are open‑ended with no specific answers.
1. In this chapter we looked at, ABANY (ABORTION--FOR ANY REASON),
one of the variables in the GSS02A data measuring people's
attitudes about abortion. There are other variables measuring different aspects
of the abortion issue. These are:
Pick one of these variables and perform the
appropriate techniques discussed in this chapter for the variable. Write up a
short narrative explaining what you found about this variable. (Looking back at
what we did with ABANY should help you with this.
Your write up should be designed to best explain what you found so do not
report all the SPSS output, just that output necessary to clearly and
accurately describe your findings.)
2. In this chapter we looked at EDUC (HIGHEST YEAR OF SCHOOL
COMPLETED). There are similar variables measuring respondent's parents’
education:
Pick one of these variables and perform the
appropriate techniques discussed in this chapter for describing the variable.
Write up a short narrative explaining what you found about this variable. (You
might want to look back at what we did with EDUC. Your write up should be
designed to best explain what you found so do not report all the SPSS output,
just that output necessary to clearly and accurately describe your findings.)
3. The GSS02A file provides answers to a wide range of
questions from a sample of respondents in the
4. One way to evaluate how close a sample is to the population
from which it was drawn is by a comparison of known variables of the population
with the same variables in the sample. The 2002 General Social Survey has
variables for which we pretty much know the
[1] If you want to know more about a variable, the label,
codes, etc. with the mouse pointer on the variable name in the variable list,
right click the mouse button and choose Variable Information.
