4.

Descriptive Statistics: Measures of

Variability and Central Tendency
Objectives
Calculate descriptive for continuous and categorical data
Edit output tables
Although measures of central tendency and variability were presented as separate
chapters in the Fundamentals text, they are presented together here because they are
options located in the same command windows in SPSS. Descriptive statistics are
calculated using the Analyze menu. Most are calculated using either the Descriptives or
Frequencies command under Descriptive Statistics. When calculating descriptives for
more complex designs including more than one independent variable, you can also use
the Means/ Compare Means or the Descriptive Statistics/ Crosstabs command which
allow you to calculate descriptive statistics of subgroups.
It is always important to take a moment to think about the type of data you are
using and what descriptive statistics will be most useful given the type. For continuous
or measurement data, you typically report measures of central tendency and measures of
variability. For categorical data (i.e., nominal data) you typically report the frequency of
each value. Though you dont typically report the frequencies for continuous data, it is
often useful to observe the frequency distributions or histograms of continuous
distributions to note if they are normal or skewed.
Descriptive Statistics
Lets begin by calculating descriptive statistics for the data in Appendix D which
can be found on the web as appendix.dav. (In some editions of these books the file is
referred to as Appendix Data Set or as Add.dat or as ADD.dat.) In this data set, I think
of ADD symptoms, IQ score, English grade, and GPA as continuous variables. Well
calculate measures of central tendency and variability for each of these.
Open appendixd.sav.
In the Analyze menu, select Descriptive Statistics and then Descriptives.

 Select each of the continuous variables

by either double clicking them, which
automatically puts them in the Variable
list, highlight them one at a time by
single clicking them and then clicking
the arrow to shift them into the variable
list, or by holding the control key down
while highlighting all of the variables of
interest and then shifting them into the
variable list all at once by clicking the
arrow. Then click Options.
Select each of the measures youve been
learning about (Mean, Std. deviation, Variance,
Range, Minimum and Maximum). Then, select
the Display Order you would prefer. This will
determine the order they appear in for the
resulting table. I like them in the order I
indicated in the Variable list. Then click
Continue.

In the main descriptives dialog box, check the box that says Save standardized
values as variables. SPSS will calculate z scores for each of the variables using the
formula you learned about and append them to the end of your data file. Click Ok.
The resulting output will look like this. Note that the variable labels are used rather
than the variable names. Remember, we specified this as the default in
Edit/Options/Output Labels.

Double click the table so you can edit it. As was the case with graphs, SPSS has many
options to edit statistics in tables as well. Lets try some of them.

Under Pivot, select Transpose Rows and Columns. Which orientation do you
prefer? I like the first since its more conventional, so I will Transpose the Rows and
Columns again to return to the original orientation.
Now, click on Format/Table properties. Take a moment to view all of the options in
this dialog box. General allows you to specify the width of row and column labels.
Footnotes allows you to chose numeric or alphabetic labels and subscript or
superscript as the position for those labels. Cell formats allows you to change the
font style and size, color, and the alignment. Borders allows you to add or remove
borders around rows, columns, and even cells. Printing allows you to select options
such as rescaling tables to fit on paper. After youve viewed the options, hit Cancel.

Now, select Format/Table

Looks. Scroll through the
TableLook Files and look at
the samples. Select one you
like and click Ok. I chose
Academic.

 The resulting table is below. I could edit each individual cell by double clicking on it
and then edit the text. For example, I could alter each statistic to include 2 decimal
places if I wanted. You try it.

Now, click on Window/SPSS Statistics Data Editor and look at the standardized
values (z scores) SPSS added to your file. A brief portion of the Data Editor appears
below. You can see that SPSS named each variable with a z. SPSS also labeled the
new variables. Check this out in Variable View.

Frequencies
Now, well use the frequencies command to help us examine the distributions of
the same continuous variables.
Select Analyze/Descriptive Statistics/Frequency.
Put the variables of interest in the
Variable list box. Unselect
Display frequency tables,
because this will be a list of the
frequency of every value. (Ignore
what looks like an error message.)
Click on Charts, select
Histogram with normal curve
and click Continue. Now, click
Statistics.
Click on Statistics. This dialog
box has all of the same options
we selected under Descriptives
earlier. However, the
Descriptives dialog box did not
include the median and mode.
Select all of the statistics of
interest and click Continue.
Then, click Ok. A sample of the
output follows.

Frequencies

Histogram

Take a moment to review the output. It looks like ADD is somewhat normally
distributed, though a bit negatively skewed. Looking at your own output, are the other
variables normally distributed? I also remember now that English grade is nominal too.
Variables were scored as A, B, C, D, and F, though coded as 1 - 4. As noted in the text,

we could analyze this as continuous data, but it seems that reporting the frequencies
rather than measures of central tendency and dispersion may be more appropriate for this
variable.
As before, you can edit the tables or the graphs by double clicking on them. One
difference we have seen between the Descriptives and Frequencies options is that
descriptives only include mean for measures of central tendency whereas Frequencies
include the mean, median, and mode. Further, Descriptives does not have any built in
graphing options, but Frequencies does.
Now lets use Frequencies to describe categorical data.
Select Analyze/Descriptive Statistics/Frequencies.
This time, put gender, level of English class, English grade, repeated a grade, social
problems, and drop out status in the variable list. Select Display frequency table.
Since there is a finite number of values, we want to know how many people fit in
every category. Click on Statistics and unselect all of the options because we decided
that measures of central tendency and variability are not useful for these data. Then
click Continue. Next, click on Charts. Click on Bar chart and select Percentages
as the Chart Values. Click Continue and then Ok. A sample of the resulting output is
below. Take a moment to review it.

Notice that the frequency tables include a column labeled Percent and another
labeled Valid percent. This is an important distinction when you have missing cases.
The percent column indicates the percent of cases in each category out of those cases for
which there is complete data on the variable. Valid percent indicates the percent of cases
in each category out of the total number of cases, even if some data are missing. For
example, imagine a sample of 100 students. Fifty cases are women, 40 are men, and 10
are missing the data. The percent of men would be 44.4%, but the valid percent of men
would be 40%. Which do you believe is the more accurate way to describe the sample?
Id argue the valid percent. Now lets move on to a more complicated type of frequency
table.
Crosstabs
Sometimes we need to know the number and percent of cases that fall in multiple
categories. This is useful when we have multiple categorical variables in a data set. For
example, in the data set we have been using, Id like to know what percent of dropout and
nondropout students had social problems. Well use crosstabs to calculate this.
Click Analyze/Descriptive Statistics/Crosstabs.

Select social problems for Rows and

dropped out for Columns. Click on
Cells and select Observed for Counts,
and select Row, Column, and Total
under Percentages. The click
Continue. Lets select Display
clustered bar charts to see if we find
this option useful. Then, click Ok. The
output follows. You can edit both the
table and the chart as you have learned.

Both the table and the graph show that of those youth with social problems, an
equal number did and did not ultimately drop out. This suggests that social problems in
ninth grade and drop out status are independent, something we can test later using chi
square.
Compare Means
Now, lets consider a case where we want to describe a continuous variable but at
different levels of a categorical variable. This is often necessary when you are comparing
group means. For example, we can compare ADD symptoms for males and females.
Lets try it together.
Select Analyze/Compare Means/Means. Notice this is the first time we havent
selected Descriptive Statistics in this chapter.
Select ADD score for the
Dependent List and Gender
for the Independent List.
Click Options. Notice that
mean, standard deviation and
number of cases are already
selected under statistics. Add
any other descriptive you are
interested in, then click
Continue and then Ok. The
output follows.

Do you think males and females differed in their ADD symptoms?

Lets try another more complicated example. This time, lets calculate descriptive
statistics for ADD symptoms broken down by gender and whether or not a child had
social problems.
Select Analyze/Compare Means/Means.

Just like before, select

ADD score for the
Dependent List, and
gender for the Layer 1
Independent List. Then
click Next. Select social
problems as the Layer 2
Independent List. Select
whatever statistics you
want under Options and
then click Continue and
Ok. The output is below.

Notice that this table gives you the marginal descriptives (i.e., the descriptive for
gender independent of social problems and vice versa) under totals and the cell
descriptives (i.e., the descriptives at each level of the variables-e.g., for boys with social
problems).
Exit SPSS. There is no need to save the Data File since we havent changed it. It is up
to you to decide whether or not you would like to save the output file for future
reference.
Weve reviewed a variety of options for calculating descriptive statistics
depending on the type of data and the kinds of questions. Weve also seen that many of
the graphs we reviewed in Chapter 3 are options in the subcommands under Descriptive
Statistics. In the following chapters you will discover that descriptive statistics are an
option embedded within many other analyses dialog boxes (e.g. t-test, ANOVA, etc).
Try the following exercises to be sure you understand all of the various options for
calculating descriptives and to help you identify your own preferences.

Exercise
1. Using merge1.sav calculate the mean, median, mode, range, variance, and
standard deviation for the following variables: self-esteem, anxiety, coping, and
health. Create a histogram for anxiety. Note how you did each.
2. Using the data in appendixd.sav, calculate the frequency and percent of females
and males who did and did not have social problems.
3. Using the data in appendixd.sav, calculate the mean, variance, and standard
deviation for GPA broken down by social problems and drop out status.