LESSON FOUR
DESCRIPTIVE STATISTICS
In this lesson, we will learn how to
obtain descriptive statistics and produce a boxplot and an error bar chart.
Example 1.
Seven subjects have taken two tests, test X and test Y. Their raw scores are as
follow:
Obtain the mean, unbiased standard
deviation, and true variance for each test.
SPSS for Windows
A. Open a
new data editor window.
B. Enter data
C. Choose the Descriptives
procedure
From
the menus choose: Analyze / Descriptive Statistics / Descriptives.
To know
more about Procedure Descriptives, click on the Help button as shown above.
The help topic will appear.
Close the help topic window when
you are done.
Next, select the variables, X and
Y, to be analyzed. (Move X and Y to the Variable(s) Box. Click OK.
D. How to list cases
From
the menus choose: Analyze / Reports / Case summaries
The
variables in your data file will be displayed on the source list. Select the
variables for which you want case listings. Click OK. The results will appear
in the Viewer window.
E. Print
the case listing: File / Print. Click on OK.
SPSS Printout
A. For
Test X
a. What is the mean and
unbiased standard deviation for test X? (Mx = 3, SD = 1.291)
b. True and Unbiased Variances
(a) If the researcher considers
the subjects to be a random sample from the target population (e.g., all the
fourth graders in a school district) the unbiased variance is a correct computational formula for the
estimate of the variance of this larger population. Notice that SPSS computes
only
unbiased
variances and unbiased standard deviations by default.
(b) If
the researcher only wants to compute the variance to describe a set of scores,
the true variance is an appropriate computational formula to use.
B. For Test Y
What is
the mean and unbiased standard deviation for test Y?
(My = 3, SD = 1.414)
C. Which distribution, X or Y, is
more variable? Y. Examine the (unbiased) standard deviation. 1.414 > 1.291
Example 2.
Produce a boxplot
that shows the
median and variability of the current salary in each job category.
Open the
data file.
Open an
existing data file -- Employee data. From the menus of the Data Editor window
choose: File / Open / Data. Click on Employee data from the list. Click
Open.
What is a
boxplot?
From the
menus, choose Help / Topics. Type boxplots in the text box. Click Display.
select Box Plots, defined. Click Display. The definition can be shown below.
Create a
boxplot.
From the
menus choose: Graphs / Boxplot.
Type of
the boxplot: Simple. It is the default.
Data in
Chart Are: Summaries for groups of cases
Click
Define.
Variable: Select Current Salary [salary]
Category Axis: Select Employment Category [jobcat]. Click OK.
Boxplot
The
boxplot provides a vertical view of the data. A boxplot plots the 25th
percentile, the median (the 50th percentile), the 75th percentile, and outlying
or extreme values.
Variability
The length of the box represents the difference
between the 25th and 75th percentiles. From the length of the box, you
can determine the
variability. The
larger the box, the greater the spread of the data. Which group has the
largest variability?
(Manager)
Central
Tendency
The
horizontal line inside the box represents the median. If the
median
is not in the center of the box,
the distribution may be skewed.
Whiskers
Whiskers: Draw lines from the
ends of the box to the largest and smallest values that are not outliers.
These lines are called whiskers (Refer to SPSS User's Guide).
Identify Outliers and Extremes
Case numbers are used to label
outliers (o) and extremes (*). The boxplot shown above detected outliers and
extremes. The outliers are cases with the values between 1.5 and 3
box-lengths from the 75th percentile or 25th percentile. The extreme values
are cases with the values more than 3 box-lengths from the 75th percentile
or 25th percentile (Refer to SPSS User's Guide).
What
might cause the outliers? Once you know the reason, you may correct the problem.
The
possible reasons may be
(1)
Recoding errors.
(2)The sample was drawn from a skewed population distribution.
(3) The sample was not drawn from the same population.
(4) The small sample size.
Optional Reading:
Outliers and
Data Screening by Mike Wulder
Example 3.
Produce an error
bar chart that shows the mean and
variability of the current salary in each job category.
Use the same data file -- Employee
data.
From the menus choose: Graphs / Error Bar
Type of
the error bar: Simple. It is the default.
Data in
Chart Are: Summaries for groups of cases
Click
Define
Variable: Select Current Salary [salary]
Category Axis: Select Employment Category [jobcat].
Bars
Represent: Click the down arrow sign. Another list will appear. Click the down
arrow sign. You will find Standard deviation. Click Standard deviation.
The multiplier box will be available now. The default is 2 (two standard
deviations around the mean). Click OK.
This chart shows the mean values
with the bar that
stretch two standard deviations on either side of the mean.
The
current salary in the manager category has the largest variability and the
highest mean. The current salary in the custodial category has the least
variability.
Example 4.
Produce a
clustered boxplot of beginning salaries
for males and females across three job categories.
Use
the same data file -- Employee data.
From the
menus choose: Graphs / Boxplot.
Type of
the boxplot: Click the picture next to Clustered.
Data in
Chart Are: Summaries for groups of cases
Click
Define
Variable: Select Beginning Salary [salbegin].
Category Axis: Select Employment Category [jobcat].
Define
Clusters by: Gender (gender) Click OK.
Compare
the
beginning salaries for males and
females across three job categories. What do you observe?
What to look for (Refer
to SPSS for Windows Base System User's Guide)
Boundaries of the box:
The 25th percentile and the 75th percentile. The larger the box, the greater the
spread of the data.
The
horizontal line inside the box represents the median.
The median is defined as that point below which fifty percent of the cases fall.
If the median is
not in the center of
the box, the distribution is skewed.
a. If
the median is closer to the
top
of the box, the distribution may be negatively skewed. Why? The scores tend
more
toward the
higher
end of the distribution.
b. If the median is closer to the
bottom of the box, the
distribution may be positively skewed, Why?
The scores tend more
toward the
lower
end of the distribution.
Whiskers:
Draw lines from the boundaries of the box to the largest and smallest
values that are not outliers. These lines are called whiskers.
If whiskers are of unequal length, the
distribution may be skewed.
Outliers and Extremes: Case
numbers are used to label outliers and extremes. The extreme values are
cases (*) with the values more than 3 box-lengths from the boundaries of the
box. The outliers are cases (o) with the values between 1.5 and 3 box-lengths
from the boundaries of the box.
What
might cause the outliers?
They may
be due to recording errors, due to the sample being drawn from a skewed
population distribution or being drawn from different populations, or simply due
to the small sample size. Once you know the reason, you may correct the problem.
Optional Reading:
Outliers and
Data Screening by Mike Wulder