Transcript SPSS2CentralTendencyandDispersion

SPSS 2: Central Tendency and
Dispersion
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Frequency Distribution
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A summary which gives
an account of the
frequency of answers in
each category of
response to a question
(e.g., how many “yes,”
“no,” or “maybe”
answers; how many
“strongly agree,”
“agree,” “neither agree
nor disagree,” “disagree,”
or “strongly disagree,”
etc. The elements
described are the
question, the categories
or response alternatives
of the question, and the
distribution of responses
Example from
our previous
practice2 data
set; this is a
perfectly
heterogeneous
distribution
(each scale
value has the
same number
of people
endorsing it)
Frequency Distributions
Example from the full St.
Barnabas data set
Note that there is a missing
case-one of the 140 Ss did not
answer this item; hence we
have percent and “valid
“percent”. Note also that this
is a much more homogeneous
distribution (has an obvious
central tendency)
What would a narrative account of this
frequency distribution say if you were writing it
up for a paper?
Conventions in the Presentation of
Tabled Frequency Data
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Categories of the response variable are listed in the left-most
column
Frequencies are listed to the right
Table is numbered and provided with a title
Frequencies are listed in ascending or descending order
Table 3
Student and Instructor Participation in Finland and U.S. Conferences
Conventions in the Presentation of
Tabled Frequency Data, cont’d
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It is convenient to the reader to provide both the raw frequencies and
the percentages for each category, plus the totals
When there are many potential categories, for example, number
grades or annual income, it is customary to group the data into
categories like 90-100, $10,000-15,000, etc. in a grouped frequency
distribution: class intervals
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With class intervals we construct a range that has an upper and lower limit.
For example, for the class interval 80-90 the actual values are 79.5-89.5.
Each class interval has a midpoint which stands for the class
Some frequency tables use the letter f to stand for frequencies and the
letters cf to stand for cumulative frequencies (total number of cases
having a score of x or lower) in the column headings, and similarly
with % and c% for cumulative data
For most grouped data, class intervals of equal size are used, but in
some cases, for example income data, intervals of unequal size may be
more appropriate
Comparing Two Distributions:
Skewness and Kurtosis
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70
60
50
40
30
20
Std. Dev = 38.52
10
Mean = 104.8
N = 400.00
0
0
0.
23 .0
0
22 .0
0
21 .0
0
20 .0
0
19 .0
0
18 .0
0
17 .0
0
16 .0
0
15 .0
0
14 .0
0
13 .0
0
12 0
0.
11 .0
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10
.0
90
.0
80
.0
70
.0
60
.0
50
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Compare the histograms for the
distribution of “horsepower” and “time to
accelerate” from the cars.sav data.
Imagine that we had created a figure called
a frequency polygon for each variable
which created a line connecting the midpoint of each class interval at the height on
the y axis representing the frequencies
within the categories. The distribution on
the bottom is much more symmetrical,
while the one on top is skewed. It is said
to be positively skewed. Most cases took
on lower values of the variables, and the
“tail” of the frequency polygon would be
longer and flatter on the right hand side of
the x axis
Distributions are also described in terms of
their peakednesss: e.g., leptokurtic,
platykurtic,. A distribution which is more
peaked near the mean is called leptokurtic.
Horsepower
70
60
50
40
30
20
Std. Dev = 2.82
10
Mean = 15.5
N = 406.00
0
8.0
10.0
9.0
12.0
11.0
14.0
13.0
16.0
15.0
18.0
17.0
20.0
19.0
22.0
21.0
Time to Accelerate from 0 to 60 mph (sec)
24.0
23.0
25.0
Kurtosis Examples
Horsepower
Vehicle Weight (lbs.)
60
50
Time to Accelerate from 0 to 60 mph (sec)
70
70
60
60
50
50
40
40
30
30
40
30
N = 406.00
0
Std. Dev = 38.52
10
Mean = 104.8
N = 400.00
0
20
Frequency
Mean = 2969.6
Frequency
20
Std. Dev = 849.83
10
Mean = 15.5
N = 406.00
0
8.0
10.0
9.0
Horsepower
Vehicle Weight (lbs.)
Std. Dev = 2.82
10
0
0.
23 .0
0
22 .0
0
21 .0
0
20 .0
0
19 .0
0
18 .0
0
17 .0
0
16 .0
0
15 .0
0
14 .0
0
13 .0
0
12 0
0.
11 0
0.
10
.0
90 0
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80 0
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70 0
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60 0
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50
.0
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52
.0
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48 0
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44
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40 0
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36
.0
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32 0
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28
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24 0
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16
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12
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80
Frequency
20
12.0 14.0
11.0
13.0
16.0 18.0
15.0 17.0
20.0
22.0 24.0
19.0 21.0
23.0
25.0
Time to Accelerate from 0 to 60 mph (sec)
Engine Displacement (cu. inches)
Statistics
100
80
N
60
Frequency
40
20
Std. Dev = 105.21
Mean = 194.0
N = 406.00
0
0
0.
45 .0
5
42 .0
0
40 .0
5
37 .0
0
35 .0
5
32 .0
0
30 .0
5
27 .0
0
25 .0
5
22 .0
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20 .0
5
17 .0
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15 .0
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12 .0
0
10
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75
.0
50 0
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25
0
0.
Engine Displacement (cu. inches)
Mean
Skewness
Std. Error of Skewnes s
Kurtos is
Std. Error of Kurtos is
Valid
Mis sing
Vehicle
Weight (lbs.)
406
0
2969.56
.468
.121
-.752
.242
Horsepower
400
6
104.83
1.044
.122
.591
.243
Time to
Accelerate
from 0 to 60
mph (sec)
406
0
15.50
.211
.121
.389
.242
Engine
Dis placement
(cu. inches )
406
0
194.04
.692
.121
-.791
.242
Miles per
Gallon
398
8
23.51
.457
.122
-.511
.244
Note that the distributions that are more peaked near the mean
have larger positive values of kurtosis
How to Obtain a Frequency
Distribution in SPSS
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Download today’s data set here
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In the Data View, you will go to Analyze/Descriptive Statistics/Frequencies; a dialog box
will appear. Make sure the “display frequency tables” box is checked.
On the left-hand side, you will highlight the variable for which you want to create a
frequency table and press the right arrow button to move the variable to the right-hand
side. (buttons work both ways) If you want a chart, you will click the charts tab and
select a chart type such as “pie” or “bar” and click continue. Then you will click OK in the
frequencies dialog box. Your output will appear in a separate Output viewer.
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A Sample Questionnaire
1. Are you ______male ______female
2. Are you (please mark [X] one.)
_____African American/Black _____Chinese/Chinese American _____Korean/Korean
American _____Mexican/MexicanAmerican _____White, nonHispanic/Caucasian (please specify ethnicity, for example, Irish, Swedish, etc.)
_____Other, please specify _________________________
For each statement below, decide whether you disagree or agree with the statement
using a scale ranging from strongly disagree to strongly agree.
3. I hesitate to use a computer for fear of making mistakes I can’t correct.
strongly agree
agree
neither agree nor disagree
disagree
strongly disagree
4. I have avoided computers because they are unfamiliar and somewhat intimidating
to me.
strongly agree
agree
neither agree nor disagree
disagree
strongly disagree
5. I have difficulty in understanding the technical aspects of computers
strongly agree
agree
neither agree nor disagree
disagree
strongly disagree
Sample Questionnaire, cont’d
6. How often is someone to share a good thought, poems, or experiences with available to
you through e-mail or the Internet?
None of the time
A little of the time
Some of the time
Most of the time
All of the time
7. How often is someone who gives you encouragement when you feel down available to you
through e-mail or the Internet?
None of the time
A little of the time
Some of the time
Most of the time
All of the time
8. How often is someone who values your advice and opinion available to you through e-mail
or the Internet?
None of the time
A little of the time
Some of the time
Most of the time
All of the time
9. How often is someone who cares about you available to you through e-mail or the
Internet?
None of the time
A little of the time
Some of the time
Most of the time
All of the time
10. How often is someone you can count on to listen to you when you need to talk/write
available to you through e-mail or the Internet?
None of the time
A little of the time
Some of the time
Most of the time
All of the time
11. How often is someone to give you advice when asked available to you through e-mail or
the Internet?
None of the time
A little of the time
Some of the time
Most of the time
All of the time
Data from Sample Questionnaire
Let’s consider the data from the Lesson3.sav data set.
When you have opened the data set in the SPSS Data
Editor, go to Data View and then select View/Value Labels to
see what the numbers mean. Uncheck Value Labels to
return to the raw numbers
Creating a Frequency Distribution
from the Sample Data Set
Select the command path Analyze/Descriptive Statistics/Frequencies
Move all the variables but ID from left to right (click the first variable and then
hold down the shift key as you click the last variable to select a set of
adjacent variables), check “display frequency tables,” click on “charts”
and select “bar” under chart type and “frequencies” under chart values,
click “Continue”, then OK.
Sample from the Output Viewer
Check to see if you got these results:
Bar Charts for the Lesson3 Data
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Now obtain a frequency distribution for the Ethnicity variable and display it in a bar chart
Analyze/Descriptive Statistics/Frequencies; move the previous variable out of the
dialogue box and move Ethnicity in
Go to Charts and select Bar Charts and then click Continue, then OK
The SPSS Output Viewer
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Attributes of the Output Viewer
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Switch between viewer and data editor with tabs on the Windows menu
bar or use alt/tab combination
Note that there is a complete outline of the output viewer contents on
the left hand side; you can use it to navigate around the charts and
tables and to cut, copy and paste charts and tables as needed within
the output viewer
Notes are hidden; click on the “plus” in the outline to view them
Charts can be copied and pasted into applications like Word,
PowerPoint, etc. where they can be edited (different editing techniques
for Copy Object and Copy) or parts can be selected, copied and pasted
into new output file (.spo); can be exported as HTML,PDF, PPT doc/rtf,
excel, txt, image formats. etc
Output file can be saved to use later
To print a chart or table, highlight it with a mouse click and press the
Printer icon; choose to print either the selection or “all visible output”
Charts can be edited in the Chart Builder by double-clicking on the
chart object
Default Chart vs. Edited Chart
Old Chart
Edited Chart
Let’s have a chart beauty contest. Double-click on the chart
“Ethnicity” and beautify it in the Chart Editor. Highlight the bars to
apply color and shading, for example, edit the axis labels, or add a
Figure number and title. You can copy the chart and paste into other
documents or editors.
Exploring Charts and Graphs
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Alternatives for visualizing data are
available from the Graphs menu; use the
Legacy option to view the various types
(these are the older individual chart types
that are now available through a single
interface, the Chart Builder)
For example, click on the bar chart icon,
then examine the various options (simple,
clustered, stacked) for bar charts.
Make a bar chart for Gender in the
Lesson3.sav data file: Graphs/Legacy
Dialogs/Bar/. Select the simple option and
“summaries for groups of cases” and click
Define. Highlight the “Gender” variable
and move it to the right into the Category
Axis box by clicking the right arrow, then
click OK. Can leave out missing values
with the Options dialog.
Note that the response categories of the
variable are along the x axis and the count
within categories are along the y axis
Stacked Bar Charts with Two
Variables
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Now make a bar chart in
which you compare men
and women with respect
to ethnicity
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Select Graph/Legacy
Dialogs/Bar. Choose
the “stacked” option,
“summaries for groups
of cases,” then press
the Define button.Move
“Gender” to the
category axis and the
ethnicity variable to the
“Define stacks by” box.
Select “N of cases” and
press OK.
Create a Frequency (or Percentage)
Polygon
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Let’s make a frequency polygon for
the variable from lesson3.sav
“someone cares about me”
Go to Graphs/Chart Builder
Click OK to get rid of the variable
definition box
Click reset to clear the preview
window
Select Line
Drag the simple line into the preview
window
Select the “someone cares about me”
variable and drag it onto the x-axis
rectangle
Click OK
Rerun the procedures but this time
we will visualize the percentages
Do everything as above but use the
Element Properties drop-down
Statistics menu to select
percentages, then Apply, then OK
Cross-tabulation
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Another way to obtain a
quick comparison of two
groups with respect to
some variable is to
perform a cross-tabulation
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Cross-tabulate “Gender”
and “Ethnicity” with
Analyze/ Descriptive
Statistics/ Crosstabs
Move gender into the
Row box and “Ethnicity”
into the Column box
Click on Cells and check
the Observed, Row,
Column, and Total boxes
Cross-tabulation, cont’d
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Distribution of Ethnicity across Gender categories There are missing values for two of the 9
subjects, so N=7. The colored dots show the appropriate values to compare. For example,
for men, 16.7 % were Chinese, none was Mexican, and 83.3% were White, but among
women, the respondents were distributed evenly (33.3% each) across ethnic categories.
Total males
Total females
Total N of subjects
Blue dots read across and compare men to women with respect to Ethnicity; pink dots read down and
compare Chinese/Mexican/White with respect to Gender
Scatter Plot
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What is the relationship
between the variables
“difficulty understanding” and
“there is usually somebody
online that values my advice”?
Do a scatter plot: Go to
Graphs/Legacy Dialogs/Scatter.
Select simple, then Define.
Move the first variable into the
y axis and the second into the
x axis box, then click OK. In
the Output viewer, double click
on the chart to bring up the
Chart Editor; go to Elements
and select “Fit Line at Total,”
then select “linear” and click
Close
Note: although there are nine subjects,
there are only five data points shown in the
scatter plot. Check the raw data to find out
why. (toggle the Value Labels to numbers
and compare comanx3 and modmos3)-look
for missing data and duplicate xy
coordinates)
View Distribution after Splitting the
Data on a Variable
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How do the responses for the variable “difficulty understanding the technical
aspects of computers” compare for men and women? Split the file on gender and
compare the resulting distributions. Go to Data/Split File. Select Compare
groups. Then select Gender and move it into the “groups based on” box and click
OK. Then Go to Analyze/Descriptive Statistics/Frequencies. Use reset to remove
previous variables. Highlight the “difficulty variable” and move it Variables Box
and click OK. Undo the split with the Data/Split File/ reset option so that further
analyses will be on the combined cases.
To restore the original order in the data editor viewer, you can sort the cases on
ID in ascending order by right clicking on the ID variable column and selecting
“sort ascending”
Difficulty Understanding
What sort of differences do
you think there appear to be
between males and females
with respect to this variable?
Gender
Male
Valid
Female
Valid
Strongly Agree
Agree
Neither
Total
Agree
Strongly Dis agree
Total
Frequency
1
4
1
6
2
1
3
Percent
16.7
66.7
16.7
100.0
66.7
33.3
100.0
Valid Percent
16.7
66.7
16.7
100.0
66.7
33.3
100.0
Cumulative
Percent
16.7
83.3
100.0
66.7
100.0
Recoding Data to Collapse
Categories of the Response Scale
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Under some circumstances we might wish to reduce the
number of response categories to a variable after we have
collected the data. To do so we need to recode the old
variables into new ones using SPSS.
Suppose that we want to collapse the agreement and
disagreement categories for variable companx1, so that you
only have three response categories, “agree,” “disagree,”
and “neither agree nor disagree:”
OLD Item companx1: I hesitate to use a computer for fear of
making mistakes I can’t correct.
strongly agree (1)
disagree(4)
agree (2)
neither agree nor disagree(3)
strongly disagree (5)
NEW Item newcomp1: I hesitate to use a computer for fear of
making mistakes I can’t correct.
agree (1)
neither agree nor disagree(2)
disagree(3)
Recoding Data to Collapse Categories of
the Response Scale, cont’d
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In SPSS, go to the the Transform menu and select
Recode/Into Different Variables
In the dialog box that opens, highlight “fear of
mistaking mistakes” and use the black arrow to move
it into the Input Variable > Output Variable box
Give the variable to be recoded a new name and label
in the Output Variable boxes: give it the name
“newcomp1” and the label “recoded Fear of Making
Mistakes”, then click on the Change button
Click on Old and New Values button to open up the
Recode into Different Variables: Old and New Values
Box to enter information as to how you want to
regroup the response categories
Recoding Data to Collapse Categories of
the Response Scale, cont’d, 2
1. Click the Range button under Old
Value and enter the old values
1 through 2 (the values for
“strongly agree” and “agree”
from the old variable) in the
boxes, then enter the new
value of 1 in the New Value
box. Use the Add button
2. Click the Value button under Old
Value and enter the value 3,
which was the value of the old
“neither agree nor disagree”
category, then enter the new
value of 2 in the New Value box
and use the Add button
3. Click the Range button under Old
Value and enter the old values
4-5 (the values for “disagree”
and “strongly disagree” from
the old variable) in the boxes,
then enter the new value of 3 in
the New Value box. Use the
Add button and then click on
Continue, and then OK in the
next dialog box
Recoding Data to Collapse Categories of
the Response Scale, cont’d, 3
• Go to the Variable View and find the new
variable, newcomp1. Review the variable
definitions and establish new value labels:
Frequency Distributions for Old and
New Variables
In Data View, go to
Analyze/Descriptive
Statistics/Frequencies to
generate frequency
distributions for the old
and new variables.
Use the reset button to
clear old instructions.
Highlight the old and new
variables in the list on
the left and move them
to the Variable window.
Click the Chart button,
select bar chart and
frequencies, Continue,
then OK.
Compare Frequency Distributions of
Old and New Variables
• Compare the frequency distributions
for the old and new variables:
Fear of Making Mistakes
Valid
Agree
Dis agree
Strongly Dis agree
Total
Frequency
2
5
2
9
Percent
22.2
55.6
22.2
100.0
Valid Percent
22.2
55.6
22.2
100.0
Cumulative
Percent
22.2
77.8
100.0
recoded Fear of Making Mistakes
Valid
Agree
Dis agree
Total
Frequency
2
7
9
Percent
22.2
77.8
100.0
Valid Percent
22.2
77.8
100.0
Cumulative
Percent
22.2
100.0
Compare Frequency Distributions of
Old and New Variables, cont’d
• Compare the bar charts for visual
evidence of the effect of recoding:
Measures of Central Tendency
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The measures of central tendency most
often seen in quantitative studies are the
mean
or M, the median (Md or Mdn),
and the mode (Mo).
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Mean-the numerical average of a set of
responses. Measure which is most stable for
random samples, which makes it suitable for
making estimates about populations from
samples. It has the property that the sum of
the deviations of the raw scores from it equals
zero.
Median- the response value for which there are
an equal number of responses both below and
above it (e.g., larger or smaller). Used with
ordinal or numerical variables
Mode-the most frequently selected (commonly
occurring) response category. The only
measure for nominal level variables but can be
used with scaled data. A distribution can be
bimodal, trimodal, etc
Identify the mean, median and mode for
the variable companx3. How close are
these three statistics? Which best
represents central tendency?
Scores have
been
reordered
from high to
low
Median versus Mode when Outliers
are Present
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The median, unlike the mean, is
not affected by extreme scores or
outliers. Consider the data on the
right, where scores on the variable
could range from 1 to 9. Which
statistic, the median or mean, best
represents central tendency? Go
here for an interactive
demonstration of the effect of skew
in the distribution on measures of
central tendency. Note that
regardless of skew the relative
positions of mode, median and
mean remain the same, with mode
at the peak of the curve, then
median, then mean, going in the
direction of the flatter tail.
Median with Even Number of Cases
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The median case is found by
adding 1 to the number of cases
and dividing by 2. Thus if there
are 11 cases the 6th case is the
median case (because there are
five cases above and five cases
below it). The median is the
value associated with that case.
When there are an even number
of cases, the median will take a
value half-way between the two
cases on either side of a
hypothetical “median case”. What
is the median response of this
group of ten responses?
Using SPSS to Find Measures of
Central Tendency
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Go to Analyze/
Descriptive Statistics/
Frequencies
Highlight the variable of
interest (say companx1,
“fear of making
mistakes,” and use the
arrow to move it into the
Variable window
Click on the Statistics
button and check Mean,
Median and Mode; click
Continue and then OK
You can also get these
measures of central
tendency and more from
Analyze/Descriptive
Statistics/Explore
Statistics
Fear of Making Mis takes
N
Valid
9
Mis sing
0
Mean
3.7778
Median
4.0000
Mode
4.00
Measures of Dispersion or
Variability
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Dispersion is the degree or amount of variability in a
set of responses to a quantitative measure such as a
questionnaire item. Measures like the standard
deviation add information to indicators of central
tendency such as the mean because they tell us how
homogeneous or heterogeneous the sample was and
allow us to estimate the likelihood of error in
estimating population parameters from a sample.
Measures of dispersion vary as a function of the level
of measurement of the data. Dispersion measures
include the index of qualitative variation, the range
and interquartile range, and the standard deviation.
Measures of Dispersion as a
Function of Level of Measurement
Measures of Dispersion
Categorical Variables
Nominal
Index of
Qualitative
Variation
Interval and Ratio Variables)
Ordinal
Range and
Interquartile
Range
Standard
Deviation
Range and
Interquartile
Range
Adapted from Kendrick,
p.214
Nominal Data: the IQV
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Index of qualitative variation
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Ranges in value from 0 to 1. Values approaching 0 mean there is little
variability in the sample; values approaching 1 mean that the sample
is very heterogeneous
Represents the differences between observed and maximum possible
differences among the respondents to the categories of a variable
Consider the variable “ethnicity”. Here’s how our 9 respondents break down (in
the data editor select View/Value Labels:
0
African American/Black
2
Chinese/Chinese American
0
Korean/Korean American
1
Mexican/Mexican American
6
White, non-Hispanic/Caucasian
0
Hispanic, not Mexican
0
Other
To calculate the IQV we will use only the categories with nonmissing data:
2
1
6
Chinese/Chinese American
Mexican/Mexican American
White, non-Hispanic/Caucasian
IQV, cont’d
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IQV= total observed differences divided by maximum
possible differences
To calculate the numerator you have to have sum p possible
products, where p is the number of “populated” categories in
the nominal scale, in our case 3
2
1
6
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Chinese/Chinese American
Mexican/Mexican American
White, non-Hispanic/Caucasian
To compute the first product, multiply the first frequency, 2, by
the sum of the frequencies below it. (p1= 2 (1+6)=14)
To compute the second product, multiply the second frequency,
1, by the sum of the frequencies below it (p2 =1(6)= 6)
The numerator is the sum of this total observed differences:
6+14 =20
Doesn’t matter in what order you list the categories
IQV, cont’d, 2
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The denominator is equal to K (K-1)/2,
where K is the number of populated
categories, times the square of N, the
∑fifj (in this case 20)
number of subjects, divided by K. The
denominator equals 3 (2)/2 times 9/3
(K (K-1)/2) (N/K)2
squared, or 3x9=27.
The value of the IQV is 20/27 or .74, which
is fairly heterogeneous. Does this seem an
accurate reflection? What would happen if
you included the four unpopulated
categories in the equation?
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The numerator would remain the same, but
the denominator would be 7(6)/2 X (9/7)2,
or about 34, which would reduce the IQV to
about .59 (more homogeneous). Does this
seem more accurate?
The problem is that there is no distribution of
values of IQV to consult to see how likely
you are to get a result of .74 or .59
Range and Interquartile Range
•
•
Range: the difference between
the lowest and highest value in
a frequency distribution. For
example, for the variable on
the right the range is 6 (8-2).
The range is really of interest
only in relation to a measure of
dispersion. For example, a
standard deviation from the
mean of 3.2 is pretty big when
the range is 6 but not so big if
the range is 100. Range is
unduly affected by outliers
Interquartile Range: the range
of scores for the middle 50% of
a distribution
•
The appropriate measure of
dispersion when the median
rather than the mean is
reported
Interquartile Range
Statistics
•
406
0
4408
2222.25
2811.00
3614.75
50
40
30
20
Open the data set cars.sav and look at
the variable “vehicle weight.” What
are the range and interquartile range
for this variable? Go to Analyze/
Descriptive Statistics/ Frequencies and
move the vehicle weight variable into
the Variable box. Click on Statistics
and select Range, Quartiles, click
Continue, then OK.
Std. Dev = 849.83
10
Mean = 2969.6
0
N = 406.00
.0
00
52
.0
00
48 0
.
00
44
.0
00
40 0
.
00
36
.0
00
32 0
.
00
28
.0
00
24 0
.
00
20
.0
00
16
.0
00
12
0
0.
80
Frequency
Vehicle Weight (lbs .)
N
Valid
Mis sing
Range
Percentiles 25
50
75
Vehicle Weight (lbs.)
60
Vehicle Weight (lbs.)
IQR = Q3 – Q 1 =3614.75 –
2222.25 = 1392.5
Boxplot
You can further visualize the data by
creating a boxplot (with the cars
dataset open, go to Graphs/Boxplot.
Highlight Simple in the dialog box,
select “summaries of separate
variables”, and click on Define. Move
the variable “vehicle weight” into the
“boxes represent” window and click
OK. The black line represents the
median. The beige box represents
the cases in the interquartile range.
The larger the box relative to the
range, the more heterogeneous the
distribution
Variance
•
•
The variance (s2) is the sum of the squared deviations from the sample
mean, divided by N-1 where N is the number of cases. (Note: in the
formula for the population variance, the denominator is N rather than N-1)
Reopen lesson3.sav and determine the variance, mean, range and
standard deviation (square root of the variance) of the three computer
anxiety variables using Analyze/Descriptive Statistics/Frequencies. All
other things being equal, the larger the variance, the greater the
heterogeneity of the sample. Which measure seems to have drawn the
most homogeneous responses?
Formula for sample
variance
Standard Deviation and Normal
Distribution
•
•
The standard deviation is the square root of the
variance. It is a measure which is very useful
because it allows you to make comparisons
between samples with respect to their variability
(how much a respondent from the sample typically
departs from the mean). The size of the SD is
generally about one-sixth the size of the value of
the range
The standard deviation becomes meaningful in the
context of the normal distribution. In a normal
distribution
•
•
•
The mean, median and mode of responses for a
variable coincide
The distribution is symmetrical in that it divides into
two equal halves at the mean, so that 50% of the
scores fall below the mean and 50% above it
68.26 % of the scores in a normal distribution fall
within plus or minus one standard deviation of the
mean. 95.44% fall within 2 SDs
•
•
Thus we are able to use the SD to assess the relative
standing of a score within a distribution, to say that it is 2
SDs above or below the average, for example
The normal distribution has a kurtosis equal to zero
Finding the Variance and Standard
Deviation with SPSS
•
Open the cars.sav file. Use the command Analyze/ Descriptive Statistics/
Frequencies, hit the reset button to clear the Variables box, and move the
variable “time to accelerate” into the Variables box. Click on Statistics and
standard deviation, range, variance, mean, skewness, kurtosis, and
quartiles, then click Continue and then OK. Note that the standard
deviation is about 1/6 the size of the range, which is typical of a normal
distribution
Variance
Standard
deviation
(s or SD-if SD
(population)
then N in the
denominator)
Histogram with Superimposed
Normal Curve
•
•
Next, create a histogram by the
command Graphs/ Chart Builder.
Bypass the first dialog Highlight the
variable “time to accelerate” and
drag the simple histogram icon into
the preview window. Drag the “time
to accelerate” variable onto the X
axis. In the Element Properties
window check Display Normal Curve
and then click “Apply” and then OK.
Note that there is a curve
superimposed on the histogram.
This is what a normal distribution of
a variable with the same mean and
standard deviation would look like.
Is the obtained sample distribution
skewed? Is the distribution more
platykurtic than the normal
distribution? (see slides 5-6)
Definition: “Kurtosis is a measure of whether
the data are peaked or flat relative to a normal
distribution. That is, data sets with high kurtosis
tend to have a distinct peak near the mean,
decline rather rapidly, and have heavy tails. Data
sets with low kurtosis tend to have a flat top near
the mean rather than a sharp peak. A uniform
distribution would be the extreme case.”