SW388R6

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Measures of Central Tendency

• Mean
• Median
• Mode
• Typical value/preferred measure

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Measures of Central Tendency

• Measures of central tendency use a single number
  or data value to indicate a typical or
  representative value for the entire distribution.
• The preferred measure of central tendency is
  based on the level of measurement of a variable.
• Statistically, measures of central tendency are
  values that minimize the amount of error we would
  make in stating a representative value for all
  cases in a distribution.

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The Mean

• The mean is the average of all data values.
• The mean should only be computed for interval
  level variables.
• The mean is the preferred measure of central
  tendency for an interval level variable, unless
  the variable is badly skewed.

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The Median

• The median is the value that divides a
  distribution of sorted data values into two
  halves.
• The median should only be computed for ordinal
  and interval level variables.
• The median is the preferred measure of central
  tendency for ordinal level variables and interval
  level variables that are badly skewed.

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The Mode

• The mode is the data value with the largest
  number of cases in a frequency distribution.
• The mode may be computed for any level variable.
  If the variable is a continuous interval level
  variable, values should be grouped before
  computing the mode.
• The mode is the preferred measure of central
  tendency for nominal level variables.
• Not all distributions have a single mode. Some
  have two (bi-modal) or more (multi-modal).

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Request measures of central tendency
To compute measures of central tendency in SPSS,
select the Descriptive Statistics Frequencies
command from the Analyze menu.
Other SPSS procedures also calculate measures of
central tendency, but include a somewhat
different list. For example, the Descriptives
procedure calculates the mean and median, but not
the mode,
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Select the variable for the analysis
First, move the variable sex to the list of
Variable(s).
Second, click on the Statistics button to select
the specific measures we want.
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Request the specific measures
First, mark the checkboxes for Mean, Median, and
Mode. Since the SPSS output for this command is
not unwieldy, it is easiest to request all
measures that we might possibly need.
Third, click on the Continue button to complete
the specifications.
Second, mark the checkbox for Skewness which we
will need for interval level variables.
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Request measures of central tendency
To complete the request for measures of central
tendency, click on the OK button.
To reduce the output, you can clear the Display
frequency tables checkbox and the frequency
distribution will not be included in the output.
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SPSS output for measures of central tendency
The table of Statistics contains the measures of
central tendency and skewness that we will need
to solve our homework problems.
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Problem 1

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic?
• The mode for the variable "RESPONDENTS SEX" is 2.
• 1. True
• 2. False
• 3. Incorrect application of a statistic

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Solution 1
The mode requires that the variable be nominal,
ordinal, or interval. The variable "RESPONDENTS
SEX" is nominal, satisfying this requirement.
The mode for the variable "RESPONDENTS SEX" is
2. The answer to the question is true.
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Problem 2

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic?
• The median for the variable "RESPONDENTS SEX" is
  2.00.
• 1. True
• 2. False
• 3. Incorrect application of a statistic

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Solution 2
The median requires that the variable be ordinal
or interval. The variable "RESPONDENTS SEX" is
nominal, which does not meet this
requirement. The answer to the question is
incorrect application of a statistic.
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Problem 3

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic?
• The typical survey respondent was a female.
• 1. True
• 2. False
• 3. Incorrect application of a statistic

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Solution 3 part 1
The mode represents a typical value for a
nominal level variable The mode for the variable
"RESPONDENTS SEX" is 2.
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Solution 3 part 2
Looking at the value labels for the variable
RESPONDENTS SEX, we see that the modal value of
2 corresponds to the label FEMALE. The answer
to the question is true.
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Problem 4

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic?
• The mode for the variable "RS HIGHEST DEGREE" is
  1.
• 1. True
• 2. False
• 3. Incorrect application of a statistic

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Solution 4
The mode for the variable "RS HIGHEST DEGREE" is
1. The value 1 corresponds to the label HIGH
SCHOOL. The answer to the question is true.
The mode requires that the variable be nominal,
ordinal, or interval. The variable "RS HIGHEST
DEGREE" is ordinal, satisfying this
requirement. It is legitimate to report the mode
for an ordinal level variable, though thit is not
the preferred measure of central tendency.
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Problem 5

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic?
• The median for the variable "RS HIGHEST DEGREE"
  is 1.41.
• 1. True
• 2. False
• 3. Incorrect application of a statistic

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Solution 5
The median for the variable "RS HIGHEST DEGREE"
is 1.00, not 1.41. The answer to the question is
false.
The median requires that the variable be ordinal
or interval. The variable "RS HIGHEST DEGREE" is
ordinal, satisfying this requirement.
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Problem 6

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic?
• The mean for the variable "AGE OF RESPONDENT" is
  32.
• 1. True
• 2. False
• 3. Incorrect application of a statistic

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Solution 6
The mean for the variable "AGE OF RESPONDENT" is
45.99, not 32. The answer to the question is
false.
The mean requires that the variable be interval.
The variable "AGE OF RESPONDENT" is interval,
satisfying this requirement.
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Problem 7

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic?
• The typical survey respondent was 45.99 years of
  age.
• 1. True
• 2. False
• 3. Incorrect application of a statistic

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Solution 7
The mean represents a typical value for an
interval level variable whose distribution is not
badly skewed. The skewness for this variable is
0.595. The mean for the variable "AGE OF
RESPONDENT" is 45.99. The answer to the question
is true.
The skewness for this variable is 0.595, meeting
the rule of thumb that the skewness for the
distribution of a variable is between -1.0 and
1.0, the variable is not badly skewed.
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Problem 8

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic?
• The mean for the variable "NUMBER OF BROTHERS AND
  SISTERS" is 3.52.
• 1. True
• 2. False
• 3. Incorrect application of a statistic

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Solution 8
The mean for the variable "NUMBER OF BROTHERS AND
SISTERS" is 3.52. The answer to the question is
true.
The mean requires that the variable be interval.
The variable "NUMBER OF BROTHERS AND SISTERS" is
interval, satisfying this requirement. Though
the mean is not the preferred measure of central
tendency for a variable that is badly skewed, it
is still a legitimate statistic to compute.
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Problem 9

• Based on the dataset GSS2000.SAV, is the
  following statement true, false, or an incorrect
  application of a statistic?
• The typical survey respondent had 3.52 brothers
  and sisters.
• 1. True
• 2. False
• 3. Incorrect application of a statistic

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Solution 9
While the mean is generally the preferred
measure of central tendency for an interval level
variable, the median is the preferred measure if
the distribution of an interval level variable is
badly skewed. The median of 3.00 should be
reported as the measure of central tendency which
best represents a typical data value for this
variable. The answer to the question is false.
The skewness for this variable is 2.391, meeting
the rule of thumb that the distribution of a
variable is badly skewed if its skewness is less
than or equal -1.0 or greater than or equal to
1.0.
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Steps in solving central tendency problems - 1
The following is a guide to the decision process
for answering central tendency homework
problems. If the problem asks about a specific
measure of central tendency
Is the level of measurement required for the
statistic satisfied?
No
Incorrect application of a statistic
Yes
False
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Steps in solving central tendency problems - 2
If the problem asks about the preferred measure
of central tendency (typical score)
Is the variable nominal level?
Is the variable ordinal level?
Is the variable interval level?
No
No
Is the value of the mode correct?
Is the value for the median correct?
Is the variable badly skewed?
Yes
False
No
No
Is the value for the mean correct?
False
True
False
True
No
True
False