Measures Of Variability
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SW388R6 Data Analysis and Computers I
Measures of Variability
Slide 1
Standard Deviation and Variance Range Index of Qualitative Variation Preferred measure
SW388R6 Data Analysis and Computers I
Measures of Variability
Slide 2
Measures of variability (also called measures of dispersion) use a single number or data value to indicate variability, or differences between individual scores and the measure of central tendency.
Measures of variability are paired with measures of central to describe or summarize a distribution. Like measures of central tendency, the preferred measure of variability is based on the level of measurement of a variable.
Statistically, measures of variability are inidcators of the errors we making in using a measure of central tendency as the best estimate of the data value for all subjects in a distribution.
SW388R6 Data Analysis and Computers I
The Standard Deviation and Variance - 1
Slide 3
The standard deviation is the measure of variability that is paired with the mean as the measure of central tendency.
The difference between each case in a distribution and the mean is called the deviation. The sum of all the deviations is equal to zero.
If we add up the squared deviations for all cases in a distribution, we produce a measure of error called the total sum of squares. If we divide the total sum of squares by the number of cases, we produce the variance. The square root of the variance is the standard deviation.
SW388R6 Data Analysis and Computers I
The Standard Deviation and Variance - 2
Slide 4
Standard deviation and variance should only be computed for interval level variables.
The standard deviation (and variance) are the preferred measure of central tendency for an interval level variable, unless the variable is badly skewed. The standard deviation is more frequently used.
SW388R6 Data Analysis and Computers I
The Range
Slide 5
The range is the the difference between the largest and smallest value in a distribution.
The range should only be computed for ordinal and interval level variables.
The range is the preferred measure of central tendency for ordinal level variables and interval level variables that are badly skewed.
SW388R6 Data Analysis and Computers I
The Index of Qualitative Variation
Slide 6
The Index of Qualitative Variation (IQV) is a measure of the distribution of cases not in the modal category.
The IQV 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 IQV is the preferred measure of central tendency for nominal level variables.
We will not use the IQV because SPSS does not compute it.
SW388R6 Data Analysis and Computers I
Request measures of variability
Slide 7
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,
To compute measures of central tendency in SPSS, select the Descriptive Statistics | Frequencies command from the Analyze menu.
SW388R6 Data Analysis and Computers I
Select the variable for the analysis
Slide 8
First, move the variable “sex” to the list of Variable(s).
Second, click on the Statistics… button to select the specific measures we want.
SW388R6 Data Analysis and Computers I
Request the specific measures
Slide 9
First, mark the checkboxes for Standard Deviation, Variance, and Range. Since the SPSS output for this command is not unwieldy, is 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.
SW388R6 Data Analysis and Computers I
Complete request for measures of variability
Slide 10
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.
SW388R6 Data Analysis and Computers I
SPSS output for measures of variability
Slide 11
The table of Statistics contains the measures of variability and skewness that we will need to solve our homework problems.
SW388R6 Data Analysis and Computers I
Problem 1
Slide 12
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The range for the variable "RS HIGHEST DEGREE" is 4. 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 1
Slide 13
The range requires that the variable be ordinal or interval. The variable "RS HIGHEST DEGREE" is ordinal, satisfying this requirement.
The range for the variable "RS HIGHEST DEGREE" is 4. The answer to the question is true.
SW388R6 Data Analysis and Computers I
Problem 2
Slide 14
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The standard deviation for the variable "RS HIGHEST DEGREE" is 1.16. 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 2
Slide 15
The standard deviation requires that the variable be interval. The variable "RS HIGHEST DEGREE" is ordinal, which does not meet this requirement. The answer to the question is incorrect application of a statistic.
SW388R6 Data Analysis and Computers I
Problem 3
Slide 16
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The range of 4 best represents the variability of the variable "RS HIGHEST DEGREE". 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 3
Slide 17
The range requires that the variable be ordinal or interval. The variable "RS HIGHEST DEGREE" is ordinal, satisfying this requirement.
The range best represents variability for an ordinal level variable. The range for the variable "RS HIGHEST DEGREE" is 4. The answer to the question is true.
SW388R6 Data Analysis and Computers I
Problem 4
Slide 18
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The range for the variable "AGE OF RESPONDENT" is 16.81. 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 4
Slide 19
The range requires that the variable be ordinal or interval. The variable "AGE OF RESPONDENT" is interval, satisfying this requirement.
The range for the variable "AGE OF RESPONDENT" is 70, not 16.81. The answer to the question is false.
SW388R6 Data Analysis and Computers I
Problem 5
Slide 20
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The variance for the variable "AGE OF RESPONDENT" is 70. 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 5
Slide 21
The variance requires that the variable be interval. The variable "AGE OF RESPONDENT" is interval, satisfying this requirement.
The variance for the variable "AGE OF RESPONDENT" is 282.46, not 70. The answer to the question is false.
SW388R6 Data Analysis and Computers I
Problem 6
Slide 22
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The standard deviation of 16.81 best represents the variability of the variable "AGE OF RESPONDENT". 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 6
Slide 23
The standard deviation for the variable "AGE OF RESPONDENT" is 16.81. The standard deviation requires that the variable be interval. The variable "AGE OF RESPONDENT" is interval, satisfying this requirement.
The answer to the question is true.
The standard deviation best represents variability for an interval level variable whose distribution is not badly skewed. The skewness for this variable is 0.595, not badly skewed by the rule of thumb that the distribution of a variable is not badly skewed if its skewness is between -1.0 and +1.0.
SW388R6 Data Analysis and Computers I
Problem 7
Slide 24
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The range of 22 best represents the dispersion of the variable "NUMBER OF BROTHERS AND SISTERS". 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 7
Slide 25
The range requires that the variable be ordinal or interval. The variable "NUMBER OF BROTHERS AND SISTERS" is interval, satisfying this requirement. The range best represents variability for an interval level variable whose distribution is badly skewed. The skewness for this variable is 2.391, badly skewed by 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.
The range for the variable "NUMBER OF BROTHERS AND SISTERS" is 22. The answer to the question is true.
SW388R6 Data Analysis and Computers I
Steps in solving variability problems - 1
Slide 26
The following is a guide to the decision process for answering central tendency homework problems. If the problem asks about a specific measure of variability
Is the level of measurement required for the statistic satisfied?
No
Incorrect application of a statistic
Yes
Is the correct value for the statistic reported? Yes True
No False
SW388R6 Data Analysis and Computers I
Steps in solving variability problems - 2
Slide 27
If the problem asks about the preferred measure of variability: Is the variable ordinal level?
No
Is the variable interval level?
No
Yes
Yes
Is the value for the range correct?
Yes
Is the variable Falsebadly skewed?
No Yes False
No
True
Is the value for the standard deviation or variance correct? No Yes False
True
Is the variable nominal level? Yes
SPSS does not calculate the IQV
Measures of Variability
Slide 1
Standard Deviation and Variance Range Index of Qualitative Variation Preferred measure
SW388R6 Data Analysis and Computers I
Measures of Variability
Slide 2
Measures of variability (also called measures of dispersion) use a single number or data value to indicate variability, or differences between individual scores and the measure of central tendency.
Measures of variability are paired with measures of central to describe or summarize a distribution. Like measures of central tendency, the preferred measure of variability is based on the level of measurement of a variable.
Statistically, measures of variability are inidcators of the errors we making in using a measure of central tendency as the best estimate of the data value for all subjects in a distribution.
SW388R6 Data Analysis and Computers I
The Standard Deviation and Variance - 1
Slide 3
The standard deviation is the measure of variability that is paired with the mean as the measure of central tendency.
The difference between each case in a distribution and the mean is called the deviation. The sum of all the deviations is equal to zero.
If we add up the squared deviations for all cases in a distribution, we produce a measure of error called the total sum of squares. If we divide the total sum of squares by the number of cases, we produce the variance. The square root of the variance is the standard deviation.
SW388R6 Data Analysis and Computers I
The Standard Deviation and Variance - 2
Slide 4
Standard deviation and variance should only be computed for interval level variables.
The standard deviation (and variance) are the preferred measure of central tendency for an interval level variable, unless the variable is badly skewed. The standard deviation is more frequently used.
SW388R6 Data Analysis and Computers I
The Range
Slide 5
The range is the the difference between the largest and smallest value in a distribution.
The range should only be computed for ordinal and interval level variables.
The range is the preferred measure of central tendency for ordinal level variables and interval level variables that are badly skewed.
SW388R6 Data Analysis and Computers I
The Index of Qualitative Variation
Slide 6
The Index of Qualitative Variation (IQV) is a measure of the distribution of cases not in the modal category.
The IQV 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 IQV is the preferred measure of central tendency for nominal level variables.
We will not use the IQV because SPSS does not compute it.
SW388R6 Data Analysis and Computers I
Request measures of variability
Slide 7
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,
To compute measures of central tendency in SPSS, select the Descriptive Statistics | Frequencies command from the Analyze menu.
SW388R6 Data Analysis and Computers I
Select the variable for the analysis
Slide 8
First, move the variable “sex” to the list of Variable(s).
Second, click on the Statistics… button to select the specific measures we want.
SW388R6 Data Analysis and Computers I
Request the specific measures
Slide 9
First, mark the checkboxes for Standard Deviation, Variance, and Range. Since the SPSS output for this command is not unwieldy, is 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.
SW388R6 Data Analysis and Computers I
Complete request for measures of variability
Slide 10
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.
SW388R6 Data Analysis and Computers I
SPSS output for measures of variability
Slide 11
The table of Statistics contains the measures of variability and skewness that we will need to solve our homework problems.
SW388R6 Data Analysis and Computers I
Problem 1
Slide 12
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The range for the variable "RS HIGHEST DEGREE" is 4. 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 1
Slide 13
The range requires that the variable be ordinal or interval. The variable "RS HIGHEST DEGREE" is ordinal, satisfying this requirement.
The range for the variable "RS HIGHEST DEGREE" is 4. The answer to the question is true.
SW388R6 Data Analysis and Computers I
Problem 2
Slide 14
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The standard deviation for the variable "RS HIGHEST DEGREE" is 1.16. 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 2
Slide 15
The standard deviation requires that the variable be interval. The variable "RS HIGHEST DEGREE" is ordinal, which does not meet this requirement. The answer to the question is incorrect application of a statistic.
SW388R6 Data Analysis and Computers I
Problem 3
Slide 16
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The range of 4 best represents the variability of the variable "RS HIGHEST DEGREE". 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 3
Slide 17
The range requires that the variable be ordinal or interval. The variable "RS HIGHEST DEGREE" is ordinal, satisfying this requirement.
The range best represents variability for an ordinal level variable. The range for the variable "RS HIGHEST DEGREE" is 4. The answer to the question is true.
SW388R6 Data Analysis and Computers I
Problem 4
Slide 18
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The range for the variable "AGE OF RESPONDENT" is 16.81. 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 4
Slide 19
The range requires that the variable be ordinal or interval. The variable "AGE OF RESPONDENT" is interval, satisfying this requirement.
The range for the variable "AGE OF RESPONDENT" is 70, not 16.81. The answer to the question is false.
SW388R6 Data Analysis and Computers I
Problem 5
Slide 20
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The variance for the variable "AGE OF RESPONDENT" is 70. 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 5
Slide 21
The variance requires that the variable be interval. The variable "AGE OF RESPONDENT" is interval, satisfying this requirement.
The variance for the variable "AGE OF RESPONDENT" is 282.46, not 70. The answer to the question is false.
SW388R6 Data Analysis and Computers I
Problem 6
Slide 22
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The standard deviation of 16.81 best represents the variability of the variable "AGE OF RESPONDENT". 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 6
Slide 23
The standard deviation for the variable "AGE OF RESPONDENT" is 16.81. The standard deviation requires that the variable be interval. The variable "AGE OF RESPONDENT" is interval, satisfying this requirement.
The answer to the question is true.
The standard deviation best represents variability for an interval level variable whose distribution is not badly skewed. The skewness for this variable is 0.595, not badly skewed by the rule of thumb that the distribution of a variable is not badly skewed if its skewness is between -1.0 and +1.0.
SW388R6 Data Analysis and Computers I
Problem 7
Slide 24
Based on the dataset GSS2000.SAV, is the following statement true, false, or an incorrect application of a statistic? The range of 22 best represents the dispersion of the variable "NUMBER OF BROTHERS AND SISTERS". 1. True 2. False 3. Incorrect application of a statistic
SW388R6 Data Analysis and Computers I
Solution 7
Slide 25
The range requires that the variable be ordinal or interval. The variable "NUMBER OF BROTHERS AND SISTERS" is interval, satisfying this requirement. The range best represents variability for an interval level variable whose distribution is badly skewed. The skewness for this variable is 2.391, badly skewed by 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.
The range for the variable "NUMBER OF BROTHERS AND SISTERS" is 22. The answer to the question is true.
SW388R6 Data Analysis and Computers I
Steps in solving variability problems - 1
Slide 26
The following is a guide to the decision process for answering central tendency homework problems. If the problem asks about a specific measure of variability
Is the level of measurement required for the statistic satisfied?
No
Incorrect application of a statistic
Yes
Is the correct value for the statistic reported? Yes True
No False
SW388R6 Data Analysis and Computers I
Steps in solving variability problems - 2
Slide 27
If the problem asks about the preferred measure of variability: Is the variable ordinal level?
No
Is the variable interval level?
No
Yes
Yes
Is the value for the range correct?
Yes
Is the variable Falsebadly skewed?
No Yes False
No
True
Is the value for the standard deviation or variance correct? No Yes False
True
Is the variable nominal level? Yes
SPSS does not calculate the IQV
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