Statistics for a Single
Measure (Univariate)
(Click icon for audio)
Dr. Michael R. Hyman, NMSU
2
Wrong Ways to Think About Statistics
3
Evidence
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Descriptive Analysis
Transformation of raw data into a form
that make them easy to understand
and interpret; rearranging, ordering,
and manipulating data to generate
descriptive information
5
Analysis Begins with Tabulation
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Tabulation
• Tabulation: Orderly arrangement of
data in a table or other summary
format
• Frequency table
• Percentages
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Frequency Table
• Numerical data arranged in a row-andcolumn format that shows the count
and percentages of responses or
observations for each category
assigned to a variable
• Pre-selected categories
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Sample Frequency Table
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Sample SPSS Frequency Output
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SPSS Histogram Output
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Base
• Number of respondents or observations
(in a row or column) used as a basis for
computing percentages
• Possible bases
– All respondents
– Respondents who were asked question
– Respondents who answered question
• Be careful with multi-response questions
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Measures of Central Tendency
• Mode - the value that occurs most often
• Median - midpoint of the distribution
• Mean - arithmetic average
– µ, population
– X , sample
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Measures of Dispersion
or Spread for Single Measure
•
•
•
•
•
Range
Inter-quartile range
Mean absolute deviation
Variance
Standard deviation
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Low Dispersion
5
4
3
2
1
150
160
170 180
190
Value on Variable
200
210
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High Dispersion
5
4
3
2
1
150
160
170
180
190
Value on Variable
200
210
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Range as a Measure of
Spread
• Difference between smallest and
largest value in a set
• Range = largest value – smallest value
• Inter-quartile range = 75th percentile –
25th percentile
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Deviation Scores
Differences between each
observed value and the mean
di  xi  x
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Average Deviation
(X i  X )
0
n
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Mean Squared Deviation
 ( Xi  X )
n
2
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Variance
Population

2
Sample
S
2
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Variance
 X  X )
S 
n 1
2
2
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Variance
• Variance is given in squared units
• Standard deviation is the square root of
variance
S

2
 Xi X 
n 1
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Summary: Central Tendency
and Dispersion
Type of Scale
Measure of
Central
Tendency
Measure of
Dispersion
Nominal
Ordinal
Interval or ratio
Mode
Median
Mean
None
Percentile
Standard deviation
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Distributions for Single
Measures
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Symmetric Distribution
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Skewed
Distributions
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Normal Distribution
• Bell shaped
• Symmetrical about its mean
• Mean identifies highest
point
• Almost all values are within
+3 standard deviations
• Infinite number of cases--a
continuous distribution
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Normal Distribution
13.59%
2.14%
34.13%
34.13%
13.59%
2.14%
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Standard Normal Curve
• The curve is bell-shaped or symmetrical
• About 68% of the observations will fall
within 1 standard deviation of the mean
• About 95% of the observations will fall
within approximately 2 (1.96) standard
deviations of the mean
• Almost all of the observations will fall
within 3 standard deviations of the mean
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Normal Curve: IQ Example
70
85
100
115
130
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Standardized Normal Distribution
• Area under curve has a probability
density = 1.0
• Mean = 0
• Standard deviation = 1
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Standardized Normal Curve
-2
-1
0
1
2
z
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Standardized Normal is Z
Distribution
–z
+z
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Standardized Scores
Used to compare an individual value
to the population mean in units of the
standard deviation
z
x

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Linear Transformation of Any Normal
Variable into a Standardized Normal
Variable




Sometimes the
scale is stretched
X
Sometimes the
scale is shrunk
z
-2
-1
0
1
2
x

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Data Transformation
• Data conversion
• Changing the original form of the
data to a new format
• More appropriate data analysis
• New variables
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Data Transformation
Summative Score =
VAR1 + VAR2 + VAR 3
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Index Numbers
• Score or observation recalibrated to
indicate how it relates to a base
number
• CPI - Consumer Price Index
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Recap
•
•
•
•
Count/frequency data
Measures of central tendency
Dispersion from central tendency
Data distributions
– Symmetric versus skewed
– (Standardized) normal
• Data transformation
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