Statistics for a Single Measure (Univariate) (Click icon for audio)

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Statistics for a Single
Measure (Univariate)
(Click icon for audio)
Dr. Michael R. Hyman, NMSU
2
Wrong Ways to Think About Statistics
3
Evidence
4
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
6
Tabulation
• Tabulation: Orderly arrangement of
data in a table or other summary
format
• Frequency table
• Percentages
7
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
8
Sample Frequency Table
9
Sample SPSS Frequency Output
10
SPSS Histogram Output
11
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
12
Measures of Central Tendency
• Mode - the value that occurs most often
• Median - midpoint of the distribution
• Mean - arithmetic average
– µ, population
– X , sample
13
Measures of Dispersion
or Spread for Single Measure
•
•
•
•
•
Range
Inter-quartile range
Mean absolute deviation
Variance
Standard deviation
14
Low Dispersion
5
4
3
2
1
150
160
170 180
190
Value on Variable
200
210
15
High Dispersion
5
4
3
2
1
150
160
170
180
190
Value on Variable
200
210
16
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
17
Deviation Scores
Differences between each
observed value and the mean
di  xi  x
18
Average Deviation
(X i  X )
0
n
19
Mean Squared Deviation
 ( Xi  X )
n
2
20
Variance
Population

2
Sample
S
2
21
Variance
 X  X )
S 
n 1
2
2
22
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
24
Distributions for Single
Measures
25
Symmetric Distribution
26
Skewed
Distributions
27
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
28
Normal Distribution
13.59%
2.14%
34.13%
34.13%
13.59%
2.14%
29
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
31
Standardized Normal Distribution
• Area under curve has a probability
density = 1.0
• Mean = 0
• Standard deviation = 1
32
Standardized Normal Curve
-2
-1
0
1
2
z
33
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

35
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

36
Data Transformation
• Data conversion
• Changing the original form of the
data to a new format
• More appropriate data analysis
• New variables
37
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
39
Recap
•
•
•
•
Count/frequency data
Measures of central tendency
Dispersion from central tendency
Data distributions
– Symmetric versus skewed
– (Standardized) normal
• Data transformation
40
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