Dr. Michael R. Hyman, NMSU Statistics for a Single Measure (Univariate)
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Transcript of Dr. Michael R. Hyman, NMSU Statistics for a Single Measure (Univariate)
Dr. Michael R. Hyman, NMSU
Statistics for a Single Measure (Univariate)
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Wrong Ways to Think About Statistics
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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
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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-and-column 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– , sampleX
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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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150 160 170 180 190 200 210
5
4
3
2
1
Value on Variable
Fre
quen
cy
Low Dispersion
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150 160 170 180 190 200 210
5
4
3
2
1
Fre
quen
cy
Value on Variable
High Dispersion
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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
xxd ii
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Average Deviation
0)(
n
XX i
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Mean Squared Deviation
n
XXi 2)(
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Variance
2
2
S
Sample
Population
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Variance
1)2
2
nXX
S
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Variance
• Variance is given in squared units
• Standard deviation is the square root of variance
1
2
n
XX iS
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Measure ofCentral Measure of
Type of Scale Tendency Dispersion
Nominal Mode NoneOrdinal Median PercentileInterval or ratio Mean Standard deviation
Summary: Central Tendency and Dispersion
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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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2.14%
13.59% 34.13% 34.13% 13.59%
2.14%
Normal Distribution
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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
31 85 115100 14570
Normal Curve: IQ Example
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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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0-1-2 2 z
Standardized Normal Curve
1
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Standardized Normal is Z Distribution
–z +z
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xz
Standardized Scores
Used to compare an individual value to the population mean in units of the standard deviation
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Linear Transformation of Any Normal Variable into a Standardized Normal
Variable
-2 -1 0 1 2
Sometimes thescale is stretched
Sometimes thescale is shrunk
X
xz
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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