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Dr. Michael R. Hyman, NMSU Statistics for a Single Measure (Univariate)
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3 Wrong Ways to Think About Statistics
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4 Evidence
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5 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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6 Analysis Begins with Tabulation
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7 Tabulation Tabulation: Orderly arrangement of data in a table or other summary format Frequency table Percentages
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8 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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9 Sample Frequency Table
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10 Sample SPSS Frequency Output
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11 SPSS Histogram Output
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12 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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13 Measures of Central Tendency Mode - the value that occurs most often Median - midpoint of the distribution Mean - arithmetic average –µ, population –, sample
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14 Measures of Dispersion or Spread for Single Measure Range Inter-quartile range Mean absolute deviation Variance Standard deviation
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15 150 160 170 180 190 200210 5432154321 Value on Variable Frequency Low Dispersion
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16 150 160 170 180 190 200210 5432154321 Frequency Value on Variable High Dispersion
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17 Range as a Measure of Spread Difference between smallest and largest value in a set Range = largest value – smallest value Inter-quartile range = 75 th percentile – 25 th percentile
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18 Deviation Scores Differences between each observed value and the mean
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19 Average Deviation
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20 Mean Squared Deviation
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21 Variance
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22 Variance
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23 Variance Variance is given in squared units Standard deviation is the square root of variance
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24 Measure of CentralMeasure of Type of ScaleTendencyDispersion NominalModeNone OrdinalMedianPercentile Interval or ratioMeanStandard deviation Summary: Central Tendency and Dispersion
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25 Distributions for Single Measures
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26 Symmetric Distribution
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27 Skewed Distributions
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28 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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29 2.14% 13.59% 34.13% 13.59% 2.14% Normal Distribution
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30 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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31 85115 100 14570 Normal Curve: IQ Example
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32 Standardized Normal Distribution Area under curve has a probability density = 1.0 Mean = 0 Standard deviation = 1
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33 0-22 z Standardized Normal Curve 1
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34 Standardized Normal is Z Distribution –z+z
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35 Standardized Scores Used to compare an individual value to the population mean in units of the standard deviation
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36 Linear Transformation of Any Normal Variable into a Standardized Normal Variable -2 -1 0 1 2 Sometimes the scale is stretched Sometimes the scale is shrunk X
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37 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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38 Data Transformation Summative Score = VAR1 + VAR2 + VAR 3
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39 Index Numbers Score or observation recalibrated to indicate how it relates to a base number CPI - Consumer Price Index
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40 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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