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Topics: Descriptive Statistics A road map Examining data through frequency distributions Measures of central tendency Measures of variability The normal curve Standard scores and the standard normal distribution

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The Role of Description Description as a purpose of research Choosing the right statistical procedures

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Raw Data: Overachievement Study

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Frequency Distributions A method of summarizing and highlighting aspects of the data in a data matrix, showing the frequency with which each value occurs. Numerical Representations: a tabular arrangement of scores Graphical Representations: a pictorial arrangement of scores

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Numerical Frequency Distributions Ungrouped Frequency Distributions Grouped Frequency Distributions Relative Frequency Distributions Cumulative Frequency Distributions

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Tabular Frequency Distributions Single-Variable (“Univariate”)

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Frequency Distribution: Major MAJOR Valid Cum Value LabelValue FrequencyPercent Percent Percent PHYSICS1.00512.512.512.5 CHEMISTRY2.00410.010.022.5 BIOLOGY3.00717.517.540.0 ENGINEERING4.00512.512.552.5 ANTHROPOLOGY5.00512.512.565.0 SOCIOLOGY6.00410.010.075.0 ENGLISH7.00717.517.592.5 DESIGN8.0037.57.5100.0 ------- ------- ------- Total40100.0100.0 Valid cases 40 Missing cases 0

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Frequency Distribution: Major Group MAJORGRP Valid Cum Value LabelValueFrequencyPercentPercent SCIENCE & ENGINEERIN1.002152.552.552.5 SOCIAL SCIENCE2.00922.522.575.0 HUMANITIES3.001025.025.0100.0 ------- ------- ------- Total40100.0100.0

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Frequency Distribution: SAT SAT ValidCum ValueFrequencyPercent Percent 1000.0025.05.05.0 1025.0012.52.57.5 1050.0025.05.012.5 1060.0012.52.515.0 1075.0012.52.517.5 1080.0012.52.520.0 1085.0012.52.522.5 1090.0025.05.027.5 1100.00717.517.545.0 1120.0025.05.050.0 1125.0037.57.557.5 1130.0012.52.560.0 1150.00512.512.572.5 1160.0025.05.077.5 1175.0037.57.585.0 1185.0012.52.587.5 1200.00512.512.5100.0 ------- ------- ------- Total40100.0100.0 Valid cases 40 Missing cases 0

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Grouped Frequency Distribution: SAT

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Graphical Frequency Distributions Bar Graphs Histograms Stem and Leaf Frequency Polygons Pie Chart

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Graphical Frequency Distributions: Single-Variable (“Univariate”)

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Bar Chart: Major

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Histogram: SAT (From Grouped Data)

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Frequency Polygon Overlay: SAT (From Grouped Data)

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Frequency Polygon: SAT (From Grouped Data)

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Frequency Polygon: SAT Scores (From Ungrouped Data)

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Cumulative Frequency Polygon: SAT Scores

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Stem and Leaf: SAT

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SAT Stem-and-Leaf Plot Frequency Stem & Leaf 3.00 10. 002 8.00 10. 55678899 13.00 11. 0000000222223 11.00 11. 55555667778 5.00 12. 00000 Stem width: 100.00 Each leaf: 1 case(s)

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Graphical Frequency Distributions Two-Variable (“Joint” or “Bivariate”)

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Relative Frequency Polygon: GPA Comparison of Majors

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Relative Frequency Polygon: GPA Comparison of Gender

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What Can Be Seen in Frequency Distributions Shape Central Tendency Variability

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Shapes of Frequency Polygons

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Shapes of Distributions

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Descriptive Statistics Central Tendency –Mode –Median –Mean Variability –Range –Standard Deviation –Variance

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Definitions: Measures of Central Tendency Mean: –“Arithmetic mean” –“Center of gravity” such that the “weight” of the scores above the mean exactly balances the “weight” of the scores below the mean Median: –The number that lies at the midpoint of the distribution of scores; divides the distribution into two equal halves Mode: –Most frequently occurring score

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Mean, Median, Mode: SAT Scores by Gender

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Mean, Median, Mode: SAT Scores by Area

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Relative Position of Mode, Median, and Mean

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Definitions: Measures of Variability Range: –Difference between highest and lowest score Inter-quartile Range: –The spread of the middle 50% of the scores –The difference between the top 25% (Upper Quartile-Q3) and the lower 25% (Lower Quartile-Q1) Standard Deviation: –The average dispersion or deviation of scores around the mean (measured in original score units) Variance: –The average variability of scores (measured in squared units of the original scores (square of the standard deviation)

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Range, Interquartile Range, and Standard Deviation: SAT Scores by Area

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Range, Interquartile Range, and Standard Deviation: SAT Scores by Gender

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Properties of Normal Distribution Bell-shaped (unimodal) Symmetric about the mean Mode, median, and mean are equal (though rarely occurs) Asymptotic (curve never touches the abscissa)

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.3413.1359.0214.3413 Normal Curve Areas Under the Curve X -1s-2s+1s+2s-3s+3s.0013 68% 95% 99%

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Definitions: Standard Scores Standard Scores: scores expressed as SD away from the mean (z-scores) Obtained by finding how far a score is above or below the mean and dividing that difference by the SD Changes mean to 0 and SD to 1, but does not change the shape (called Standard Normal Distribution)

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Uses of Standard Normal Distribution What proportion of scores falls between the mean and a given raw score What proportion of scores falls above or below a given raw score What proportion of scores falls between two raw scores What raw score fall above (or below) a certain percentage of scores

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