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Topics: Descriptive Statistics

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. The Role of Description. Description as a purpose of research

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Topics: Descriptive Statistics

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  1. 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

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

  3. Raw Data: Overachievement Study

  4. 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

  5. Numerical Frequency Distributions • Ungrouped Frequency Distributions • Grouped Frequency Distributions • Relative Frequency Distributions • Cumulative Frequency Distributions

  6. Tabular Frequency Distributions Single-Variable (“Univariate”)

  7. Frequency Distribution: Major MAJOR Valid Cum Value Label Value Frequency Percent Percent Percent PHYSICS 1.00 5 12.5 12.5 12.5 CHEMISTRY 2.00 4 10.0 10.0 22.5 BIOLOGY 3.00 7 17.5 17.5 40.0 ENGINEERING 4.00 5 12.5 12.5 52.5 ANTHROPOLOGY 5.00 5 12.5 12.5 65.0 SOCIOLOGY 6.00 4 10.0 10.0 75.0 ENGLISH 7.00 7 17.5 17.5 92.5 DESIGN 8.00 3 7.5 7.5 100.0 ------- ------- ------- Total 40 100.0 100.0 Valid cases 40 Missing cases 0

  8. Frequency Distribution: Major Group MAJORGRP Valid Cum Value Label Value Frequency Percent Percent SCIENCE & ENGINEERIN 1.00 21 52.5 52.5 52.5 SOCIAL SCIENCE 2.00 9 22.5 22.5 75.0 HUMANITIES 3.00 10 25.0 25.0 100.0 ------- ------- ------- Total 40 100.0 100.0

  9. Frequency Distribution: SAT SAT Valid Cum Value Frequency Percent Percent 1000.00 2 5.0 5.0 5.0 1025.00 1 2.5 2.5 7.5 1050.00 2 5.0 5.0 12.5 1060.00 1 2.5 2.5 15.0 1075.00 1 2.5 2.5 17.5 1080.00 1 2.5 2.5 20.0 1085.00 1 2.5 2.5 22.5 1090.00 2 5.0 5.0 27.5 1100.00 7 17.5 17.5 45.0 1120.00 2 5.0 5.0 50.0 1125.00 3 7.5 7.5 57.5 1130.00 1 2.5 2.5 60.0 1150.00 5 12.5 12.5 72.5 1160.00 2 5.0 5.0 77.5 1175.00 3 7.5 7.5 85.0 1185.00 1 2.5 2.5 87.5 1200.00 5 12.5 12.5 100.0 ------- ------- ------- Total 40 100.0 100.0 Valid cases 40 Missing cases 0

  10. Grouped Frequency Distribution: SAT

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

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

  13. Bar Chart: Major

  14. Histogram: SAT(From Grouped Data)

  15. Frequency Polygon Overlay: SAT(From Grouped Data)

  16. Frequency Polygon: SAT(From Grouped Data)

  17. Frequency Polygon: SAT Scores(From Ungrouped Data)

  18. Cumulative Frequency Polygon: SAT Scores

  19. Stem and Leaf: SAT

  20. Stem and Leaf: SAT 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)

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

  22. Relative Frequency Polygon: GPAComparison of Majors

  23. Relative Frequency Polygon: GPA Comparison of Gender

  24. What Can Be Seen in Frequency Distributions • Shape • Central Tendency • Variability

  25. Shapes of Frequency Polygons

  26. Shapes of Distributions

  27. Descriptive Statistics • Central Tendency • Mode • Median • Mean • Variability • Range • Standard Deviation • Variance

  28. 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

  29. Mean, Median, Mode:SAT Scores by Gender

  30. Mean, Median, Mode:SAT Scores by Area

  31. Relative Position of Mode, Median, and Mean

  32. 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)

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

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

  35. 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)

  36. Normal Curve Areas Under the Curve .3413 .3413 .1359 .1359 68% .0214 .0214 95% .0013 .0013 99% -3s -2s -1s +1s +2s +3s X

  37. 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)

  38. 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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