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Frequency Distributions in Statistics

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Learn about frequency, percentage, and cumulative distributions in statistics, with practical examples and graphing techniques. Explore the importance of normal curves and data interpretation.

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Frequency Distributions in Statistics

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  1. FREQUENCY DISTRIBUTIONS & GRAPHING

  2. Statistics • Ordering & Grouping of Information • N = 50, a test score of 83% • where does it fit in the class?? • N = 25, a GRF value of 27 N/kg • where does it fit in the sample??

  3. Nomenclature • Frequency: number of cases or subjects or occurrences • represented with f • ie f = 12 for a score of 25 • 12 occurrences of 25 in the sample

  4. Nomenclature • Percentage: number of cases or subjects or occurrences expressed per 100 • represented with P or % • ie f = 12 for a score of 25 when n = 25 • P = 12/25*100 = 48%

  5. Caveat (Warning) • Should report the f when presenting percentages • ie 80% of the elementary students came from a family with an income < $25,000 • different interpretation if n = 5 compared to n = 100 • report in literature as • f = 4 (80%) OR • 80% (f = 4) OR 80% (n = 4)

  6. Numerator Monster Headline: State Farm distributes more than $1 billion in repayments to policyholders More than $34 million in Illinois Pantagraph, 6/13/00

  7. Numerator Monster Pantagraph, 6/13/00

  8. Numerator Monster Pantagraph, 6/13/00

  9. Frequency Distribution of Test Scores • 40 items on exam • Most students scored >34 • skewed • more scores at one end of the scale

  10. Cumulative frequency • Cumulative Frequency: how many subjects in and below a given score (SPSS???)

  11. Cumulative percentage • Cumulative Percentage: what % of subjects in and above a given score

  12. Cumulative percentage • Cumulative Percentage: what % of subjects in and below a given score

  13. Eyeball data check: Graphing with SPSS • Stem and Leaf plot (quick viewing of data distribution) • Bar Chart: each score separate bar • Histogram: malleable bar graph • Polygon: line graph (not available on SPSS)

  14. Fast look at shape of distribution shows f numerically & graphically stem is value, leaf is f Stem and Leaf(SPSS: Explore command) Frequency Stem & Leaf 2.00 Extremes (=<25.0) 2.00 28 . 00 2.00 29 . 00 1.00 30 . 0 1.00 31 . 0 3.00 32 . 000 1.00 33 . 0 6.00 34 . 000000 3.00 35 . 000 4.00 36 . 0000 8.00 37 . 00000000 Stem width: 1 Each leaf: 1 case(s)

  15. Format of Bar Chart Y axis (ordinate) f X axis (abcissa) scores/categories

  16. Test score data as Bar Chart

  17. Rules for Frequency Distributions • 10 to 20 groups of score intervals

  18. Rules for Frequency Distributions • 10 to 20 groups of score intervals • intervals of the same size

  19. Rules for Frequency Distributions • 10 to 20 groups of score intervals • intervals of the same size • score interval size calculated as (Max - Min) / # score intervals

  20. Rules for Frequency Distributions • 10 to 20 groups of score intervals • intervals of the same size • score interval size calculated as (Max - Min) / # score intervals • Need more information about calculating by hand?

  21. Format of Histogram Y axis (ordinate) f Can be manipulated X axis (abcissa) Scores / categories

  22. Test score data as Histogram

  23. Test score data as revised Histogram

  24. Shapes of Distributions • Scores arranged from low to high creates a distribution • with large n, distribution becomes smoother (a curve) • characteristic curves are named

  25. Normal Curve (bell-shaped) f high low scores

  26. Bell Curve Importance f high low scores Height or weight or VO2 or GRF or Vision or Skill

  27. Bell-curve importance • Natural data typically follows the normal curve • height, weight, strength, aerobic capacity, vitamin intake • The Happiness Meter • Basis of inferential statistics

  28. Skewed Distribution f Positive Skewness high low scores

  29. Skewed Distribution f Negative Skewness high low scores

  30. Male & Female Strength

  31. Interesting Distribution Assessment f Expected distribution of agent-paid claims (State Farm Ins.) high low $$ amount

  32. Interesting Distribution Assessment f Observed distribution of an agent-paid claims (suspicious.) high low $$ amount

  33. Table 4.1 data as Histogram Normal Curve

  34. Table 4.1 data as Histogram

  35. Always plot out data as a first step in the analysis

  36. Percentiles(Percentile Ranks) • Percentile: Values above and below which certain percentages of cases fall. • ie Doctors • 25% earn < $100,000 • 25% earn > $225,000 • How to divide scores into these ranks?

  37. Percentiles (Quartiles) Data from Table 4.1

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