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Variability

Variability. Quantitative Methods in HPELS 440:210. Agenda. Introduction Frequency Range Interquartile range Variance/SD of population Variance/SD of sample Selection. Introduction. Statistics of variability: Describe how values are spread out

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Variability

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  1. Variability Quantitative Methods in HPELS 440:210

  2. Agenda • Introduction • Frequency • Range • Interquartile range • Variance/SD of population • Variance/SD of sample • Selection

  3. Introduction • Statistics of variability: • Describe how values are spread out • Describe how values cluster around the middle • Several statistics  Appropriate measurement depends on: • Scale of measurement • Distribution

  4. Basic Concepts • Measures of variability: • Frequency • Range • Interquartile range • Variance and standard deviation • Each statistic has its advantages and disadvantages

  5. Agenda • Introduction • Frequency • Range • Interquartile range • Variance/SD of population • Variance/SD of sample • Selection

  6. Frequency • Definition: The number/count of any variable • Scale of measurement: • Appropriate for all scales • Only statistic appropriate for nominal data • Statistical notation: f

  7. Frequency • Advantages: • Ease of determination • Only statistic appropriate for nominal data • Disadvantages: • Terminal statistic

  8. Calculation of the Frequency  Instat • Statistics tab • Summary tab • Group tab • Select group • Select column(s) of interest • OK

  9. Agenda • Introduction • Frequency • Range • Interquartile range • Variance/SD of population • Variance/SD of sample • Selection

  10. Range • Definition: The difference between the highest and lowest values in a distribution • Scale of measurement: • Ordinal, interval or ratio

  11. Range • Advantages: • Ease of determination • Disadvantages: • Terminal statistic • Disregards all data except extreme scores

  12. Calculation of the Range  Instat • Statistics tab • Summary tab • Describe tab • Calculates range automatically • OK

  13. Agenda • Introduction • Frequency • Range • Interquartile range • Variance/SD of population • Variance/SD of sample • Selection

  14. Interquartile Range • Definition: The difference between the 1st quartile and the 3rd quartile • Scale of measurement: • Ordinal, interval or ratio • Example: Figure 4.3, p 107

  15. Interquartile Range • Advantages: • Ease of determination • More stable than range • Disadvantages: • Disregards all values except 1st and 3rd quartiles

  16. Calculation of the Interquartile Range  Instat • Statistics tab • Summary tab • Describe tab • Choose additional statistics • Choose interquartile range • OK

  17. Agenda • Introduction • Frequency • Range • Interquartile range • Variance/SD of population • Variance/SD of sample • Selection

  18. Variance/SD  Population • Variance: • The average squared distance/deviation of all raw scores from the mean • The standard deviation squared • Statistical notation: σ2 • Scale of measurement: • Interval or ratio • Advantages: • Considers all data • Not a terminal statistic • Disadvantages: • Not appropriate for nominal or ordinal data • Sensitive to extreme outliers

  19. Variance/SD  Population • Standard deviation: • The average distance/deviation of all raw scores from the mean The square root of the variance Statistical notation: σ • Scale of measurement: • Interval or ratio • Advantages and disadvantages: • Similar to variance

  20. Calculation of the Variance  Population • Why square all values? • If all deviations from the mean are summed, the answer always = 0

  21. Example: 1, 2, 3, 4, 5 Mean = 3 Variations: 1 – 3 = -2 2 – 3 = -1 3 – 3 = 0 4 – 3 = 1 5 – 3 = 2 Sum of all deviations = 0 Sum of all squared deviations Variations: 1 – 3 = (-2)2 = 4 2 – 3 = (-1)2 = 1 3 – 3 = (0)2 = 0 4 – 3 = (1)2 = 1 5 – 3 = (2)2 = 4 Sum of all squared deviations = 10 Calculation of the Variance  Population Variance = Average squared deviation of all points  10/5 = 2

  22. Calculation of the Variance  Population • Step 1: Calculate deviation of each point from mean • Step 2: Square each deviation • Step 3: Sum all squared deviations • Step 4: Divide sum of squared deviations by N

  23. Calculation of the Variance  Population • σ2 = SS/number of scores, where SS = • Σ(X - )2 • Definitional formula (Example 4.3, p 112) • or • ΣX2 – [(ΣX)2] • Computational formula (Example 4.4, p 112)

  24. Computational formula Step 4: Divide by N

  25. Computation of the Standard Deviation  Population • Take the square root of the variance

  26. Agenda • Introduction • Frequency • Range • Interquartile range • Variance/SD of population • Variance/SD of sample • Selection

  27. Variance/SD  Sample • Process is similar with two distinctions: • Statistical notation • Formula

  28. Statistical Notation DistinctionsPopulation vs. Sample • σ2 = s2 • σ = s •  = M • N = n

  29. Formula DistinctionsPopulation vs. Sample • s2 = SS / n – 1, where SS = • Σ(X - M)2 • Definitional formula • ΣX2 - [(ΣX)2] • Computational formula Why n - 1?

  30. N vs. (n – 1)  First Reason • General underestimation of population variance • Sample variance (s2) tend to underestimate a population variance (σ2) • (n – 1) will inflate s2 • Example 4.8, p 121

  31. Actual population σ2 = 14 Average biased s2 = 63/9 = 7 Average unbiased s2 = 126/9 = 14

  32. N vs. (n – 1)  Second Reason • Degrees of freedom (df) • df = number of scores “free” to vary • Example: • Assume n = 3, with M = 5 • The sum of values = 15 (n*M) • Assume two of the values = 8, 3 • The third value has to be 4 • Two values are “free” to vary • df = (n – 1) = (3 – 1) = 2

  33. Computation of the Standard Deviation of Sample  Instat • Statistics tab • Summary tab • Describe tab • Calculates standard deviation automatically • OK

  34. Agenda • Introduction • Frequency • Range • Interquartile range • Variance/SD of population • Variance/SD of sample • Selection

  35. Selection • When to use the frequency • Nominal data • With the mode • When to use the range or interquartile range • Ordinal data • With the median • When to sue the variance/SD • Interval or ratio data • With the mean

  36. Textbook Problem Assignment • Problems: 4, 6, 8, 14.

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