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Descriptive Statistics

Descriptive Statistics. Describing Numerical Data:. Numerical Descriptive Statistics. Numerical Descriptive Statistics. Numerical Summary Measures:. Population Parameters Sample Statistics. Numerical Summary Measures. Measures of Central Tendency:. Mean Median Mode. Mean.

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Descriptive Statistics

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  1. Descriptive Statistics Describing Numerical Data: Numerical Descriptive Statistics

  2. Numerical Descriptive Statistics Numerical Summary Measures: Population Parameters Sample Statistics

  3. Numerical Summary Measures Measures of Central Tendency: Mean Median Mode

  4. Mean Most common measure of central tendency The arithmetic average Acts as ‘balance point’ Affected by extreme values (‘outliers’)

  5. Mean Formula (sample mean): Let X = sample mean n = sample size Xi = value of measurement i (i=1...n) then

  6. Dataset I: Score = 2726/50 = 54.52

  7. Dataset I: Score 30 40 50 60 70 80 mean = 54.52

  8. Median Measure of central tendency Middle value in an ordered array If odd n, middle value of ordered array If even n, average of 2 middle values Not affected by extreme values

  9. Example 1: 45 56 60 65 72 Median = 60

  10. Example 2: 30 45 56 60 65 72 Median = (56+60)/2 = 58

  11. Dataset I: Score Positions: 1 2 …25 26 … 49 50 Test Scores: 55 55 Median = (55+55)/2 = 55

  12. Dataset I: Score Mean vs Median 30 40 50 60 70 80 mean = 54.52 median = 55

  13. Mean vs Median 40% 30% 15% 10% 5% 30 40 50 60 70 80 medianmean

  14. Mean vs Median 30 40 50 60 70 80 mean median

  15. Mean vs Median 30 40 50 60 70 80 mean = median = 55

  16. Mean vs Median 30 40 50 60 70 80 mean = median = 55

  17. Mode Measure of central tendency Value that occurs most often Not affected by extreme values May be no mode or several modes May be used for numerical & categorical data

  18. Dataset I: Score Mode = 52, 55 (each test score value occurs 4 times)

  19. Dataset I: Score Mean vs Median vs Mode 30 40 50 60 70 80 mode = 52 mean = 54.52 median = 55 = mode

  20. Quantitative Data Discrete Data: finite, countable number of values Continuous Data: infinite number of values

  21. Numerical Summary Measures Measures of Non-Central Location: Percentile Quartiles

  22. Percentiles Divide the data into hundreds Example: 95th percentile has 95% of the data below it & 5% of the data above it

  23. 25% 25% 25% 25% Q1 Q2 Q3 Quartiles Measure of noncentral location Split ordered data into 4 quarters

  24. Quartiles Finding quartiles: Q2 = median of the entire dataset Q1 = median of the first half of dataset Q3 = median of the second half of dataset

  25. Dataset I: Score Dataset I: Score Positions: 1 2 …13…25 26…38… 49 50 Test Scores: 30 40 … 51 …55 55 … 59 … 67 79 55

  26. Quartiles Data set I: ________________________ L Q1 Q2 Q3 H 51 55 59

  27. Five Number Summary Data Set I: 25% 25% 25% 25% _______________________ L Q1 Q2 Q3 H 30 51 55 59 79

  28. Numerical Summary Measures Measures of Dispersion: range interquartile range variance / standard deviation coefficient of variation

  29. Measures of Dispersion Dataset A Dataset B ________ ________ n = 5 n = 5 mean = 40 mean = 40

  30. Measures of Dispersion Dataset A Dataset B 20 39 30 40 40 40 50 40 60 41 ___________ ___________ n = 5 n = 5 mean = 40 mean = 40

  31. Range Measure of dispersion Difference between largest & smallest observations Formula:

  32. Example: Dataset A Dataset B 20 39 30 40 40 40 50 40 60 41 ___________ ___________ n = 5 n = 5 mean = 40 mean = 40 range = 40 range = 2

  33. Interquartile Range Measure of dispersion Also called midspread Spread in middle 50% of data Not affected by extreme values Formula:

  34. Example: Dataset A Dataset B 20 39 30 40 40 40 50 40 60 41 ___________ ___________ n = 5 n = 5 mean = 40 mean = 40 range = 40 range = 2 IQR = 20 IQR = 0

  35. Variance & Standard Deviation Measure of dispersion Most commonly used measures Consider how data are distributed Show variation about mean (X or )

  36. Variance & Standard Deviation Calculation based on: deviation where deviation = Xi - X

  37. Deviations: (Xi - X) Dataset A Dataset B -20 +20 -1 +1 -10 +10 20 30 40 50 60 39 40 41 mean = 40 mean = 40

  38. Sample Variance Formula: n - 1 in denominator! (Use N if population variance)

  39. Sample Standard Deviation

  40. Example: Dataset A Dataset B 20 39 30 40 40 40 50 40 60 41 ___________________ ____________________ mean = 40 mean = 40 range = 40 range = 2 IQR = 20 IQR = 0 s² = 250 s² = .5 s = 15.84 s = .707

  41. Interpreting Standard Deviation Empirical Rule¹: mean± 1 s  encompasses about 68% of data mean± 2 s  encompasses about 95% of data mean± 3 s  encompasses about 100% of data ¹ Applies to Mound-shaped distributions only

  42. Example: Dataset C: Mound shaped: mean = 50 and s = 10 20 30 40 50 60 70 80 68% 95% 100%

  43. Dataset 1: Score Score has a bell-shaped distribution, but does it have a true NORMAL probability distribution?  Check the Empirical Rule

  44. Dataset I: Score Score: x = 54.52 s = 7.678 x ± 1s  54.52 ± 1(7.678) 46.84 to 62.20 Q: What percent of test scores fall in this interval?

  45. Dataset I: Score Ans: (40/50) x 100 = 80%  68% Score does not have a true NORMAL probability distribution!

  46. Dataset I: Score Score has a heavy-tailed distribution (also called an outlier-prone distribution)

  47. Bell-shaped distributions Bell-shaped distributions can be: 1. Normal distributions 2. Heavy-tailed distributions, or 3. Light-tailed distributions

  48. Coefficient of Variation Measure of relative dispersion Always a % Shows variation relative to mean Used to compare 2 or more groups Formula (sample):

  49. Stock AB Average Return $50 $12 Standard Dev. $10 $4 Coefficient of Variation Example: Given the following summary information on two stocks, A and B, listed on the American Stock Exchange, which shows more volatililty?

  50. Measures of Skewness & Heavy-tailedness SKEW1 Kurtosis

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