Data Transformation

14. Data Transformation • Data conversion • Changing the original form of the data to a new format • More appropriate data analysis • New variables

15. Data Transformation Summative Score = VAR1 + VAR2 + VAR 3

21. Descriptive Analysis • The transformation of raw data into a form that will make them easy to understand and interpret; rearranging, ordering, and manipulating data to generate descriptive information

22. Tabulation • Tabulation - Orderly arrangement of data in a table or other summary format • Frequency table • Percentages

23. Frequency Table • The arrangement of statistical data in a row-and-column format that exhibits the count of responses or observations for each category assigned to a variable

24. Central Tendency Measure of Central Measure of Type of Scale Tendency Dispersion Nominal Mode None Ordinal Median Percentile Interval or ratio Mean Standard deviation

25. Base • The number of respondents or observations (in a row or column) used as a basis for computing percentages

26. Index Numbers • Score or observation recalibrated to indicate how it relates to a base number • CPI - Consumer Price Index

27. Measures of Central Tendency • Mean - arithmetic average • µ, Population; , sample • Median - midpoint of the distribution • Mode - the value that occurs most often

28. Population Mean

29. Sample Mean

30. Measures of Dispersion or Spread • Range • Mean absolute deviation • Variance • Standard deviation

31. The Range as a Measure of Spread • The range is the distance between the smallest and the largest value in the set. • Range = largest value – smallest value

32. Deviation Scores • The differences between each observation value and the mean:

33. Low Dispersion Verses High Dispersion 5 4 3 2 1 Low Dispersion Frequency 150 160 170 180 190 200 210 Value on Variable

34. Low Dispersion Verses High Dispersion 5 4 3 2 1 High dispersion Frequency 150 160 170 180 190 200 210 Value on Variable

35. AverageDeviation

36. Mean Squared Deviation

37. The Variance

38. Variance

39. Variance • The variance is given in squared units • The standard deviation is the square root of variance:

40. Sample Standard Deviation

41. The Normal Distribution • Normal curve • Bell shaped • Almost all of its values are within plus or minus 3 standard deviations • I.Q. is an example

42. Normal Distribution 13.59% 13.59% 34.13% 34.13% 2.14% 2.14%

43. Normal Curve: IQ Example 70 145 85 115 100

44. Standardized Normal Distribution • Symetrical about its mean • Mean identifies highest point • Infinite number of cases - a continuous distribution • Area under curve has a probability density = 1.0 • Mean of zero, standard deviation of 1

45. Standard Normal Curve • The curve is bell-shaped or symmetrical • About 68% of the observations will fall within 1 standard deviation of the mean • About 95% of the observations will fall within approximately 2 (1.96) standard deviations of the mean • Almost all of the observations will fall within 3 standard deviations of the mean

46. A Standardized Normal Curve z 1 2 -2 -1 0

47. The Standardized Normal is the Distribution of Z –z +z

48. Standardized Scores

49. Standardized Values • Used to compare an individual value to the population mean in units of the standard deviation

50. Linear Transformation of Any Normal Variable into a Standardized Normal Variable s s m X m Sometimes the scale is stretched Sometimes the scale is shrunk -2 -1 0 1 2