Location scores

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Introduction to computing z-scores and other standardized scores

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Location scores

  1. 1.      Prayer A bit more about standard deviations z-Scores: the basics Standardizing distributions Tuesday: • More on standardized distributions /T-scores • Using R • The STORY in your data
  2. 2. LuAnn, S., J. Walter and D. Antosh. (2007) Dieting behaviors of young women post-college graduation. College Student Journal 41:4.
  3. 3. Make fear and greed work for you Wall Street constantly swings between these two emotions. You can either get caught in the frenzy - or profit from it. By Janice Revell, Money Magazine senior writer Last Updated: July 21, 2009: 10:56 AM ET “Making matters worse, the big stock bet would be far riskier on a year-to-year basis than other strategies. The most common measure of portfolio risk is standard deviation, which tells you how much an investment's short-term returns bounce around its long-term average. Since 1926 stocks have returned average gains of 9.6% a year, with a standard deviation of 21.5 percentage points, according to Ibbotson Associates. That means that about two-thirds of the time, the annual return on stocks landed 21.5 percentage points below or above the average - that is, in any given year, your results would range from a 12% loss to a 31% gain. You'd need either an iron stomach or a steady supply of Zantac to stay the course. And if you happened to be at or near retirement when one of those really bad years hit, you might have to rethink your plans.” http://money.cnn.com/2009/07/20/pf/funds/fear_greed.moneymag/
  4. 4.  Explain how z-scores provide a description of a location in a distribution  Transform an X score into a z-score  Transform z-scores back into X scores, when the mean and standard deviation are given.  Use z-scores to make comparisons across variables and individuals.  Describe the effects when an entire data set is standardized by transforming all the scores to z-scores, including the impact on the shape, mean and standard deviation, and its comparability to other standardized distributions.  Use z-scores to transform a distribution into a standardized distribution.  Use SPSS to create standardized scores for a distribution.
  5. 5.  Exact location is described by z-score • Sign tells whether score is located above or below the mean • Number tells distance between score and mean in standard deviation units
  6. 6. 64 46 67 58 70 70 73 82 76 94
  7. 7. Learning Check • A z-score of z = +1.00 indicates a position in a distribution ____ A • Above the mean by 1 point B • Above the mean by a distance equal to 1 standard deviation C • Below the mean by 1 point D • Below the mean by a distance equal to 1 standard deviation
  8. 8. Learning Check - Answer • A z-score of z = +1.00 indicates a position in a distribution ____ A • Above the mean by 1 point B • Above the mean by a distance equal to 1 standard deviation C • Below the mean by 1 point D • Below the mean by a distance equal to 1 standard deviation
  9. 9. Learning Check • Decide if each of the following statements is True or False. T/F • A negative z-score always indicates a location below the mean T/F • A score close to the mean has a z-score close to 1.00
  10. 10. Answer True • Sign indicates that score is below the mean False • Scores close to 0 have z-scores close to 0.00
  11. 11. z  X  X   Numerator is a deviation score  Denominator expresses deviation in standard deviation units
  12. 12. z X  X  so X  X  z   Numerator is a deviation score  Denominator expresses deviation in standard deviation units
  13. 13. Learning Check • For a population with μ = 50 and σ = 10, what is the X value corresponding to z=0.4? A • 50.4 B • 10 C • 54 D • 10.4
  14. 14. Learning Check - Answer • For a population with μ = 50 and σ = 10, what is the X value corresponding to z=0.4? A • 50.4 B • 10 C • 54 D • 10.4
  15. 15. Learning Check • Decide if each of the following statements is True or False. T/F • If μ = 40 and X = 50 corresponds to z=+2.00, then σ = 5 points T/F • If σ = 20, a score above the mean by 10 points will have z = 1.00
  16. 16. Answer True • If 2σ = 10 then σ = 5 False • Why?
  17. 17.  All z-scores are comparable to each other  Scores from different distributions can be converted to z-scores  The z-scores (standardized scores) allow the comparison of scores from two different distributions along
  18. 18. • Every X value can be transformed to a z-score • Characteristics of z-score transformation – Same shape as original distribution – Mean of z-score distribution is always 0. – Standard deviation is always 1.00 • A z-score distribution is called a standardized distribution
  19. 19. Learning Check • Last week Andi had exams in Chemistry and in Spanish. On the chemistry exam, the mean was µ = 30 with σ = 5, and Andi had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 6 and Andi had a score of X = 65. For which class should Andi expect the better grade? A • Chemistry B • Spanish C • There is not enough information to know
  20. 20. Learning Check - Answer • Last week Andi had exams in Chemistry and in Spanish. On the chemistry exam, the mean was µ = 30 with σ = 5, and Andi had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 6 and Andi had a score of X = 65. For which class should Andi expect the better grade? A • Chemistry B • Spanish C • There is not enough information to know
  21. 21. Concepts Equations Interpretation
  22. 22.  All z-scores are comparable to each other  Scores from different distributions can be converted to z-scores  The z-scores (standardized scores) allow the comparison of scores from two different distributions along
  23. 23.  Process of standardization is widely used • SAT has Mean = 500 and σ = 100 • IQ has Mean = 100 and σ = 15 Point  Standardizing a distribution has two steps • Original raw scores transformed to z-scores • The z-scores are transformed to new X values so that the specific μ and σ are attained.
  24. 24. This form of standardized score, with M = 50 and  = 10, is known as a T-score.
  25. 25.  Interpretation of research results depends on determining if (treated) sample is noticeably different from the population  One technique for defining noticeably different uses z-scores.
  26. 26. Learning Check • Last week Andi had exams in Chemistry and in Spanish. On the chemistry exam, the mean was µ = 30 with σ = 5, and Andi had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 6 and Andi had a score of X = 65. For which class should Andi expect the better grade? A • Chemistry B • Spanish C • There is not enough information to know
  27. 27. Learning Check - Answer • Last week Andi had exams in Chemistry and in Spanish. On the chemistry exam, the mean was µ = 30 with σ = 5, and Andi had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 6 and Andi had a score of X = 65. For which class should Andi expect the better grade? A • Chemistry B • Spanish C • There is not enough information to know
  28. 28. Learning Check TF • Decide if each of the following statements is True or False. T/F • Transforming an entire distribution of scores into z-scores will not change the shape of the distribution. T/F • If a sample of n = 10 scores is transformed into z-scores, there will be five positive zscores and five negative z-scores.
  29. 29. Concepts Equations Interpretation

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