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Chapter 5 z-Scores: Location of Scores and Standardized Distributions PowerPoint Lecture Slides Essentials of Statistics for the Behavioral Sciences Eighth Edition by Frederick J. Gravetter and Larry B. Wallnau Chapter 5 Learning Outcomes 1 • Understand z-score as location in distribution 2 • Transform X value into z-score 3 • Transform z-score into X value 4 • Describe effects of standardizing a distribution 5 • Transform scores to standardized distribution Tools You Will Need • The mean (Chapter 3) • The standard deviation (Chapter 4) • Basic algebra (math review, Appendix A) 5.1 Purpose of z-Scores • Identify and describe location of every score in the distribution • Standardize an entire distribution • Take different distributions and make them equivalent and comparable Figure 5.1 Two Exam Score Distributions 5.2 z-Scores and Location in a Distribution • 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 Figure 5.2 Relationship Between z-Scores and Locations 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 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 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 Learning Check - Answer True • Sign indicates that score is below the mean False • Scores quite close to the mean have z-scores close to 0.00 Equation (5.1) for z-Score z X • Numerator is a deviation score • Denominator expresses deviation in standard deviation units Determining a Raw Score From a z-Score • z X so X z • Algebraically solve for X to reveal that… • Raw score is simply the population mean plus (or minus if z is below the mean) z multiplied by population the standard deviation Figure 5.3 Visual Presentation of the Question in Example 5.4 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 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 Learning Check • Decide if each of the following statements is True or False. T/F • If μ = 40 and 50 corresponds to z = +2.00 then σ = 10 points T/F • If σ = 20, a score above the mean by 10 points will have z = 1.00 Learning Check - Answer False • If z = +2 then 2σ = 10 so σ = 5 False • If σ = 20 then z = 10/20 = 0.5 5.3 Standardizing a Distribution • 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 Figure 5.4 Visual Presentation of Question in Example 5.6 Figure 5.5 Transforming a Population of Scores Figure 5.6 Axis Re-labeling After z-Score Transformation Figure 5.7 Shape of Distribution After z-Score Transformation z-Scores Used for Comparisons • All z-scores are comparable to each other • Scores from different distributions can be converted to z-scores • z-scores (standardized scores) allow the direct comparison of scores from two different distributions because they have been converted to the same scale 5.4 Other Standardized Distributions • Process of standardization is widely used – SAT has μ = 500 and σ = 100 – IQ has μ = 100 and σ = 15 Points • 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 predetermined μ and σ are attained. Figure 5.8 Creating a Standardized Distribution Learning Check • A score of X=59 comes from a distribution with μ=63 and σ=8. This distribution is standardized to a new distribution with μ=50 and σ=10. What is the new value of the original score? A • 59 B • 45 C • 46 D • 55 Learning Check - Answer • A score of X=59 comes from a distribution with μ=63 and σ=8. This distribution is standardized to a new distribution with μ=50 and σ=10. What is the new value of the original score? A • 59 B • 45 C • 46 D • 55 5.5 Computing z-Scores for a Sample • Populations are most common context for computing z-scores • It is possible to compute z-scores for samples – Indicates relative position of score in sample – Indicates distance from sample mean • Sample distribution can be transformed into z-scores – Same shape as original distribution – Same mean M and standard deviation s 5.6 Looking Ahead to Inferential Statistics • Interpretation of research results depends on determining if (treated) a sample is “noticeably different” from the population • One technique for defining “noticeably different” uses z-scores. Figure 5.9 Conceptualizing the Research Study Figure 5.10 Distribution of Weights of Adult Rats 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 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 Learning Check • 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. Learning Check Answer True • Each score location relative to all other scores is unchanged so the shape of the distribution is unchanged False • Number of z-scores above/below mean will be exactly the same as number of original scores above/below mean Equations? Concepts? Any Questions ?