Z-Scores Quantitative Methods in HPELS HPELS 6210

Agenda Introduction Location of a raw score Standardization of distributions Direct comparisons Statistical analysis

Introduction Z-scores use the mean and SD to transform raw scores  standard scores What is a Z-score?  A signed value (+/- X)  Sign: Denotes if score is greater (+) or less (-) than the mean  Value (X): Denotes the relative distance between the raw score and the mean Figure 5.2, p 141

Introduction Purpose of Z-scores: 1. Describe location of raw score 2. Standardize distributions 3. Make direct comparisons 4. Statistical analysis

Agenda Introduction Location of a raw score Standardization of distributions Direct comparisons Statistical analysis

Z-Scores: Locating Raw Scores Useful for comparing a raw score to entire distribution Calculation of the Z-score:  Z = X - µ /  where X = raw score µ = population mean  = population standard deviation

Z-Scores: Locating Raw Scores Example 5.3, 5.4 p 144

Z-Scores: Locating Raw Scores Can also determine raw score from a Z-score:  X = µ + Z 

Agenda Introduction Location of a raw score Standardization of distributions Direct comparisons Statistical analysis

Z-Scores: Standardizing Distributions Useful for comparing dissimilar distributions Standardized distribution: A distribution comprised of standard scores such that the mean and SD are predetermined values Z-Scores:  Mean = 0  SD = 1 Process:  Calculate Z-scores from each raw score

Z-Scores: Standardizing Distributions Properties of Standardized Distributions: 1. Shape: Same as original distribution 2. Score position: Same as original distribution 3. Mean: 0 4. SD: 1 Figure 5.3, p 145

Z-Scores: Standardizing Distributions Example 5.5 and Figure 5.5, p 147

µ=3=2µ=3=2

Agenda Introduction Location of a raw score Standardization of distributions Direct comparisons Statistical analysis

Z-Scores: Making Comparisons Useful when comparing raw scores from two different distributions Example (p 148):  Suppose Bob scored X=60 on a psychology exam and X=56 on a biology test. Which one should get the higher grade?

Z-Score: Making Comparisons Required information:  µ of each distribution of raw scores   of each distribution of raw scores Calculate Z-scores from each raw score

Psychology Exam Distribution: µ = 50  = 10 Z = X - µ /  Z = 60 – 50 / 10 Z = 1.0 Biology Exam Distribution: µ = 48  = 4 Z = X - µ /  Z = / 4 Z = 2.0 Based on the relative position (Z-score) of each raw score, it appears that the Biology score deserves the higher grade

Agenda Introduction Location of a raw score Standardization of distributions Direct comparisons Statistical analysis

Z-Scores: Statistical Analysis Appropriate usage of the Z-score as a statistic:  Descriptive  Parametric

Z-Scores: Statistical Analysis Review: Experimental Method Process: Manipulate one variable (independent) and observe the effect on the other variable (dependent)  Independent variable: Treatment  Dependent variable: Test or measurement

Z-Scores: Statistical Analysis Figure 5.8, p 153

Z-Score: Statistical Analysis Value = 0  No treatment effect Value > or < 0  Potential treatment effect  As value becomes increasingly greater or smaller than zero, the PROBABILITY of a treatment effect increases

Textbook Problem Assignment Problems: 1, 2, 9, 17, 23