Z-Scores

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Z-Scores

• Quantitative Methods in HPELS
• HPELS 6210

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Agenda

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

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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

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Introduction

• Purpose of Z-scores
• Describe location of raw score
• Standardize distributions
• Make direct comparisons
• Statistical analysis

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Agenda

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

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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

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Z-Scores Locating Raw Scores

• Example 5.3, 5.4 p 144

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Z-Scores Locating Raw Scores

• Can also determine raw score from a Z-score
• X µ Z?

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Agenda

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

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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

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Z-Scores Standardizing Distributions

• Properties of Standardized Distributions
• Shape Same as original distribution
• Score position Same as original distribution
• Mean 0
• SD 1
• Figure 5.3, p 145

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Z-Scores Standardizing Distributions

• Example 5.5 and Figure 5.5, p 147

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µ 3 ? 2
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Agenda

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

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Z-Scores Making Comparisons

• Useful when comparing raw scores from two
  different distributions
• Example (p 148)
• Suppose Bob scored X60 on a psychology exam and
  X56 on a biology test. Which one should get the
  higher grade?

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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

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Psychology Exam Distribution µ 50 ? 10
Biology Exam Distribution µ 48 ? 4
Z X - µ / ? Z 60 50 / 10 Z 1.0
Z X - µ / ? Z 56 - 48 / 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
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Agenda

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

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Z-Scores Statistical Analysis

• Appropriate usage of the Z-score as a statistic
• Descriptive
• Parametric

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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

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Z-Scores Statistical Analysis

• Figure 5.8, p 153

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Z-Score Statistical Analysis

• Value 0 ? No treatment effect
• Value gt or lt 0 ? Potential treatment effect
• As value becomes increasingly greater or smaller
  than zero, the PROBABILITY of a treatment effect
  increases

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Textbook Problem Assignment

• Problems 1, 2, 9, 17, 23