Z-Scores

Quantitative Methods in HPELS

440:210

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 distribution2. Score position: Same as original

distribution3. Mean: 04. SD: 1 Figure 5.3, p 145

Z-Scores: Standardizing Distributions

Example 5.5 and Figure 5.5, p 147

µ = 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 = 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

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:DescriptiveParametric

Z-Scores: Statistical Analysis Review: Experimental Method Process: Manipulate one variable

(independent) and observe the effect on the other variable (dependent) Independent variable: TreatmentDependent 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