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Chapter 5 - z-Scores: Location of Scores and Standardized Distributions -…
Chapter 5 - z-Scores: Location of Scores and Standardized Distributions
5-1 Introduction
Purpose of z-scores (standard score):
identify and describe the exact location of each score in a distribution
to standardize an entire distribution
raw scores
- the original, unchanged scores that are the direct result of measurement
5-2 z-Scores and Locations in a Distribution
z-score transforms each X value into a signed number (+ or -)
the sign tells whether the score is located above (+) or below (-) the mean
the number tells the distance between the score and the mean in terms of numbers of standard deviations
z-score
- specifies the precise location of each X value within a distribution
The z-score Formula for a Population
formula for transforming scores into z-scores is
provides a structured equation to organize the calculations when the numbers are more difficult
Determining a Raw Score (X) from a z-Score
Computing z-Scores for Samples
each X value in a sample can be transformed into a z-score:
each z-score can be transformed back into an X value:
5-3 Other Relationships between z, X, the Mean, and the Standard Deviation
z-scores establish a relationship between the score, the mean, and the standard deviation
5-4 Using z-Scores to Standardize a Distribution
Population Distributions
z-score transformation - process of transforming every X value in a population into a corresponding z-score.
distribution of z-scores will have the following properties:
shape - exact same shape as the original distribution of scores
the mean - will always have a mean of zero
the standard deviation - always have a SD of 1
Sample Distributions
transformed distribution of z-scores will have:
the same shape as the original sample of scores
Mean of 0
SD of 1
Using z-scores for Making Comparisons
When any distribution is transformed into z-scores, the resulting distribution will always have a mean of 0 and SD of 1
Standardized distribution
- composed of scores that have been transformed to create predetermined values for mean and SD; used to make dissimilar distributions comparable
Ex of standardized distribution with 0 mean and 1 SD = a z-score distribution
Advantage: makes it possible to compare different scores or different individuals even though they come from completely distributions
5-5 Other Standardized Distributions Based on z-Scores
it is common to standardize a distribution by transforming the scores into a new distribution with a predetermined mean and standard deviation that are positive whole numbers
Goal: to create a new (standardized) distribution that has "simple" values but does not change locations within the distribution
Procedure:
Transform original scores into z-scores
Transform z-scores into new X values
5-6 Looking Ahead to Inferential Statistics
interpretation of research results depends on whether the sample is noticeably different from the population
To decide whether a sample is noticeably different, use z-scores.
