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z-Scores: Location of Scores and Standardized Distributions, Transform…
z-Scores: Location of Scores and Standardized Distributions
Z-Score - What is it and how does it work
• Transforms raw scores (X values)
• Uses mean as reference point
• Uses standard deviation as yardstick
• Sign indicates above (+) or below (-) mean
• Number indicates distance in SD units
Two Main Purposes
Location: Identify exact position of each score in distribution
Standardization: Create equivalent distributions for comparison (e.g., IQ scores)
The z-score accomplishes this goal by transforming each X value into a signed number (+ or −) so that
the sign tells whether the score is located above (+) or below (−) the mean, and
the number tells the distance between the score and the mean in terms of the number of standard deviations.
in a distribution of IQ scores with μ and σ, a score of would be transformed into . The z-score value indicates that the score is located above the mean (+) by a distance of 2 standard deviations (30 points).
Population Formulas:
Population Formula z = (X - μ) / σ
Population Equation 5.2: X = μ + z(σ
Raw score- original, unchanged scores that are the direct result of measurement and don't provide position information by themselves
Sample Formulas
Sample Equation: z = (X - M) / s
Sample Formula: X = M + z(s)
Z-Score Transformation - relabeling of X values in a population into precise X-value locations within a distribution
Standardized Distribution - Distributions with predetermined mean and standard deviation values (like IQ tests with μ=100, σ=15, or SAT with μ=500, σ=100) that make dissimilar distributions comparable
Standardization Process
Z-SCORE DISTRIBUTION PROPERTIES
• Shape: Identical to original
• Mean: Always μ = 0
• Standard Deviation: Always σ = 1
• Individual positions: Unchanged
• Advantage: Direct comparison possible
• Contains negative values & decimals
Practical Applications
• Compare scores from different distributions
• Educational testing (IQ, SAT)
• Identify unusual scores
• Grade assignments across courses
Research Applications
• Evaluate treatment effects
• Compare treated sample to population
• Inferential statistics foundation
• Determine statistical significance
Deviation Score -The numerator in the z-score formula (X - μ) or (X - M), which measures the distance in points from the mean. The sign shows direction, and when divided by the standard deviation, it creates the z-score.
Standardized Score - z-scores are one type of standardized score, which are the result of the transformation process. These can have any predetermined mean and standard deviation while maintaining the original distribution shape and enabling cross-test comparisons.
Transform original scores to z-scores
Transform z-scores to new distribution
Goal: Predetermined mean & SD
(e.g., SAT: μ=500, σ=100)