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Z-Scores: Location of Scores and Standard Distributions - Coggle Diagram
Z-Scores: Location of Scores and Standard Distributions
Purpose of Z-Scores
Z-scores tell the exact location of the original X value within the distribution
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 the number of standard deviations
Z-scores form a standardized distribution that can be directly compared to other distributions that have also been transformed into z-scores
Z-score formula for a population: z- X-μ/σ
The numerator of the equation, X-μ is a deviation score
Determining a Raw Score (X) from a z-score
X = μ+zσ.. zσ is the deviation of X and determines both the direction and size of the distance from the mean
Z Scores for Samples: z = X-M/s
Other Relationships between z, X, the Mean and the Standard Deviation
Z-scores establish a relationship between the score, mean, and standard deviation. This relationship can answer a variety of different questions abt scores and the distributions in which they're located
Ex: Standard deviation for the population and mean for the sample
Using z-Scores to Standardized a Distribution
Transforming every X value in a population into a corresponding z-value gives it characteristics to make the z-score transformation a useful tool
Shape: the distribution of z-scores will have exactly the same shape as the original distribution of scores
The mean: the z-score distribution will always have a mean of zero
Standard deviation: the distribution of z-scores will always have a standard deviation of σ=1
Sample Distributions
the distribution for the sample of z-scores will have the same shape as the original sample of scores
the sample of z-scores will have a mean of Mz=0
the sample of z-scores will have a standard deviation of Sz = 1
Using z-scores for Making Comparisons
Because all z-score distributions have the same mean (0) and the same standard deviation (1), the z-score distribution is called a standardized distribution
Other Standardized Distributions Based on the z-Scores
Z-score distributions contain both negative and positive values and decimals. So people sometimes standardize a distribution by transforming those scores into a new distribution with a predetermined mean and standard deviation that are positive whole numbers
Ex: psychological or educational testing
New values for the mean and standard deviation
The original scores are transformed into z-scores
The z-scores are then transformed into a new X values so the specific mean and standard deviation are attained