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INTRODUCTION TO z-SCORES In summary, the process of transforming X values into z-scores serves two useful purposes: 1- Each z-score will tell the exact location of the original X value with in the distribution. 2- The z-scores will form a standardized distribution that can be directly compared to other distributions that also have been transformed into z-scores.
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INTRODUCTION TO z-SCORES
The purpose of z-scores, or standard scores, is to identify and describe the exact location of every score in a distribution .
INTRODUCTION TO z-SCORES
In summary, the process of transforming X values into z-scores serves two useful purposes:
1- Each z-score will tell the exact location of the original X value with in the distribution.
2- The z-scores will form a standardized distribution that can be directly compared to other distributions that also have been transformed into z-scores.
z-SCORES AND LOCATION IN DISTRIBUTION One of the primary purposes of a z-scores
is to describe the exact location of a score within a distribution. The z-score accomplishes this goal by transforming each X value into a signed number (+ or -) so that
1- The sign tells whether the score is located above (+) or below (-) the mean, and
2- The number tells the distance between the score and the mean in terms of the number of standard deviations.
z-SCORES AND LOCATION IN DISTRIBUTION DEFINITION : A z-score specifies the precise location of
each X value within a distribution. The sign of the z-score (+ or -) signifies
whether the score is above the mean (positive) or below the mean (negative).
z-SCORES AND LOCATION IN DISTRIBUTION FIGURE 5.2: The relationship between
z-score values and locations in a population distribution.
The z-SCORE FORMULA The relationship between X values and
z-scores can be expressed symbolically in a formula. The formula for transforming raw scores is
z = X- μ
σ
USING z-SCORES TOSTANDARDIZE A DISTRIBUTION
1- Shape FIGURE 5.4: An entire population of scores is transformed into z-scores. The
transformation does not change the shape of the population but the mean is transformed into a value of 0 and the standard deviation is transformed to a value of 1.
USING z-SCORES TO STANDARDIZE A DISTRIBUTION
2- The Mean 3-The Standard Deviation DEFINTION A standardized distribution is composed
of scores that have been transformed to create predetermined values for μ and σ . standardized distributions are used to make dissimilar distribution comparable
USING z-SCORES TOSTANDARDIZE A DISTRIBUTION
FIGURE5.5 Following a z-score transformation the X-axis is relabeled in z-score
units. The distance that is equivalent to 1 standard deviation on the X-axis ( σ = 10 points in this example ) corresponds to 1 point on
the z-score scale .
TABLE 5.1
JOE MARIA
Raw score x = 64 43 Steps1: compute z-score z = +0.5 -1.0 Steps2: standardized score 55 40
OTHER STANDARDIZED DISTRIBUTION BASED ON z-SCORES TRANSFORMING z-SCORES TO A
DIATRIBUTION WITH A PREDETERMINED μ AND σ .
A FORMULA FOR FINDING THE STANDARDIZED SCORE
X= μ + z σ Standard score = μ new + z σ new
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