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 = Steps1: compute z-score z = 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