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