Transcript z-scores : location of scores and standardized distribution

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
X- μ
z=
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
Steps2: standardized score
z = +0.5
55
-1.0
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