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PRED 354 TEACH. PROBILITY &
STATIS. FOR PRIMARY MATH
Lesson 4
Z-SCORES
CORRECTIONS
1.
Having gap in the distribution
2.
Real limits
3.
Grouped-frequency distributions
CORRECTIONS
Real Limits
Assigment
2,5  38,46%
3 -> 42,3%
46,15% --- ??
X
f
c%
10
2
9
7
8
2
7
1
6
0
5
2
53,84
4
0
46,15
3
2
46,15
2
8
38,46
1
2
7,69
CORRECTIONS
Grouped frequency distribution tables




Rule 1. The grouped frequency distribution
table should have about 10 class intervals.
Rule 2. The width of each interval should be
relatively simple number.
Rule 3. The bottom score in each class
interval should be a multiple of the width.
Rule 4. All intervals should be the same
width.
CORRECTIONS
X
width
Bottom
score
25-35
25
15-25
15
5-15
11
5
Quiz
ANNOUNCED IN THE CLASS
Z-scores: location of scores and
standardized distributions
A z-score specifies the precise location of
each X value within a distribution.
1.
2.
The sign tells whether the score is located
above (+) or below (-) the mean, and
The number tells the distance between the
score and the mean in terms of the number
of standard deviations.
Z-scores: location of scores and
standardized distributions
z = (X - µ) / σ
EX. The distribution of SAT verbal scores for high school
seniors has a mean of µ=500 and a standard deviation
of σ=100. He took the SAT and scored 430 on the
verbal subtest. Locate his score in the distribution by
using a z-score.
Z-scores: location of scores and
standardized distributions
EX. A distribution of exam scores has a mean of 50 and
standard deviation of 8.
What raw score corresponds to z=-1/2?
What is z-score of X=68?
PURPOSE: to convert each individual score into a
standardized z-score, so that the resulting z-score
provides a meaningful description of exact location of
the individual score within the distribution.
Z-scores: location of scores and
standardized distributions
Whenever you are working with z-scores,
you should imagine or draw a picture
similar to this figure. Although you should
realize that not all distributions are normal,
we will use the normal shape as an
example when showing z-scores.
Charateristics of a z-scores
distribution
1. Shape
2. The mean
3. The standard deviation
EX. A population of N=6 scores consists of the
following values:
0, 6, 5, 2, 3, 2
Find a) z-scores, b) graphs of the distributions
c)means d) σs
Using z-scores to make comparisons
Suppose, for example, that Ali recevied a
score of X=60 on math exam (µ=50, σ=10)
and a score of X=56 on biology test (µ=48,
σ=4).
Which exam score is better?
Why is it possible to compare scores from
different
distributions
after
each
distribution is tranformed into z-scores?
Transforming z-scores to a
predetermined µ and σ
the goal is to create a new standardized
distribution that has simple values for the
mean and standardized deviation, but does
not change any inidividual’s location within
the distribution.
Ex: An instuctor gives an exam to a physic class. For this
exam, the distribution of raw scores has a mean of
µ = 57 and σ = 14. The instructor would like to
simplify the distribution by tranforming all scores
into a new standardized distribution with µ = 50 and
σ = 10.
What is Ali’s and Veli’s new scores if they took 64
and 43, respectively?
Transforming z-scores to a
predetermined µ and σ
EX: For the
Scores:
2, 4, 6, 10, 13
following
population,
Tranform this distribution for µ=50, σ=20.
Why are z-scores important?
1.
Probability
2.
Evaluating treatment effects
3.
Measuring relationship