Transcript z-scores: Location of Scores and Standardized Distributions

Chapter 5
z-Scores: Location of Scores and
Standardized Distributions
PowerPoint Lecture Slides
Essentials of Statistics for the
Behavioral Sciences
Eighth Edition
by Frederick J. Gravetter and Larry B. Wallnau
Chapter 5 Learning Outcomes
1
• Understand z-score as location in distribution
2
• Transform X value into z-score
3
• Transform z-score into X value
4
• Describe effects of standardizing a distribution
5
• Transform scores to standardized distribution
Tools You Will Need
• The mean (Chapter 3)
• The standard deviation (Chapter 4)
• Basic algebra (math review, Appendix A)
5.1 Purpose of z-Scores
• Identify and describe location of every
score in the distribution
• Standardize an entire distribution
• Take different distributions and make them
equivalent and comparable
Figure 5.1
Two Exam Score Distributions
5.2 z-Scores and Location in a
Distribution
• Exact location is described by z-score
– Sign tells whether score is located above or below
the mean
– Number tells distance between score and mean in
standard deviation units
Figure 5.2 Relationship Between
z-Scores and Locations
Learning Check
• A z-score of z = +1.00 indicates a position in a
distribution ____
A
• Above the mean by 1 point
B
• Above the mean by a distance equal to 1
standard deviation
C
• Below the mean by 1 point
D
• Below the mean by a distance equal to 1
standard deviation
Learning Check - Answer
• A z-score of z = +1.00 indicates a position in a
distribution ____
A
• Above the mean by 1 point
B
• Above the mean by a distance equal to 1
standard deviation
C
• Below the mean by 1 point
D
• Below the mean by a distance equal to 1
standard deviation
Learning Check
• Decide if each of the following statements
is True or False.
T/F
• A negative z-score always indicates
a location below the mean
T/F
• A score close to the mean has a
z-score close to 1.00
Learning Check - Answer
True
• Sign indicates that score is below
the mean
False
• Scores quite close to the mean
have z-scores close to 0.00
Equation (5.1) for z-Score
z
X 

• Numerator is a deviation score
• Denominator expresses deviation in standard
deviation units
Determining a Raw Score
From a z-Score
•
z
X 

so
X    z
• Algebraically solve for X to reveal that…
• Raw score is simply the population mean plus
(or minus if z is below the mean) z multiplied
by population the standard deviation
Figure 5.3 Visual Presentation
of the Question in Example 5.4
Learning Check
• For a population with μ = 50 and σ = 10, what
is the X value corresponding to z = 0.4?
A • 50.4
B • 10
C • 54
D • 10.4
Learning Check - Answer
• For a population with μ = 50 and σ = 10, what
is the X value corresponding to z = 0.4?
A • 50.4
B • 10
C • 54
D • 10.4
Learning Check
• Decide if each of the following statements
is True or False.
T/F
• If μ = 40 and 50 corresponds to
z = +2.00 then σ = 10 points
T/F
• If σ = 20, a score above the mean
by 10 points will have z = 1.00
Learning Check - Answer
False
• If z = +2 then 2σ = 10 so σ = 5
False
• If σ = 20 then z = 10/20 = 0.5
5.3 Standardizing a Distribution
• Every X value can be transformed to a z-score
• Characteristics of z-score transformation
– Same shape as original distribution
– Mean of z-score distribution is always 0.
– Standard deviation is always 1.00
• A z-score distribution is called a
standardized distribution
Figure 5.4 Visual Presentation of
Question in Example 5.6
Figure 5.5 Transforming a
Population of Scores
Figure 5.6 Axis Re-labeling
After z-Score Transformation
Figure 5.7 Shape of Distribution
After z-Score Transformation
z-Scores Used for Comparisons
• All z-scores are comparable to each other
• Scores from different distributions can be
converted to z-scores
• z-scores (standardized scores) allow the direct
comparison of scores from two different
distributions because they have been
converted to the same scale
5.4 Other
Standardized Distributions
• Process of standardization is widely used
– SAT has μ = 500 and σ = 100
– IQ has μ = 100 and σ = 15 Points
• Standardizing a distribution has two steps
– Original raw scores transformed to z-scores
– The z-scores are transformed to new X values so
that the specific predetermined μ and σ are
attained.
Figure 5.8 Creating a
Standardized Distribution
Learning Check
• A score of X=59 comes from a distribution with
μ=63 and σ=8. This distribution is standardized to
a new distribution with μ=50 and σ=10. What is
the new value of the original score?
A • 59
B • 45
C • 46
D • 55
Learning Check - Answer
• A score of X=59 comes from a distribution with
μ=63 and σ=8. This distribution is standardized to
a new distribution with μ=50 and σ=10. What is
the new value of the original score?
A • 59
B • 45
C • 46
D • 55
5.5 Computing z-Scores
for a Sample
• Populations are most common context for
computing z-scores
• It is possible to compute z-scores for samples
– Indicates relative position of score in sample
– Indicates distance from sample mean
• Sample distribution can be transformed into
z-scores
– Same shape as original distribution
– Same mean M and standard deviation s
5.6 Looking Ahead to
Inferential Statistics
• Interpretation of research results depends on
determining if (treated) a sample is
“noticeably different” from the population
• One technique for defining “noticeably
different” uses z-scores.
Figure 5.9 Conceptualizing
the Research Study
Figure 5.10 Distribution of
Weights of Adult Rats
Learning Check
• Last week Andi had exams in Chemistry and in Spanish.
On the chemistry exam, the mean was µ = 30 with σ = 5,
and Andi had a score of X = 45. On the Spanish exam, the
mean was µ = 60 with σ = 6 and Andi had a score of X =
65. For which class should Andi expect the better grade?
A
• Chemistry
B
• Spanish
C
• There is not enough information to know
Learning Check - Answer
• Last week Andi had exams in Chemistry and in Spanish.
On the chemistry exam, the mean was µ = 30 with σ = 5,
and Andi had a score of X = 45. On the Spanish exam, the
mean was µ = 60 with σ = 6 and Andi had a score of X =
65. For which class should Andi expect the better grade?
A
• Chemistry
B
• Spanish
C
• There is not enough information to know
Learning Check
• Decide if each of the following statements
is True or False.
T/F
• Transforming an entire distribution of
scores into z-scores will not change the
shape of the distribution.
T/F
• If a sample of n = 10 scores is transformed
into z-scores, there will be five positive zscores and five negative z-scores.
Learning Check Answer
True
• Each score location relative to all other
scores is unchanged so the shape of the
distribution is unchanged
False
• Number of z-scores above/below mean
will be exactly the same as number of
original scores above/below mean
Equations?
Concepts?
Any
Questions
?