- The document discusses transforming raw scores into z-scores and standardized distributions. It covers how to calculate z-scores, what z-scores represent in terms of distance from the mean, and how standardizing distributions makes them comparable. The key aspects of standardizing include transforming the raw scores to have a mean of 0 and standard deviation of 1 while maintaining the same shape.

1. PowerPoint Lecture Slides
Essenfials of Statisflcs for tfae Behavioral
Sciences
Seventh Edition
by Frederick J. Gravetter and Larry B. Wallnau

2. • Understand z-score as location in distribution
• Transform Xvalue into z-score
• Transform z-score into X value
• Describe effects of standardizing a distribution
• Transform scores to standardized distribution

3. • Identify and describe location of every
score in the distribution
• Standardize an entire distribution
• Takes different distributions and makes them
equivalent and comparable

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x=76
a4 4a ae to s2 sa in æ ao sy ód Ó OB 7s 72 7d zb 7a g0 s2 8d 8ó aa so m 9d 90 9a

5. • 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

6. Figure 5.2 Relationship of z-scores
and locations
—
2 —
1 0 +1 +2

7. • A z-score of z = +1.00 indicates a position
in a distribution
• Above the mean by 1 point
• Above the mean by a distance equal to 1
standard deviation
• Below the mean by 1 point
• Below the mean by a distance equal to 1
standard deviation

8. • Decide if each of the following statements
is True or False.
• A negative z-score always indicates
a location below the mean
• A score close to the mean has a
z-score close to 1.00

9. • A z-score of z +1.00 indicates a position
in a distribution
• Above the mean by a distance equal to 1
standard deviation

10. • Sign ïndicates that score is below
the mean
• Scores close to 0 have z-scores
close to 0.00

11. 80 90 100 110 120
—
2 —
1 0 +1 +2

12. • If 2o = 10 then o = 5
• Why

14. 59
ó points

15. 29 43 57 71
I ‹ I
—
2 —
1 0 +1
30 d0 50 ó0
Joe
85
< Original scores (g 57 ana ‹r 14}
z — z-Scores (ç = 0 and rr = I )
+ 2
70
‹— Standardizea scores (y —50 and ‹r — 0)

16. • Numerator is a deviation score
• Denominator expresses deviation in
standard deviation units

17. • For a population with g = 50 and o = 10,
what is the X value corresponding to
z=0.4?
• 50.4
• 10
•54
•10.4

18. —1.5 —
1.0 —0.5 0 +0.5 +l .0 +l .5
Frequency
Frequency

19. 80
Population of scores
(Kvolues)
10
110 120
Transform ło z
-2
PoDulałion of z-scores
zvciIues)
+2

20. Extreme
(z Deyond —2.00}
200
Representative
individuals
(znear 0)
Population
of
nontreateo rots
380 400 420
J.00
Extreme
individuals
(z Deyond +2.00)
440
00 2.00
X - 418
X

21. • Decide if each of the following statements
is True or False.
• Transforming an entire distribution of
scores into z-scores will not change the
shape of the distribution.
• If a sample of n = 10 scores is transformed
into z-scores, there will be five positive z
scores and five negative z-scores.

22. • A score of X=59 comes from a distribution with
y=63 and o=8. This distribution is standardized
so that the new distribution has y=63 and o=8.
What is the new value of the original score?
• 45

23. • A score of X=59 comes from a distribution with
y=63 and o=8. This distribution is standardized
so that the new distribution has y=63 and o=8.
What is the new value of the original score?
• 59
• 45
• 46
• 55

24. • Numerator is a deviation score
• Denominator expresses deviation in
standard deviation units

25. • Decide if each of the following statements
is True or False.
• If y = 40 and X = 50 corresponds
to z=+2.00, then o = 10 points
• If o 20, a score above the mean
by 10 points will have z = 1.00

26. • All z-scores are comparable to each other
• Scores from different distributions can be
converted to z-scores
• The z-scores (standardized scores) allow the
comparison of scores from two different
distributions along

27. • Interpretation of research results depends on
determining if (treated) sample is noticeably
different from the population
• One technique for defining noticeably
different uses z-scores.

28. • 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

29. 5.4 Other Standardized Distributions
• Process of standardization is widely used
—AT has g = 500 and cr = 100
—IQ has y = 100 and o = 15 Point
• 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 specificy and o are attained.

30. • Last week Andi had exams in Chemistry and in
Spanish. On the chemistry exam, the mean was
g = 30 with rr = 5, and Andi had a score of X = 45.
On the Spanish exam, the mean was y = 60 with
o = 6 and Andi had a score of X = 65. For which
class should Andi expect thebetter grade?
• Chemistry
• Spanish
• There is not enough information to know

31. • 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