chapter 5 Z-Scores (Essentials of Statistics for the Behavioral Sciences)

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?

1. Chapter 5 z-Scores
PowerPoint Lecture Slides

Essentials of Statistics for the Behavioral
Sciences
Seventh Edition

by Frederick J. Gravetter and Larry B. Wallnau

2. Chapter 5 Learning Outcomes

3. Concepts to review
•The mean (Chapter 3)

•The standard deviation (Chapter 4)

•Basic algebra (math review, Appendix A)

4. 5.1 Purpose of z-Scores
•Identify and describe location of every
score in the distribution

•Standardize an entire distribution
•Takes different distributions and makes them
equivalent and comparable

5. Figure 5.1 Two distributions of
exam scores

6. 5.2 Locations and Distributions
•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

7. Figure 5.2 Relationship of z-scores
and locations

8. Learning Check
•A z-score of z = +1.00 indicates a position
in a distribution ____

9. Learning Check - Answer
•A z-score of z = +1.00 indicates a position
in a distribution ____

10. Learning Check
•Decide if each of the following statements
is True or False.

11. Answer

12. Equation for z-score
•Numerator is a deviation score

•Denominator expresses deviation in
standard deviation units

13. Determining raw score
from z-score
•Numerator is a deviation score

•Denominator expresses deviation in
standard deviation units

14. Figure 5.3 Example 5.4

15. Learning Check
•For a population with μ = 50 and σ = 10,
what is the X value corresponding to
z=0.4?

16. Learning Check - Answer
•For a population with μ = 50 and σ = 10,
what is the X value corresponding to
z=0.4?

17. Learning Check
•Decide if each of the following statements
is True or False.

18. Answer

19. 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

20. Figure 5.4 Transformation of a
Population of Scores

21. Figure 5.5 Axis Re-labeling

22. Figure 5.6 Shape of Distribution
after z-Score Transformation

23. z-Scores for Comparisons
•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

24. 5.4 Other Standardized Distributions
•Process of standardization is widely used
– AT has μ = 500 and σ = 100
–IQ has μ = 100 and σ = 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 specific μ and σ are attained.

25. Figure 5.7 Creating a Standardized
Distribution

26. Learning Check
•A score of X=59 comes from a distribution with
μ=63 and σ=8. This distribution is standardized
so that the new distribution has μ=63 and σ=8.
What is the new value of the original score?

27. Learning Check
•A score of X=59 comes from a distribution with
μ=63 and σ=8. This distribution is standardized
so that the new distribution has μ=63 and σ=8.
What is the new value of the original score?

28. 5.5 Computing z-Scores for Samples
•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

29. 5.6 Looking to Inferential Statistics
•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.

30. Figure 5.8 Diagram of Research Study

31. Figure 5.9 Distributions of weights

32. 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?

33. 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?

34. Learning Check TF
•Decide if each of the following statements
is True or False.