Introduction to computing z-scores and other standardized scores

2. 
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Prayer
A bit more about standard deviations
z-Scores: the basics
Standardizing distributions
Tuesday:
• More on standardized distributions /T-scores
• Using R
• The STORY in your data

3. LuAnn, S., J. Walter and D. Antosh. (2007) Dieting behaviors of young women post-college graduation.
College Student Journal 41:4.

4. Make fear and greed work for you
Wall Street constantly swings between these two emotions.
You can either get caught in the frenzy - or profit from it.
By Janice Revell, Money Magazine senior writer
Last Updated: July 21, 2009: 10:56 AM ET
“Making matters worse, the big stock bet would be far riskier on a year-to-year basis than
other strategies. The most common measure of portfolio risk is standard deviation, which
tells you how much an investment's short-term returns bounce around its long-term
average. Since 1926 stocks have returned average gains of 9.6% a year, with a standard
deviation of 21.5 percentage points, according to Ibbotson Associates. That means that
about two-thirds of the time, the annual return on stocks landed 21.5 percentage points
below or above the average - that is, in any given year, your results would range from a
12% loss to a 31% gain. You'd need either an iron stomach or a steady supply of Zantac to
stay the course. And if you happened to be at or near retirement when one of those really
bad years hit, you might have to rethink your plans.”
http://money.cnn.com/2009/07/20/pf/funds/fear_greed.moneymag/

5.  Explain how z-scores provide a description of a location in a distribution
 Transform an X score into a z-score
 Transform z-scores back into X scores, when the mean and standard
deviation are given.
 Use z-scores to make comparisons across variables and individuals.
 Describe the effects when an entire data set is standardized by
transforming all the scores to z-scores, including the impact on the shape,
mean and standard deviation, and its comparability to other standardized
distributions.
 Use z-scores to transform a distribution into a standardized distribution.
 Use SPSS to create standardized scores for a distribution.

7.  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

9. 64
46
67
58
70
70
73
82
76
94

10. 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

11. 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

12. 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

13. Answer
True
• Sign indicates that score is below
the mean
False
• Scores close to 0 have z-scores
close to 0.00

15. z 
X  X

 Numerator is a deviation score
 Denominator expresses deviation in
standard deviation units

16. z
X  X

so X  X  z 
 Numerator is a deviation score
 Denominator expresses deviation in
standard deviation units

18. 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

19. 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

20. Learning Check
• Decide if each of the following statements
is True or False.
T/F
• If μ = 40 and X = 50 corresponds
to z=+2.00, then σ = 5 points
T/F
• If σ = 20, a score above the mean
by 10 points will have z = 1.00

21. Answer
True
• If 2σ = 10 then σ = 5
False
• Why?

22.  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

23. • 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

28. 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

29. 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

30. Concepts
Equations
Interpretation

31.  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

32.  Process of standardization is widely used
• SAT has Mean = 500 and σ = 100
• IQ has Mean = 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.

33. This form of standardized score, with
M = 50 and  = 10, is known as a T-score.

34.  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.

37. 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

38. 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

39. Learning Check TF
• 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.

40. Concepts
Equations
Interpretation