Chapter 5: z-Scores. 5.1 Purpose of z-Scores Identify and describe location of every score in the...
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Transcript of Chapter 5: z-Scores. 5.1 Purpose of z-Scores Identify and describe location of every score in the...
Chapter 5: z-Scores
5.1 Purpose of z-Scores
• Identify and describe location of every score in the distribution
• Take different distributions and make them equivalent and comparable
Figure 5.1Two Exam Score Distributions
5.2 z-Scores and Location in a Distribution
• Exact location is described by z-score– Sign tells…
– Number tells…
Figure 5.2 Relationship Between z-Scores and Locations
Learning Check
• A z-score of z = +1.00 indicates a position in a distribution ____
•Above the mean by 1 point
A
•Above the mean by a distance equal to 1 standard deviation
B
•Below the mean by 1 point
C
•Below the mean by a distance equal to 1 standard deviation
D
Learning Check
• Decide if each of the following statements is True or False.
•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
T/F
Equation (5.1) for z-Score
• Numerator is a…
• Denominator expresses…
Determining a Raw Score From a z-Score
• so
• 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
X
z
Learning Check
• For a population with μ = 50 and σ = 10, what is the X value corresponding to z = 0.4?
•50.4
A
•10
B
•54
C
•10.4
D
Learning Check
• Decide if each of the following statements is True or False.
•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
T/F
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
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
5.5 Computing z-Scoresfor 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 location for mean M and standard deviation
s
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?
•Chemistry
A
•Spanish
B
•There is not enough information to know
C