Chapter 5z-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
• Understand z-score as location in distribution1
• Transform X value into z-score2
• Transform z-score into X value3
• Describe effects of standardizing a distribution4
• Transform scores to standardized distribution5
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 ____
• Above the mean by 1 pointA
• Above the mean by a distance equal to 1 standard deviationB
• Below the mean by 1 pointC
• Below the mean by a distance equal to 1 standard deviation D
Learning Check - Answer
• A z-score of z = +1.00 indicates a position in a distribution ____
• Above the mean by 1 pointA
• Above the mean by a distance equal to 1 standard deviationB
• Below the mean by 1 pointC
• 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 meanT/F
• A score close to the mean has a z-score close to 1.00T/F
Learning Check - Answer
• Sign indicates that score is below the meanTrue
• Scores quite close to the mean have z-scores close to 0.00
False
Equation (5.1) for z-Score
Xz
• Numerator is a deviation score
• Denominator expresses deviation in standard deviation units
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
Xz zX
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?
• 50.4A
• 10B
• 54C
• 10.4D
Learning Check - Answer
• For a population with μ = 50 and σ = 10, what is the X value corresponding to z = 0.4?
• 50.4A
• 10B
• 54C
• 10.4D
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 pointsT/F
• If σ = 20, a score above the mean by 10 points will have z = 1.00T/F
Learning Check - Answer
• If z = +2 then 2σ = 10 so σ = 5 False
• If σ = 20 then z = 10/20 = 0.5False
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 OtherStandardized 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 aStandardized 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?
• 59A
• 45B
• 46C
• 55D
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?
• 59A
• 45B
• 46C
• 55D
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 mean M and standard deviation s
5.6 Looking Ahead toInferential 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?
• ChemistryA
• SpanishB
• There is not enough information to knowC
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?
• ChemistryA
• SpanishB
• There is not enough information to knowC
Learning Check
• 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.
T/F
• If a sample of n = 10 scores is transformed into z-scores, there will be five positive z-scores and five negative z-scores.
T/F
Learning Check Answer
• Each score location relative to all other scores is unchanged so the shape of the distribution is unchanged
True
• Number of z-scores above/below mean will be exactly the same as number of original scores above/below mean
False
AnyQuestions
?
Concepts?
Equations?