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

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Concepts?

Equations?