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 pointA

•Above the mean by a distance equal to 1 standard deviation

B

•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 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

Xz

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

• 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?

•ChemistryA

•SpanishB

•There is not enough information to knowC