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 5.1 Two 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 = 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

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 = 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-Scores for 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