Chapter 5 z-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 1 Understand z-score as location in distribution 2 Transform X value into z-score 3 Transform z-score into X value 4 Describe effects of standardizing a distribution 5 Transform scores to standardized distribution

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 FIGURE 5.1 Two distributions of exam scores. For both distributions, μ = 70, but for one distribution, σ = 3 and for the other, σ = 12. The relative position of X = 76 is very different for the two 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 FIGURE 5.2 The relationship between z-score values and locations in a population distribution.

Learning Check A z-score of z = +1.00 indicates a position in a distribution ____ A Above the mean by 1 point B Above the mean by a distance equal to 1 standard deviation C Below the mean by 1 point D Below the mean by a distance equal to 1 standard deviation

Learning Check - Answer A z-score of z = +1.00 indicates a position in a distribution ____ A Above the mean by 1 point B Above the mean by a distance equal to 1 standard deviation C Below the mean by 1 point D Below the mean by a distance equal to 1 standard deviation

Learning Check Decide if each of the following statements is True or False. T/F A negative z-score always indicates a location below the mean A score close to the mean has a z-score close to 1.00

Learning Check - Answer True Sign indicates that score is below the mean False Scores quite close to the mean have z-scores close to 0.00

Equation (5.1) for z-Score Numerator is a deviation score Denominator expresses deviation in standard deviation units Remember that z-scores identify a specific location of a score in terms of deviations from the mean and relative to the standard deviation.

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 Equations 5.1 and 5.2.

Figure 5.3 Visual Presentation of the Question in Example 5.4 FIGURE 5.3 A visual presentation of the question in Example 5.4. If 2 standard deviations correspond to a 6-point distance, then one standard deviation must equal 3 points.

Learning Check For a population with μ = 50 and σ = 10, what is the X value corresponding to z = 0.4? A 50.4 B 10 C 54 D 10.4

Learning Check - Answer For a population with μ = 50 and σ = 10, what is the X value corresponding to z = 0.4? A 50.4 B 10 C 54 D 10.4

Learning Check Decide if each of the following statements is True or False. T/F If μ = 40 and 50 corresponds to z = +2.00 then σ = 10 points If σ = 20, a score above the mean by 10 points will have z = 1.00

Learning Check - Answer False If z = +2 then 2σ = 10 so σ = 5 If σ = 20 then z = 10/20 = 0.5

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 5.4 A visual presentation of the question in Example 5.6. The 12-point distance from 42 to 54 corresponds to 3 standard deviations. Therefore, the standard deviation must be σ = 4. Also, the score X = 42 is below the mean by one standard deviation, so the mean must be μ = 46.

Figure 5.5 Transforming a Population of Scores FIGURE 5.5 An entire population of scores is transformed into z-scores. The transformation does not change the shape of the population, but the mean is transformed into a value of 0 and the standard deviation is transformed to a value of 1.

Figure 5.6 Axis Re-labeling After z-Score Transformation FIGURE 5.6 Following a z-score transformation, the X-axis is relabeled in z-score units. The distance that is equivalent to 1 standard deviation of the X-axis (σ = 10 points in this example) corresponds to 1 point on the z-score scale

Figure 5.7 Shape of Distribution After z-Score Transformation FIGURE 5.7 Transforming a distribution of raw scores (a) into z-scores (b) will not change the shape of the distribution.

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 Other Standardized 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 a Standardized Distribution FIGURE 5.8 The distribution of exam scores from Example 5.7. The original distribution was standardized to produce a new distribution with μ = 50 and σ = 10. Note that each individual is identified by an original score, a z-score, and a new, standardized score. For example, Joe has an original score of 43, a z-score of -1.00, and a standardized score of 40.

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? A 59 B 45 C 46 D 55

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? A 59 B 45 C 46 D 55

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 mean M and standard deviation s

5.6 Looking Ahead to Inferential 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 5.9 A diagram of a research study. The goal of the study is to evaluate the effect of a treatment. A sample is selected from the populations and the treatment is administered to the sample. If, after treatment, the individuals are noticeably different for the individuals in the original population, then we have evidence that the treatment does have an effect.

Figure 5.10 Distribution of Weights of Adult Rats FIGURE 5.10 The distribution of weights for the population of adult rats. Note that individuals with z-scores near 0 are typical or representative. However, individuals with z-scores beyond +2.00 and -2.00 are extreme and noticeably different from most of the others in the distribution.

Learning Check Chemistry Spanish 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? A Chemistry B Spanish C There is not enough information to know

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? A Chemistry B Spanish C There is not enough information to know

Learning Check Decide if each of the following statements is True or False. T/F Transforming an entire distribution of scores into z-scores will not change the shape of the distribution. If a sample of n = 10 scores is transformed into z-scores, there will be five positive z-scores and five negative z-scores.

Learning Check Answer True Each score location relative to all other scores is unchanged so the shape of the distribution is unchanged False Number of z-scores above/below mean will be exactly the same as number of original scores above/below mean

Any Questions? Concepts? Equations?