Figure 5-3 The relationship between z-score values and locations in a population distribution. One S.D. Two S.D.

Transforming back and forth between X and z-score The basic z-score formula for computing the z-score for any value of X. z = (X – μ) ∕ σ Example 5.3: μ = 86, σ = 7, convert X = 95 to z-score z = (95 – 86) ∕ 7 = 1.29 Equation for computing the value of X corresponding to any specific z-score. X = μ + zσ Example:5.3: μ = 60, σ = 5, convert z = -3.00 to X value X = 60 + (-3.00)(5) = 45

Other Relationships between z, X, m, and s In most cases, we simply transform scores (X values) into z-scores, or change z-scores back into X values. However, you should realize that a z-score establishes a relationship between the score, mean, and standard deviation. This relationship can be used to answer a variety of different questions about scores and the distributions in which they are located.

Other Relationships between z, X, m, and s In a population with a mean of μ = 65, a score of X = 59 corresponds to z = −2.00. What is the standard deviation for the population? z = (X – μ) ∕ σ -2.00 =(59 – 65) / σ σ = 3 Figure 5.4 A visual representation of the question in Example 5.4. If 2 standard deviations correspond to a 6-point distance, then 1 standard deviation must equal 3 points.

Other Relationships between z, X, m, and s Example 5.6 In a population distribution, a score of X = 54 corresponds to z = +2.00 and a score of X = 42 corresponds to z = −1.00. What are the values for the mean and the standard deviation for the population?

Other Relationships between z, X, m, and s FIGURE 5.5 A visual presentation of the question in Example 5.6. The 12-point distance from 42–54 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.

Using z-Scores to Standardize a Distribution It is possible to transform every X value in a distribution into a corresponding z-score. If every X value is transformed into a z-score, then the distribution of z-scores will have the following properties: The distribution of z-scores will have exactly the same shape as the original distribution of scores. The z-score distribution will always have a mean of zero. The distribution of z-scores will always have a standard deviation of 1.

A z-score Transformation Figure 5.6 An entire population of scores is transformed into z-scores. does not change the shape of the population mean is transformed into a value of 0 convert X = 100 to z-score (X=100 is at the mean) z = (X – μ) ∕ σ z = 100-100 ∕ 10 = 0 standard deviation is transformed to a value of 1 Convert X = 110 to z-score (X=110 is one S.D.) z = (X – μ) ∕ σ z = 110-100 ∕ 10 = 1

Figure 5-7 A z-score Transformation Following a z-score transformation, the X-axis is relabeled in z-score units. The distance that is equivalent to 1 standard deviation on the X-axis (σ = 10 points in this example) corresponds to 1 point on the z-score scale.

Figure 5-8 Transforming a Distribution of Raw Scores Using example 5 Figure 5-8 Transforming a Distribution of Raw Scores Using example 5.8 Transforming a distribution of raw scores (top) into z-scores (bottom) will not change the shape of the distribution.

z-Scores as a Standardized Distribution A standardized distribution is composed of scores that have been transformed to create predetermined values for μ and σ. The advantage of standardizing distributions is that two (or more) different distributions can be made the same. For example, one distribution has μ = 100 and σ = 10, and another distribution has μ = 40 and σ = 6. When these distribution are transformed to z-scores, both will have μ = 0 and σ = 1. individual scores from different distributions can be directly compared z-score of +1.00 specifies the same location in all z-score distributions

Using z-scores to make a comparison For example Dave wants to compare Psychology to Biology has a score of 60 on the Psychology exam has a score of 56 on the Biology exam in which course is he doing better? Psychology exam has μ = 50 and μ = 10 z = (X – μ) ∕ σ z = (60 – 50) / 10 = +1.00 Biology exam has μ = 48 and μ = 4 z = (X – μ) ∕ σ z = (56 – 48) / 4 = +2.00

Other Standardized Distributions Based on z-Scores transforming X values into z-scores creates a standardized distribution z-scores are confusing consist of many decimal values negative numbers more convenient to standardize a distribution into numerical values that are more familiar than z-scores select the mean and standard deviation for example IQ Pick a convenient mean such as 100 Pick a convenient standard deviation such as 15 Setp1: convert original scores to z-scores Step 2: then convert z-scores to a new distribution with selected mean and standard deviation

Using z-Scores for Making Comparisons An instructor gives an exam to a psychology class. For this exam, the distribution of raw scores has a mean of μ = 57 with σ = 14. The instructor would like to simplify the distribution by transforming all scores into a new, standardized distribution with μ = 50 and σ = 10. To demonstrate this process, we will consider what happens to two specific students: Maria, who has a raw score of X = 64 in the original distribution Joe, whose original raw score is X = 43.

Table 5.1 A demonstration of how two individual scores are changed when a distribution is standardized. See Example 5.9. Exam with µ = 57, σ = 14 Convert original score to z-score Maria: z = (X – μ) ∕ σ z = (64-57) ∕ 14 = +0.50 Convert z-score to new standardized score X = μ + zσ X = 50 + (+0.5)(10) = 55 Joe: z = (X – μ) ∕ σ z = (43-57) ∕ 14 = -1.00 X = μ + zσ X = 50 + (-1.0)(10) = 40

Figure 5-9 The distribution of exam scores from Example 5 Figure 5-9 The distribution of exam scores from Example 5.6 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, Maria has an original score of 64, a z-score of +0.5 and a standardized score of 55. 55 +0.5 64 ↑ Maria

Looking Ahead to Inferential Statistics A typical research study begins with a question about how a treatment will affect the individuals in a population. Because it is usually impossible to study an entire population, the researcher selects a sample and administers the treatment to the individuals in the sample.

Figure 5-10 A diagram of a research study Figure 5-10 A diagram of a research study. The goal of the study is to evaluate the effective-ness of a treatment. One individual is selected from the population and the treatment is administered to that individual. If, after treatment, the individual is noticeably different from the original population, then we have evidence that the treatment does have an effect.

Normal Curve with Standard Deviation | + or - one s.d. |

Figure 5-11 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 or -2.00 are extreme and noticeably different from most of the others in the distribution. 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 or −2.00 are extreme and noticeably different from most of the others in the distribution. Z =0.90 z = 2.50

A z-score Transformation using SPSS