INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE Instructor: Dr. John J. Kerbs, Associate Professor Joint Ph.D. in Social Work and Sociology Chapter 5: Z-Scores Location of Scores and Standardized Distributions

z-Scores and Location By itself, a raw score or X value provides very little information about how that particular score compares with other values in the distribution. A score of X = 53, for example, may be a relatively low score, or an average score, or an extremely high score depending on the mean and standard deviation for the distribution from which the score was obtained. If the raw score is transformed into a z-score, however, the value of the z-score tells exactly where the score is located relative to all the other scores in the distribution.

Distribution Examples: Same μ and Different σ If you received a 76 on an exam, in which class would you prefer to have this score? Figure 5.2 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.

z-Scores and Location (continued) The process of changing an X value into a z-score involves creating a signed number, called a z-score The sign of the z-score (+ or –) identifies whether the X value is located above the mean (positive) or below the mean (negative). The numerical value of the z-score corresponds to the number of standard deviations between X and the mean of the distribution. Thus, a score that is located two standard deviations above the mean will have a z-score of +2.00. And, a z-score of +2.00 always indicates a location above the mean by two standard deviations.

Relationship between z-score Values & Locations in Population Distributions Note that mean is transformed into a value of 0 and the standard deviation is transformed into a value of 1. Transforming populations of scores into z-scores: Note that distribution shape does not change below. Figure 5.3 The relationship between z-score values and locations in a population distribution. 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.

Practice Interpreting z-Scores For the following z-scores, please describe the score’s location in each distribution. z = 1.75 z = - 0.50 z = 0.75 z = - 1.25 Identify the z-score value for the following locations in a distribution. Below the mean by 3 standard deviations Above the mean by ¼ of a standard deviation Below the mean by 1 standard deviations

Transforming Back and Forth Between X and z The basic z-score definition is usually sufficient to complete most z-score transformations. However, the definition can be written in mathematical notation to create a formula for computing the z-score for any value of X. X – μ z = ──── σ

Practice z-Score Calculations X – μ z = ──── σ With the formula above, please calculate: z for a distribution with μ = 20, σ = 6, X = 18 z for a distribution with μ = 20, σ = 6, X = 22 z for a distribution with μ = 20, σ = 6, X = 26 z for a distribution with μ = 20, σ = 6, X = 32

Transforming Back and Forth Between X and z (continued) Also, the terms in the formula can be regrouped to create an equation for computing the value of X corresponding to any specific z-score. X = μ + zσ

Practice z-Score Calculations X = μ + zσ With the formula above, please calculate: X for a distribution with μ = 20, σ = 6, z = 1.5 X for a distribution with μ = 20, σ = 6, z = -1.25 X for a distribution with μ = 20, σ = 6, z = 1/3 X for a distribution with μ = 20, σ = 6, z = -0.5 If μ = 50, X = 42 and z = - 2.00, what is the standard deviation (σ) for the distribution?

Relationship between z-score Values & Locations in Population Distributions The distance that is equal to 1 standard deviation on the x-axis (σ =10) corresponds to 1 point on the z-score scale. 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 on the X-axis (σ = 10 points in this example) corresponds to 1 point on the z-score scale.

The Three Properties of z-Scores 1. Shape The distribution of z-scores will have the exact same shape as the original distribution If the original distribution is negatively skewed, then the z-scores distribution will be negatively skewed If the original distribution is positively skewed, then the z-scores distribution will be positively skewed If the original distribution is normally distributed (symetrical), then the z-scores distribution will be normally distributed

The Three Properties of z-Scores (Continued) 2. The Mean The z-score distribution will always have a mean of 0 (i.e., μ = 0). By definition, this is why all positive z-scores are above the mean By definition, this is why all negative z-scores are below the mean

The Three Properties of z-Scores (Continued) 3. The Standard Deviation (σ) The z-score distribution will always have a standard deviation of 1 (i.e., σ = 1). Because all z-score distributions have the same mean and the same standard deviation, the z-score distribution is called a standardized distribution. Standardized distributions are used to make dissimilar distributions comparable.

z-scores and Locations In addition to knowing the basic definition of a z-score and the formula for a z-score, it is useful to be able to visualize z-scores as locations in a distribution. Remember, z = 0 is in the center (at the mean), and the extreme tails correspond to z-scores of approximately –2.00 on the left and +2.00 on the right. Although more extreme z-score values are possible, most of the distribution is contained between z = –2.00 and z = +2.00. Remember: about 95% of all scores fall within + or – 2 standard deviations from the mean

z-scores and Locations (Continued) The fact that z-scores identify exact locations within a distribution means that z-scores can be used as descriptive statistics and as inferential statistics. As descriptive statistics, z-scores describe exactly where each individual is located. As inferential statistics, z-scores determine whether a specific sample is representative of its population, or is extreme and unrepresentative.

z-Scores as a Standardized Distribution When an entire distribution of X values is transformed into z-scores, the resulting distribution of z-scores will always have a mean of zero and a standard deviation of one. The transformation does not change the shape of the original distribution and it does not change the location of any individual score relative to others in the distribution.

Transforming Raw Scores to z-scores: No Change in Distribution Shape Figure 5.7 Transforming a distribution of raw scores (a) into z-scores (b) will not change the shape of the distribution. z

z-Scores as a Standardized Distribution (Continued) 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.

z-Scores as a Standardized Distribution (Continued) Please convert the following population of N=6 scores (0, 12, 10, 4, 6, 4) into a standardized distribution Step 1: Calculate the mean μ = ΣX/N Step 2: Calculate the standard deviation (σ) Step 3: Calculate z-score for each value of X

z-Scores as a Standardized Distribution (Continued) Please convert the following population of N=6 scores (0, 12, 10, 4, 6, 4) into a standardized distribution Step 1: Calculate the mean μ = ΣX/N = 36/6 = 6 Because the mean is even, you can use the definitional formula of the SS in Step 2

z-Scores as a Standardized Distribution (Continued) Please convert the following population of N=6 scores (0, 12, 10, 4, 6, 4) into a standardized distribution Step 2: Calculate the standard deviation (σ) σ = SS = Σ(X - μ)2 = (0-6)2 + (12-6)2 + (10-6)2 + (4-6)2 + (6-6)2 + (4-6)2 = 36 + 36 + 16 + 4 + 0 + 4 = 96

z-Scores as a Standardized Distribution (Continued) Please convert the following population of N=6 scores (0, 12, 10, 4, 6, 4) into a standardized distribution Step 3: Calculate z-score for each value of X X-score (x-μ) where μ = 6 σ z-score = (x-μ) /σ - 6 4 z = - 1.50 12 6 z = 1.50 10 z = 1.00 - 2 z = - 0.50 z = 0.00

z-Scores as a Standardized Distribution (Continued) Because z-score distributions all have the same mean and standard deviation, individual scores from different distributions can be directly compared. A z-score of +1.00 specifies the same location in all z-score distributions.

z-Scores and Samples It is also possible to calculate z-scores for samples. The definition of a z-score is the same for either a sample or a population, and the formulas are also the same except that the sample mean and standard deviation are used in place of the population mean and standard deviation.

z-Scores and Samples Example Thus, for a score X from a sample, you can calculate the z-score as follows: X – M z = ───── s Using z-scores to standardize a sample also has the same effect as standardizing a population. Specifically, the mean of the z-scores will be zero (M z = 0) and the standard deviation of the z-scores will be equal to 1.00 (s z = 1) provided the standard deviation is computed using the sample formula (dividing n – 1 instead of n). Each z-score can be transformed into an X value as follows: X = M + z s

z-Scores and Samples Example Please use the formula below to calculate the following: X – M z = ───── s z for a distribution with M = 40, s = 12, X = 43 z for a distribution with M = 40, s = 12, X = 34 z for a distribution with M = 40, s = 12, X = 58 z for a distribution with M = 40, s = 12, X = 28 Answers: z = 0.25, - 0.50, 1.50, - 1.00

z-Scores and Samples Example Please use the formula below to calculate the following: X = M + z s X for a distribution with M = 80, s = 20, z = - 1.00 X for a distribution with M = 80, s = 20, z = 1.50 X for a distribution with M = 80, s = 20, z = - 0.50 X for a distribution with M = 80, s = 20, z = 0.80 Answers: X = 60, 110, 70, 96

Other Standardized Distributions Based on z-Scores Although transforming X values into z-scores creates a standardized distribution, many people find z-scores burdensome because they consist of many decimal values and negative numbers. Therefore, it is often more convenient to standardize a distribution into numerical values that are simpler than z-scores.

Other Standardized Distributions Based on z-Scores (Continued) To create a simpler standardized distribution, you first select the mean and standard deviation that you would like for the new distribution. This is your choice: e.g., μ = 50 and σ = 10 or μ = 100 and σ = 10 Then, z-scores are used to identify each individual's position in the original distribution and to compute the individual's position in the new distribution.

Other Standardized Distributions Based on z-Scores (Continued) Suppose, for example, that you want to standardize a distribution so that the new mean is μ = 50 and the new standard deviation is σ = 10. An individual with z = –1.00 in the original distribution would be assigned a score of X = 40 (below μ by one standard deviation) in the standardized distribution. Repeating this process for each individual score allows you to transform an entire distribution into a new, standardized distribution.

Other Standardized Distributions Based on z-Scores (Continued) Suppose, for example, that you want to standardize a distribution so that the new mean is μ = 50 and the new standard deviation is σ = 10. In the original distribution, μ = 68 and σ = 15 What is the z-score for an x value of 83 in the original distribution: Z = (83-68)/15 = 15/15 = + 1.00 An individual with an 83 in the original distribution would be given a z-score of + 1.00. In the new distribution (μ = 50, σ = 10), the original score of 83 would be assigned a value of 60 in the new standardized distribution (μ+1 σ = 50 +10 = 60), which is one standard deviation above the mean.