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Chapter 5 Z-Scores
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Review ► We have finished the basic elements of descriptive statistics. ► Now we will begin to develop the concepts and skills that form the foundation for inferential statistics.
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Review (cont.) ► Chapter 5: Present a method for describing the exact location of an individual score relative to the other scores in a distribution. ► Chapter 6: Determine probability values associated with different locations in a distribution of scores ► Chapter 7: Apply skills from Ch. 5 & 6 to sample means instead of individual scores.
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Z-Scores ► Purpose of z-scores, or standard scores, is to identify and describe the exact location of every score in a distribution. ► To do this, we use the mean and standard deviation.
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Z-Scores ► A statistical technique that uses the mean and the standard deviation to transform each score (X value) into a z-score or a standard score. ► Purpose of a z-score or a standard score is to identify the exact location of every score in a distribution.
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Why are z-scores useful? ► If you got a 76 on a test, how did you do? ► You would need more information. ► You need to know the other scores in the distribution. ► What is the mean?
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Why are z-scores useful? (cont.) ► Knowing the mean is not enough. ► You also need to know the standard deviation. ► The relative location within the distribution depends on the mean and the statistical deviation as well as your score.
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Figure 5.1 Two distributions of exam scores Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning Score is in the extreme right hand tail – one of the highest in the distribution Score is only slightly above average.
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Purpose for z-scores ► A score by itself does not necessarily provide much information about its position within a distribution These are called raw scores ► To make raw scores more meaningful, they are often transformed into new values that contain more information. ► This transformation is one purpose for z- scores.
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Purpose for z-scores (cont.) ► We can transform scores into z-scores to find out exactly where the original scores are located. ► A second purpose is to standardize an entire distribution. IQ scores ► All are standardized with a mean of 100 and s.d. of 15 ► An IQ score of 95 is slightly below average and an IQ score of 145 is extremely high no matter what IQ test
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To describe the exact location of the score within a distribution ► A z-score transforms an X score into a signed +/- number + above the mean - below the mean The number tells the distance between the score and the mean in terms of the number of standard deviations.
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Example ► In a distribution of standardized IQ scores with and and a score of X=130 ► The score of X=130 could be transformed into z= +2.00 ► z value indicates + (above the mean) ► by a distance of 2 standard deviations (30 points)
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Definition ► A z-score Specifies the precise location of each X value with a distribution The sign of +/- signifies whether the score is above or below the mean The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between the X and
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Z-scores ► Consist of two parts +/- Magnitude ► Both parts are necessary to describe completely where a raw score is located within a distribution.
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Figure 5.2 The relationship between z-scores and locations in a distribution Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning
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► a z-score of z = +1.00 corresponds to a position exactly 1 standard deviation above the mean. ► a z-score of z = +2.00 corresponds to a position exactly 2 standard deviations above the mean. ► The numerical value tells you the number of standard deviations from the mean.
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Figure 5.1 Two distributions of exam scores Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning Now use a z-score to describe the position of X=76. Z= +2.00 The score is located above the mean by exactly 2 s.d. Z= +0.50 The score is located above the mean by 1/2 s.d.
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Learning Check ► A negative z-score always indicates a location below the mean. ► What z-score value identifies each of the following locations in a distribution? Above the mean by 2 s.d. Below the mean by ½ s.d. Above the mean by ¼ s.d. Below the mean by 3 s.d.
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Learning Check ► For a population with = 50 and = 10, find the z-score for each of the following scores: X = 55 X = 40 X = 30 Z= +0.50 Z= - 1.00 Z = - 2.00
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Figure 5.4 A z-score transformation Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning = 50 10 40 30
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Learning Check ► For a population with 50 and 10, find the X value corresponding to each of the following z-scores: z = + 1.00 z = - 0.50 z = + 2.00 X= 60 X = 45 X = 70
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Figure 5.4 A z-score transformation Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning = 50 10 40 30 6070
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Formula for transforming z-scores ► z = X –
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Example ► A distribution of scores has a mean of and a standard deviation of 10. What z-score corresponds to a score of X=120 in this distribution? Z = X –
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Transforming z-scores into X values ► X = + z = 60 + (-2.00)(5) = 60 + (-2.00)(5) = 60 + (-10.00) = 60 + (-10.00) = 50 = 50 Z = -2.00
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Using z-scores to Standardize a Distribution ► When 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 zero; The s.d. is transformed into a value of 1.
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Standardized Distribution ► A standardized distribution is composed of scores that have been transformed to create predetermined values for and ► Standardized distributions are used to make dissimilar distributions comparable.
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Using z-scores to Make Comparisons ► Example: pg. 112-113 ► Psychology score = 60 50 and 10 ► Biology score = 56 48 and 4 ► z= X – ► z= X –
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