Chapter 5 Z-Scores

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.

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.

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.

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.

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?

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.

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.

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.

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

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.

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= ► z value indicates + (above the mean) ► by a distance of 2 standard deviations (30 points)

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 

Z-scores ► Consist of two parts  +/-  Magnitude ► Both parts are necessary to describe completely where a raw score is located within a distribution.

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

► a z-score of z = corresponds to a position exactly 1 standard deviation above the mean. ► a z-score of z = corresponds to a position exactly 2 standard deviations above the mean. ► The numerical value tells you the number of standard deviations from the mean.

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= The score is located above the mean by exactly 2 s.d. Z= The score is located above the mean by 1/2 s.d.

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.

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= Z= Z =

Figure 5.4 A z-score transformation Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning =

Learning Check ► For a population with  50 and  10, find the X value corresponding to each of the following z-scores:  z =  z =  z = X= 60 X = 45 X = 70

Figure 5.4 A z-score transformation Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the Wadsworth Group, a division of Thomson Learning =

Formula for transforming z-scores ► z = X –  

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 –  

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

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.

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.

Using z-scores to Make Comparisons ► Example: pg ► Psychology score = 60   50 and  10 ► Biology score = 56   48 and  4 ► z= X –    ► z= X –   