Chapter 5Chapter 5

Z-ScoresZ-Scores

ReviewReview

►We have finished the basic elements We have finished the basic elements of descriptive statistics.of descriptive statistics.

►Now we will begin to develop the Now we will begin to develop the concepts and skills that form the concepts and skills that form the foundation for inferential statistics.foundation for inferential statistics.

Review (cont.)Review (cont.)

►Chapter 5: Present a method for describing Chapter 5: Present a method for describing the exact location of an individual score the exact location of an individual score relative to the other scores in a relative to the other scores in a distribution.distribution.

►Chapter 6: Determine probability values Chapter 6: Determine probability values associated with different locations in a associated with different locations in a distribution of scoresdistribution of scores

►Chapter 7: Apply skills from Ch. 5 & 6 to Chapter 7: Apply skills from Ch. 5 & 6 to sample means instead of individual scores.sample means instead of individual scores.

Z-ScoresZ-Scores

►Purpose of z-scores, or standard Purpose of z-scores, or standard scores, is to identify and describe the scores, is to identify and describe the exact location of every score in a exact location of every score in a distribution.distribution.

►To do this, we use the mean and To do this, we use the mean and standard deviation.standard deviation.

Z-ScoresZ-Scores

►A statistical technique that uses the A statistical technique that uses the mean and the standard deviation to mean and the standard deviation to transform each score (X value) into a transform each score (X value) into a z-score or a standard score.z-score or a standard score.

►Purpose of a z-score or a standard Purpose of a z-score or a standard score is to identify the exact location score is to identify the exact location of every score in a distribution.of every score in a distribution.

Why are z-scores useful?Why are z-scores useful?

► If you got a 76 on a test, how did you If you got a 76 on a test, how did you do?do?

►You would need more information.You would need more information.►You need to know the other scores in You need to know the other scores in

the distribution.the distribution.►What is the mean?What is the mean?

Why are z-scores useful? Why are z-scores useful? (cont.)(cont.)

►Knowing the mean is not enough.Knowing the mean is not enough.►You also need to know the standard You also need to know the standard

deviation.deviation.►The relative location within the The relative location within the

distribution depends on the mean and distribution depends on the mean and the statistical deviation as well as your the statistical deviation as well as your score.score.

Figure 5.1Figure 5.1

Two distributions of exam scoresTwo distributions of exam scores

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Score is in the extreme right hand tail – one of the highest in the distribution

Score is only slightly above average.

Purpose for z-scoresPurpose for z-scores

►A score by itself does not necessarily A score by itself does not necessarily provide much information about its provide much information about its position within a distributionposition within a distribution These are called raw scoresThese are called raw scores

►To make raw scores more meaningful, To make raw scores more meaningful, they are often transformed into new they are often transformed into new values that contain more information.values that contain more information.

►This transformation is one purpose for This transformation is one purpose for z-scores.z-scores.

Purpose for z-scores (cont.)Purpose for z-scores (cont.)►We can transform scores into z-scores We can transform scores into z-scores

to find out exactly where the original to find out exactly where the original scores are located. scores are located.

►A second purpose is to standardize an A second purpose is to standardize an entire distribution.entire distribution. IQ scoresIQ scores

►All are standardized with a mean of 100 and All are standardized with a mean of 100 and s.d. of 15s.d. of 15

►An IQ score of 95 is slightly below average and An IQ score of 95 is slightly below average and an IQ score of 145 is extremely high no matter an IQ score of 145 is extremely high no matter what IQ testwhat IQ test

To describe the exact location of To describe the exact location of the score within a distributionthe score within a distribution

►A z-score transforms an X score into a A z-score transforms an X score into a signed +/- numbersigned +/- number + above the mean+ above the mean - below the mean- below the mean The number tells the distance between The number tells the distance between

the score and the mean in terms of the the score and the mean in terms of the number of standard deviations.number of standard deviations.

ExampleExample

► In a distribution of standardized IQ In a distribution of standardized IQ scores with scores with and and and a and a score of X=130score of X=130

►The score of X=130 could be transformed The score of X=130 could be transformed into z= +2.00into z= +2.00

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

points)points)

DefinitionDefinition

►A z-scoreA z-score Specifies the precise location of each X Specifies the precise location of each X

value with a distributionvalue with a distribution The sign of +/- signifies whether the score The sign of +/- signifies whether the score

is above or below the meanis above or below the mean The numerical value of the z-score The numerical value of the z-score

specifies the distance from the mean by specifies the distance from the mean by counting the number of standard counting the number of standard deviations between the X and deviations between the X and

Z-scoresZ-scores

►Consist of two partsConsist of two parts +/-+/- MagnitudeMagnitude

►Both parts are necessary to describe Both parts are necessary to describe completely where a raw score is completely where a raw score is located within a distribution.located within a distribution.

Figure 5.2Figure 5.2

The relationship between z-scores and locations in a distributionThe relationship between z-scores and locations in a distribution

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► a z-score of z = +1.00 corresponds to a z-score of z = +1.00 corresponds to a position exactly 1 standard deviation a position exactly 1 standard deviation above the mean.above the mean.

►a z-score of z = +2.00 corresponds to a a z-score of z = +2.00 corresponds to a position exactly 2 standard deviations position exactly 2 standard deviations above the mean.above the mean.

►The numerical value tells you the The numerical value tells you the number of standard deviations from the number of standard deviations from the mean.mean.

Figure 5.1Figure 5.1

Two distributions of exam scoresTwo distributions of exam scores

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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.

Learning CheckLearning Check

►A negative z-score always indicates a A negative z-score always indicates a location below the mean.location below the mean.

►What z-score value identifies each of What z-score value identifies each of the following locations in a distribution?the following locations in a distribution? Above the mean by 2 s.d.Above the mean by 2 s.d. Below the mean by ½ s.d.Below the mean by ½ s.d. Above the mean by ¼ s.d.Above the mean by ¼ s.d. Below the mean by 3 s.d.Below the mean by 3 s.d.

Learning CheckLearning Check

►For a population with For a population with = 50 and = 50 and = 10, = 10, find the z-score for each of the following find the z-score for each of the following scores:scores: X = 55X = 55 X = 40X = 40 X = 30X = 30

Z= +0.50

Z= - 1.00

Z = - 2.00

Figure 5.4Figure 5.4

A z-score transformationA z-score transformation

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= 50

10

4030

Learning CheckLearning Check

►For a population with For a population with 50 and 50 and 10, find the X value corresponding to 10, find the X value corresponding to each of the following z-scores:each of the following z-scores: z = + 1.00z = + 1.00 z = - 0.50z = - 0.50 z = + 2.00z = + 2.00

X= 60

X = 45

X = 70

Figure 5.4Figure 5.4

A z-score transformationA z-score transformation

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= 50

10

4030 60 70

Formula for transforming z-Formula for transforming z-scoresscores

►z = X – z = X –

ExampleExample

►A distribution of scores has a mean of A distribution of scores has a mean of and a standard deviation of and a standard deviation of 10.10.

What z-score corresponds to a score of What z-score corresponds to a score of X=120 in this distribution?X=120 in this distribution?

Z = X – Z = X –

Transforming z-scores into X Transforming z-scores into X valuesvalues

►X = X = ++zz = 60 + (-2.00)(5)= 60 + (-2.00)(5)

= 60 + (-10.00)= 60 + (-10.00)

= 50= 50

Z = -2.00Z = -2.00

Using z-scores to Standardize a Using z-scores to Standardize a DistributionDistribution

►When an entire population of scores is When an entire population of scores is transformed into z-scorestransformed into z-scores The transformation does not change the The transformation does not change the

shape of the population but;shape of the population but; The mean is transformed into a value of The mean is transformed into a value of

zero;zero; The s.d. is transformed into a value of 1.The s.d. is transformed into a value of 1.

Standardized DistributionStandardized Distribution

►A standardized distribution is A standardized distribution is composed of scores that have been composed of scores that have been transformed to create predetermined transformed to create predetermined values for values for and and

►Standardized distributions are used to make Standardized distributions are used to make dissimilar distributions comparable.dissimilar distributions comparable.

Using z-scores to Make Using z-scores to Make ComparisonsComparisons

► Example: pg. 112-113Example: pg. 112-113► Psychology score = 60Psychology score = 60

50 and 50 and 1010

► Biology score = 56Biology score = 56 48 and 48 and 44

► z= X – z= X –

► z= X – z= X –