course: just 3900 introductory statistics for criminal justice instructor:

COURSE: JUST 3900INTRODUCTORY STATISTICS

FOR CRIMINAL JUSTICE

Instructor:Dr. John J. Kerbs, Associate Professor

Joint Ph.D. in Social Work and Sociology

Chapter 5: Z-Scores Chapter 5: Z-Scores Location of Scores and Location of Scores and

Standardized DistributionsStandardized Distributions

z-Scores and Location z-Scores and Location By itself, a raw score or By itself, a raw score or XX value provides value provides

very little information about how that very little information about how that particular score compares with other particular score compares with other values in the distribution. values in the distribution. A score of A score of XX = 53, for example, may be a relatively = 53, for example, may be a relatively

low score, or an average score, or an extremely high low score, or an average score, or an extremely high score depending on the mean and standard deviation score depending on the mean and standard deviation for the distribution from which the score was obtained.for the distribution from which the score was obtained.

If the raw score is transformed into a If the raw score is transformed into a zz-score, -score,

however, the value of the however, the value of the zz-score tells exactly where -score tells exactly where the score is located relative to all the other scores in the score is located relative to all the other scores in the distribution. the distribution.

Distribution Examples:Distribution Examples:Same Same μμ and Different and Different σσ

If you received a 76 on an exam, in

which class would you prefer

to have this score?

The process of changing an The process of changing an XX value into a value into a zz--score involves creating a signed number, score involves creating a signed number, called a called a zz-score-score The sign of the The sign of the zz-score (+ or –) identifies whether -score (+ or –) identifies whether

the the X X value is located above the value is located above the mean (positive) or mean (positive) or below the mean (negative). below the mean (negative).

The numerical value of the The numerical value of the zz-score corresponds to -score corresponds to the number of standard deviations between the number of standard deviations between XX and the and the mean of the distribution.mean of the distribution.

Thus, a score that is located two standard deviations Thus, a score that is located two standard deviations above the mean will have a above the mean will have a zz-score of +2.00. And, a -score of +2.00. And, a zz-score of +2.00 always indicates a location above the -score of +2.00 always indicates a location above the mean by two standard deviations. mean by two standard deviations.

zz-Scores and Location (continued)-Scores and Location (continued)

Relationship between z-score Values & Relationship between z-score Values & Locations in Population DistributionsLocations in Population Distributions

Transforming populations of scores into z-

scores: Note that distribution

shape does not change below.

Note that mean is

transformed into a value of 0 and the

standard deviation is transformed into a value

of 1.

Practice Interpreting z-ScoresPractice Interpreting z-Scores For the following z-scores, please describe the score’s For the following z-scores, please describe the score’s

location in each distribution.location in each distribution.

zz = 1.75 = 1.75 zz = - 0.50 = - 0.50 zz = 0.75 = 0.75 zz = - 1.25 = - 1.25

Identify the z-score value for the following locations in a Identify the z-score value for the following locations in a distribution.distribution.

Below the mean by 3 standard deviationsBelow the mean by 3 standard deviations Above the mean by ¼ of a standard deviationAbove the mean by ¼ of a standard deviation Below the mean by 1 standard deviationsBelow the mean by 1 standard deviations

Transforming Back and Forth Transforming Back and Forth Between Between XX and and zz

The basic The basic zz-score definition is usually -score definition is usually sufficient to complete most sufficient to complete most zz-score -score transformations. However, the definition can transformations. However, the definition can be written in mathematical notation to create a be written in mathematical notation to create a formula for computing the formula for computing the zz-score for any -score for any value of value of XX. .

X X – – μμzz = = ────────

σσ

Practice z-Score CalculationsPractice z-Score Calculations

X X – – μμ

zz = = ────────

σσ• With the formula above, please calculate:With the formula above, please calculate:

• zz for a distribution with for a distribution with μ = 20, σ = 6, μ = 20, σ = 6, X = X = 1818




Transforming Back and Forth Transforming Back and Forth Between Between XX and and z z (continued)(continued)

Also, the terms in the formula can be Also, the terms in the formula can be regrouped to create an equation for regrouped to create an equation for computing the value of computing the value of XX corresponding to corresponding to any specific any specific zz-score.-score.

X = μ + zσX = μ + zσ

Practice z-Score CalculationsPractice z-Score Calculations

X = μ + zσ• With the formula above, please calculate:With the formula above, please calculate:

• XX for a distribution with for a distribution with μ = 20, σ = 6, μ = 20, σ = 6, z = 1.5z = 1.5• XX for a distribution with for a distribution with μ = 20, σ = 6, μ = 20, σ = 6, z = -1.25z = -1.25• XX for a distribution with for a distribution with μ = 20, σ = 6, μ = 20, σ = 6, z = 1/3z = 1/3• XX for a distribution with for a distribution with μ = 20, σ = 6, μ = 20, σ = 6, z = -0.5z = -0.5• If μ = 50, X = 42 and z = - 2.00, what is the If μ = 50, X = 42 and z = - 2.00, what is the

standard deviation (standard deviation (σ) for the distribution?) for the distribution?

Relationship between z-score Values & Relationship between z-score Values & Locations in Population DistributionsLocations 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.

The Three Properties of z-ScoresThe Three Properties of z-Scores1. 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-ScoresThe Three Properties of z-Scores(Continued)(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-ScoresThe Three Properties of z-Scores(Continued)(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.

zz-scores and Locations-scores and Locations In addition to knowing the basic definition of a In addition to knowing the basic definition of a zz--

score and the formula for a score and the formula for a zz-score, it is useful to be -score, it is useful to be able to visualize able to visualize zz-scores as locations in a -scores as locations in a distribution.distribution.

Remember, Remember, zz = 0 is in the center (at the mean), and = 0 is in the center (at the mean), and the extreme tails correspond to z-scores of the extreme tails correspond to z-scores of approximately –2.00 on the left and +2.00 on the approximately –2.00 on the left and +2.00 on the right. right.

Although more extreme Although more extreme zz-score values are possible, -score values are possible, most of the distribution is contained between most of the distribution is contained between zz = – = –2.00 and 2.00 and zz = +2.00. = +2.00. Remember: about 95% of all scores fall within + or – 2 Remember: about 95% of all scores fall within + or – 2

standard deviations from the meanstandard deviations from the mean

zz-scores and Locations -scores and Locations (Continued)(Continued)

The fact that The fact that zz-scores identify exact locations -scores identify exact locations within a distribution means that within a distribution means that zz-scores can -scores can be used as descriptive statistics and as be used as descriptive statistics and as inferential statistics. inferential statistics. As descriptive statistics, As descriptive statistics, zz-scores describe -scores describe

exactly where each individual is located. exactly where each individual is located. As inferential statistics, As inferential statistics, zz-scores determine -scores determine

whether a specific sample is representative of its whether a specific sample is representative of its population, or is extreme and unrepresentative. population, or is extreme and unrepresentative.

zz-Scores as a Standardized -Scores as a Standardized DistributionDistribution

When an entire distribution of When an entire distribution of XX values is values is transformed into transformed into zz-scores, the resulting -scores, the resulting distribution of distribution of zz-scores will always have a -scores will always have a mean of zero and a standard deviation of one. mean of zero and a standard deviation of one.

The transformation does not change the The transformation does not change the shape of the original distribution and it does shape of the original distribution and it does not change the location of any individual not change the location of any individual score relative to others in the distribution.score relative to others in the distribution.

Transforming Raw Scores to z-scores:Transforming Raw Scores to z-scores:No Change in Distribution ShapeNo Change in Distribution Shape

z

zz-Scores as a Standardized -Scores as a Standardized Distribution (Continued)Distribution (Continued)

The advantage of standardizing The advantage of standardizing distributions is that two (or more) different distributions is that two (or more) different distributions can be made the same. distributions can be made the same. For example, one distribution has For example, one distribution has μμ = 100 and = 100 and

σσ = 10, and another distribution has = 10, and another distribution has μμ = = 40 and 40 and σσ = 6. = 6. When these distribution are transformed to When these distribution are transformed to zz--

scores, both will have scores, both will have μμ = 0 and = 0 and σσ = 1. = 1.


Please convert the following population of Please convert the following population of NN=6 scores (0, 12, 10, 4, 6, 4) into a =6 scores (0, 12, 10, 4, 6, 4) into a standardized distributionstandardized distribution

Step 1: Calculate the mean Step 1: Calculate the mean μμ = = ΣΣX/NX/N

Step 2: Calculate the standard deviation (Step 2: Calculate the standard deviation (σσ))Step 3: Calculate z-score for each value of XStep 3: Calculate z-score for each value of X



Step 1: Calculate the mean Step 1: Calculate the mean μμ = = ΣΣX/N = 36/6 = 6 X/N = 36/6 = 6 Because the mean is even, you can use Because the mean is even, you can use

the definitional formula of the SS in Step 2the definitional formula of the SS in Step 2



Step 2: Calculate the standard deviation (Step 2: Calculate the standard deviation (σσ)) σσ = = SS = SS = ΣΣ((XX - - μμ))22 = (0-6) = (0-6)22 + (12-6) + (12-6)22 + (10-6) + (10-6)22 + (4-6) + (4-6)22 + +

(6-6) (6-6)22 + (4-6) + (4-6)22

= 36 + 36 + 16 + 4 + 0 + 4= 36 + 36 + 16 + 4 + 0 + 4 = 96= 96 σσ = =



Step 3: Calculate z-score for each value of XStep 3: Calculate z-score for each value of XX-score (x-μ) where μ = 6 σ z-score = (x-μ) /σ

0 - 6 4 z = - 1.50

12 6 4 z = 1.50

10 4 4 z = 1.00

4 - 2 4 z = - 0.50

6 0 4 z = 0.00

4 - 2 4 z = - 0.50


Because Because zz-score distributions all have the -score distributions all have the same mean and standard deviation, same mean and standard deviation, individual scores from different distributions individual scores from different distributions can be directly compared. can be directly compared.

A A zz-score of +1.00 specifies the same -score of +1.00 specifies the same location in all location in all zz-score distributions.-score distributions.

zz-Scores and Samples -Scores and Samples

It is also possible to calculate It is also possible to calculate zz-scores for -scores for samples. samples.

The definition of a The definition of a zz-score is the same for -score is the same for either a sample or a population, and the either a sample or a population, and the formulas are also the same formulas are also the same exceptexcept that the that the sample mean and standard deviation sample mean and standard deviation are used are used in place of the in place of the population mean and standard population mean and standard deviationdeviation. .

zz-Scores and Samples -Scores and Samples ExampleExample

Thus, for a score X from a sample, you can calculate the z-Thus, for a score X from a sample, you can calculate the z-score as follows:score as follows:

XX – – MMzz = = ──────────

ss Using Using zz-scores to standardize a sample also has the same -scores to standardize a sample also has the same

effect as standardizing a population. effect as standardizing a population. Specifically, the mean of the Specifically, the mean of the zz-scores will be zero (-scores will be zero (M M z z = 0) = 0)

and the standard deviation of the and the standard deviation of the zz-scores will be equal to -scores will be equal to 1.00 (1.00 (s s z z = 1= 1) ) provided the standard deviation is computed provided the standard deviation is computed

using the using the sample formulasample formula (dividing n – 1 instead of n)(dividing n – 1 instead of n). . Each z-score can be transformed into an X value as Each z-score can be transformed into an X value as

follows: follows: XX = = MM + z + z ss


Please use the formula below to calculate the following:Please use the formula below to calculate the following:

XX – – MMzz = = ────────── ss

• zz for a distribution with for a distribution with M = 40, s = 12, M = 40, s = 12, X = X = 4343




• Answers: z = 0.25, - 0.50, 1.50, - 1.00Answers: z = 0.25, - 0.50, 1.50, - 1.00


Please use the formula below to calculate the following:Please use the formula below to calculate the following:

XX = = MM + z + z s s

• XX for a distribution with for a distribution with M = 80, s = 20, M = 80, s = 20, z = - 1.00z = - 1.00• XX for a distribution with for a distribution with M = 80, s = 20, M = 80, s = 20, z = z =

1.501.50• XX for a distribution with for a distribution with M = 80, s = 20, M = 80, s = 20, z = - z = -

0.500.50• XX for a distribution with for a distribution with M = 80, s = 20, M = 80, s = 20, z = z =

0.800.80

• Answers: X = 60, 110, 70, 96Answers: X = 60, 110, 70, 96

Other Standardized Distributions Other Standardized Distributions Based on Based on zz-Scores -Scores

Although transforming Although transforming XX values into values into zz--scores creates a standardized distribution, scores creates a standardized distribution, many people find many people find zz-scores burdensome -scores burdensome because they consist of many decimal because they consist of many decimal values and negative numbers. values and negative numbers.

Therefore, it is often more convenient to Therefore, it is often more convenient to standardize a distribution into numerical standardize a distribution into numerical values that are simpler than values that are simpler than zz-scores. -scores.

Other Standardized Distributions Other Standardized Distributions Based on Based on zz-Scores (Continued)-Scores (Continued)

To create a simpler standardized To create a simpler standardized distribution, you first select the mean and distribution, you first select the mean and standard deviation that you would like for standard deviation that you would like for the new distribution. This is your choice:the new distribution. This is your choice: e.g., e.g., μμ = 50 and = 50 and σσ = 10 = 10 oror μμ = 100 and = 100 and σσ = 10 = 10

Then, Then, zz-scores are used to identify each -scores are used to identify each individual's position in the original individual's position in the original distribution and to compute the individual's distribution and to compute the individual's position in the new distribution. position in the new distribution.

Other Standardized Distributions Other Standardized Distributions Based on z-Scores (Continued)Based on z-Scores (Continued)

Suppose, for example, that you want to Suppose, for example, that you want to standardize a distribution so that the new mean is standardize a distribution so that the new mean is μμ = 50 and the new standard deviation is = 50 and the new standard deviation is σσ = 10. = 10.

An individual with An individual with zz = –1.00 in the original = –1.00 in the original distribution would be assigned a score of distribution would be assigned a score of XX = 40 = 40 (below (below μμ by one standard deviation) in the by one standard deviation) in the standardized distribution. standardized distribution.

Repeating this process for each individual score Repeating this process for each individual score allows you to transform an entire distribution into a allows you to transform an entire distribution into a new, standardized distribution. new, standardized distribution.

Other Standardized Distributions Other Standardized Distributions Based on z-Scores (Continued)Based on z-Scores (Continued)

Suppose, for example, that you want to Suppose, for example, that you want to standardize a distribution so that the new mean is standardize a distribution so that the new mean is μμ = 50 and the new standard deviation is = 50 and the new standard deviation is σσ = 10. = 10.

In the original distribution, In the original distribution, μμ = 68 and = 68 and σσ = 15 = 15 What is the z-score for an x What is the z-score for an x value of 83 in the original value of 83 in the original

distribution: distribution: Z Z = (83-68)/15 = 15/15 = + 1.00= (83-68)/15 = 15/15 = + 1.00

An individual with an 83 in the original distribution would An individual with an 83 in the original distribution would be given a be given a z-z-score of + 1.00. In the new distribution score of + 1.00. In the new distribution ((μμ = 50, = 50, σσ = 10), the original score of 83 would be = 10), the original score of 83 would be assigned a value of 60 in the new standardized assigned a value of 60 in the new standardized distribution (distribution (μ+1 σ = 50 +10 = 60)μ+1 σ = 50 +10 = 60), which is one , which is one standard deviation above the mean.standard deviation above the mean.

course: just 3900 introductory statistics for criminal justice instructor:

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score of x

specific zscore

zscore transformations

z continuedalso

zscore values locations

following zscores

zscores location of

x value