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PSYC 5: Chapter 5PSYC 5: Chapter 5

z-scores & Standardized Distributionsz-scores & Standardized DistributionsLearning objectives:Learning objectives:

Define z-scoresDefine z-scoresDescribe the benefit of using z-scoresDescribe the benefit of using z-scoresCalculate z-scoresCalculate z-scores

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• The symbol means “plus or minus.” Therefore, 1 means +1 and/or -1.

The The absolute valueabsolute value of a number is the size of a number is the size of that number, regardless of its sign. That of that number, regardless of its sign. That is, the absolute value of +2 is 2 and the is, the absolute value of +2 is 2 and the absolute value of -2 is 2.absolute value of -2 is 2.

New Statistical NotationNew Statistical Notation

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Understanding Understanding zz-Scores-Scores

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Frequency Distribution of Frequency Distribution of Attractiveness ScoresAttractiveness Scores

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zz-Scores-Scores

Like any raw score, a Like any raw score, a zz-score is a -score is a locationlocation on the distribution. A on the distribution. A zz-score also -score also automatically communicates the raw automatically communicates the raw score’s distance from the meanscore’s distance from the mean

A A zz-score describes a raw score’s location -score describes a raw score’s location in terms of how far above or below the in terms of how far above or below the mean it is when measured in standard mean it is when measured in standard deviationsdeviations

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X

Xz

zz-Score Formula-Score Formula

The formula for computing a The formula for computing a zz-score for a -score for a raw score in a sample israw score in a sample is

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• When a z-score and the associated and are known, this information can be used to calculate the original raw score. The formula for this is

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))(( XzX

Computing a Raw ScoreComputing a Raw Score

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Interpreting Interpreting zz-Scores-ScoresUsing the Using the zz-Distribution-Distribution

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A A zz-Distribution-Distribution

A A zz-distribution-distribution is the distribution produced is the distribution produced

by transforming all raw scores in the data by transforming all raw scores in the data

into into zz-scores.-scores.

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zz-Distribution of Attractiveness -Distribution of Attractiveness ScoresScores

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Characteristics of the Characteristics of the zz-Distribution-Distribution

1.1. A A zz-distribution always has the same -distribution always has the same shape as the raw score distributionshape as the raw score distribution

2.2. The mean of any The mean of any zz-distribution always -distribution always equals 0equals 0

3.3. The standard deviation of any The standard deviation of any zz-distribution always equals 1-distribution always equals 1

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Comparison of Two Comparison of Two zz-Distributions, -Distributions, Plotted on the Same Set of AxesPlotted on the Same Set of Axes

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Relative FrequencyRelative Frequency

Relative frequency can be computed using Relative frequency can be computed using the proportion of the total area under the the proportion of the total area under the curve.curve.

The relative frequency of a particular The relative frequency of a particular zz-score will be the same on all normal -score will be the same on all normal zz-distributions.-distributions.

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The Standard Normal CurveThe Standard Normal Curve

The The standard normal curvestandard normal curve is a perfect is a perfect

normal normal zz-distribution that serves as our -distribution that serves as our

model of the model of the zz-distribution that would result -distribution that would result

from any approximately normal raw score from any approximately normal raw score

distributiondistribution

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Proportions of Total Area Under Proportions of Total Area Under the Standard Normal Curvethe Standard Normal Curve

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PercentilePercentile

The standard normal curve also can be used The standard normal curve also can be used

to determine a score’s percentile. to determine a score’s percentile.

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Proportions of the Standard Normal Curve at Proportions of the Standard Normal Curve at Approximately the 2nd PercentileApproximately the 2nd Percentile

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Using Using zz-Scores to Describe -Scores to Describe Sample MeansSample Means

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Sampling Distribution of MeansSampling Distribution of Means

A distribution which shows all possible A distribution which shows all possible

sample means that occur when an infinite sample means that occur when an infinite

number of samples of the same size number of samples of the same size NN are are

randomly selected from one raw score randomly selected from one raw score

population is called the population is called the sampling sampling

distribution of meansdistribution of means..

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Central Limit TheoremCentral Limit Theorem

The The central limit theoremcentral limit theorem tells us the tells us the

sampling distribution of meanssampling distribution of means1.1. forms an approximately normal distribution,forms an approximately normal distribution,

2.2. has a has a equal to the equal to the of the underlying raw score of the underlying raw score

population, andpopulation, and

3.3. has a standard deviation that is mathematically has a standard deviation that is mathematically

related to the standard deviation of the raw score related to the standard deviation of the raw score

population.population.

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NX

X

Standard Error of the MeanStandard Error of the Mean

The standard deviation of the sampling The standard deviation of the sampling

distribution of means is called the distribution of means is called the standard standard

error of the meanerror of the mean. The formula for the true . The formula for the true

standard error of the mean isstandard error of the mean is

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X

Xz

zz-Score Formula for -Score Formula for a Sample Meana Sample Mean

The formula for computing a The formula for computing a zz-score for a -score for a sample mean issample mean is