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z Scores & the Normal Curve

Model

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The normal distribution and standard

deviations

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deviations

Approximately 68% of scores will fall within one

standard deviation of the mean

In a normal distribution:

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deviations

Approximately 95% of scores will fall within two

standard deviations of the mean


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deviations

Approximately 99% of scores will fall within three

standard deviations of the mean


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Using standard deviation units todescribe individual scores

Here is a distribution with a mean of 100 and and

standard deviation of 10:

100 110 1209080-1 sd 1 sd 2 sd-2 sd

What score is one sd below the mean? 90

What score is two sd above the mean? 120

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Using standard deviation units to

describe individual scores

Here is a distribution with a mean of 100 and and

standard deviation of 10:

100 110 1209080-1 sd 1 sd 2 sd-2 sd

How many standard deviations below the mean is a score of 90? 1

2H

ow many standard deviations above the mean is a score of 120?

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Z scores

What is a z-score?A z score is a raw score expressed in

standard deviation units.

z scores are

sometimes calledstandard scores

S

XX

z

!Here is the formula for a z score:

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Computational Formula

z = (X M)/SX

Score minus the mean divided by thestandard deviation

Different formula for the population

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Using z scores to compare two raw scores

from different distributions

You score 80/100 on a statistics test and your friend also scores 80/100 on

their test in another section. Hey congratulations you friend sayswe are

both doing equally well in statistics. What do you need to know if the two

scores are equivalent?

the mean?

What if the mean of both tests was 75?

You also need to know thestandard deviation

What would you say about the two test scores if the S in your

class was 5 and the S in your friends class is 10?

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Calculating z scoresWhat is the z score for your test: raw

score = 80; mean = 75, S= 5?

S

XXz

!1

5

7580!

!z

What is the z score of your friends test:

raw score = 80; mean = 75, S= 10?

S

XXz

!5.

10

7580!

!z

Who do you think did better on their test? Why do you think this?

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Why z-scores?

Transforming scores in order to makecomparisons, especially when using differentscales

Gives information about the relative standingof a score in relation to the characteristics of

the sample or population Location relative to mean

Relative frequency and percentile

Slug, Binky and Biff example p 133

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What does it tell us?

z-score describes the location of the

raw score in terms of distance from themean, measured in standard deviations

Gives us information about the locationof that score relative to the averagedeviation of all scores

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Fun facts about z scores

Any distribution of raw scores can be converted to a

distribution of z scores

positive z scores represent raw scores that are

__________ (above or below) the mean?above

negative z scores represent raw scores that are

__________ (above or below) the mean?below

the mean of a distribution has a z score of____?

zero

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Computing Raw Score when

Know z-score

X = (z) (SX) + M

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Z-score Distribution

Mean of zero Zero distance from the mean

Standard deviation of 1 The z-score has two parts:

The number

The sign Negative z-scores arent bad

Z-score distribution always has sameshape as raw score

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Uses of the z-score

Comparing scores from different

distributions Interpreting individual scores

Describing and interpreting sample means

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Comparing Different Variables

Standardizes different scores

Example in text: Statistics versus English test performance

Can plot different distributions on samegraph

increased height reflects larger N

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Determining Relative Frequency

Proportion of time a score occurs

Area under the curve The negative z-scores have a relative

frequency of .50

The positive z-scores have a relativefrequency of .50

68% scores +/- 1 z-score

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The Standard Normal Curve

Theoretically perfect normal curve

Use to determine the relative frequencyof z-scores and raw scores

Proportion of the area under the curveis the relative frequency of the z-score

Rarely have z-scores greater than 3(.26% of scores above 3, 99.74%between +/- 3)

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Application of Normal Curve

Model

Can determine the proportion of scores

between the mean and a particular score Can determine the number of people

within a particular range of scores bymultiplying the proportion by N

Can determine percentile rank

Can determine raw score given thepercentile

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Using the z-

Table

Important when dealing with decimal z-scores

Table I of Appendix B (p. 488 491) Gives information about the area between the

mean and the z and the area beyond z in thetail

Use z-scores to define psychological attributes

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Using z-scores to Describe

Sample Means Useful for evaluating the sample and for inferential

statistical procedures

Evaluate the sample means relative standing Sampling distribution of means could be created by

plotting all possible means with that sample size andis always approximately a normal distribution

Sometimes the mean will be higher, sometimeslower

The mean of the sampling distribution always equalsthe mean of the underlying raw scores of the

population (most of the means will be around Q)

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Central LimitTheorem

Used for creating a theoretical sampling distribution

A statistical principle that defines the mean as equal

to Q, SD that is equal to W, and the shape of thedistribution which is approximately normal

Obtain information without having to actually samplethe population

Interpretation is the same: if close to mean occursmore frequently

Compute z-scores to indicate relative frequency ofthe sample mean

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Standard Error of the MeanAverage amount that the sample means

deviate from the Q

Population standard error: WM = WX/square root of N

Larger N produces more representative

samples Determine on average how much the means

differ from the Q

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Calculating z-score for sample

mean

Z = (M - Q)/WM

Determine relative frequency of sample means Use the standard normal curve and z-tables to

describe relative frequency of sample means

Interpretation is identical: larger the z, thesmaller the relative frequency

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