z Scores(1)
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Transcript of z Scores(1)
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z Scores & the Normal Curve
Model
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The normal distribution and standard
deviations
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The normal distribution and standard
deviations
Approximately 68% of scores will fall within one
standard deviation of the mean
In a normal distribution:
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The normal distribution and standard
deviations
Approximately 95% of scores will fall within two
standard deviations of the mean
In a normal distribution:
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The normal distribution and standard
deviations
Approximately 99% of scores will fall within three
standard deviations of the mean
In a normal distribution:
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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