z Scores

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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
In a normal distribution
Approximately 68 of scores will fall within one
standard deviation of the mean
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The normal distribution and standard deviations
In a normal distribution
Approximately 95 of scores will fall within two
standard deviations of the mean
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The normal distribution and standard deviations
In a normal distribution
Approximately 99 of scores will fall within
three standard deviations of the mean
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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
120
90
80
-1 sd
1 sd
2 sd
-2 sd
What score is one sd below the mean?
90
120
What score is two sd above the mean?
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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
120
90
80
-1 sd
1 sd
2 sd
-2 sd
1
How many standard deviations below the mean is a
score of 90?
How many standard deviations above the mean is a
score of 120?
2
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Z scores
A z score is a raw score expressed in standard
deviation units.
What is a z-score?
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Computational Formula

• z (X M)/SX
• Score minus the mean divided by the standard
  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 the standard 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 scores
What is the z score for your test raw score
80 mean 75, S 5?
What is the z score of your friends test raw
score 80 mean 75, S 10?
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 make comparisons,
  especially when using different scales
• Gives information about the relative standing of
  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 the mean, measured in
  standard deviations
• Gives us information about the location of that
  score relative to the average deviation 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

the mean of a distribution has a z score of ____?
zero
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
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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 same shape 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 same graph
• 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 relative frequency
  of .50
• 68 scores /- 1 z-score

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The Standard Normal Curve

• Theoretically perfect normal curve
• Use to determine the relative frequency of
  z-scores and raw scores
• Proportion of the area under the curve is 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 by multiplying the
  proportion by N
• Can determine percentile rank
• Can determine raw score given the percentile

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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 the tail
• 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 and is always approximately a normal
  distribution
• Sometimes the mean will be higher, sometimes
  lower
• The mean of the sampling distribution always
  equals the mean of the underlying raw scores of
  the population (most of the means will be around
  ?)

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Central Limit Theorem

• Used for creating a theoretical sampling
  distribution
• A statistical principle that defines the mean as
  equal to ?, SD that is equal to ?, and the shape
  of the distribution which is approximately normal
• Obtain information without having to actually
  sample the population
• Interpretation is the same if close to mean
  occurs more frequently
• Compute z-scores to indicate relative frequency
  of the sample mean

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Standard Error of the Mean

• Average amount that the sample means deviate from
  the ?
• Population standard error
• ?M ?X/square root of N
• Larger N produces more representative samples
• Determine on average how much the means differ
  from the ?

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Calculating z-score for sample mean

• Z (M - ?)/?M
• 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, the
  smaller the relative frequency