The Normal Distribution

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The Normal Distribution
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The Normal Distribution

• Distribution any collection of scores, from
  either a sample or population
• Can be displayed in any form, but is usually
  represented as a histogram
• Normal Distribution specific type of
  distribution that assumes a characteristic bell
  shape and is perfectly symmetrical

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The Normal Distribution

• Can provide us with information on likelihood of
  obtaining a given score
• 60 people scored a 6 6/350 .17 17
• 9 people scored a 1 3

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The Normal Distribution

• Why is the Normal Distribution so important?
• Almost all of the statistical tests that we will
  be covering (Z-Tests, T-Tests, ANOVA, etc.)
  throughout the course assume that the population
  distribution, that our sample is drawn from (but
  for the variable we are looking at), is normally
  distributed
• Also, many variables that psychologists and
  health professionals look at are normally
  distributed
• Why this is requires a detailed examination of
  the derivation of our statistics, that involves
  way more detail than you need to use the
  statistic.

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The Normal Distribution

• Ordinate
• Density what is measured on the ordinate (more
  on this in Ch. 7)
• Abscissa

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The Normal Distribution

• Mathematically defined as
• Since ? and e are constants, we only have to
  determine µ (the population mean) and s (the
  population standard deviation) to graph the
  mathematical function of any variable we are
  interested in
• Dont worry, understanding this is not necessary
  to understanding the normal distribution, only a
  helpful aside for the mathematically inclined

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The Normal Distribution

• Using this formula, mathematicians have
  determined the probabilities of obtaining every
  score on a standard normal distribution (see
  Table E.10 in your book)
• To determine these probabilities for the variable
  youre interested in we must plug in your
  variable to the formula
• Note This assumes that your variable fits a
  normal distribution, if not, your results will be
  inaccurate

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The Normal Distribution

• However, this table refers to a Standard Normal
  Distribution
• ? 0 s 1
• How do you get your variable to fit?

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The Normal Distribution

• Z-Scores
• Range from 8 to -8
• Represent the number of standard deviations your
  score is from the mean
• i.e. z 1 is a score that is 1 standard
  deviation above the mean and z -3 is a score 3
  standard deviations below the mean
• Now we can begin to use the table to determine
  the probability that our z score will occur using
  table E.10

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The Normal Distribution
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The Normal Distribution

• Mean to Z

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The Normal Distribution
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The Normal Distribution

• Larger Portion

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The Normal Distribution

• Smaller Portion

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• Reminder Z-Scores represent of standard
  deviations from the mean
• For this distribution, if µ 50 and s 10, what
  score does z -3 represent? z 2.5?

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• z -.1 z 1.645
• (z -.1, Mean to Z) (z 1.645, Mean to
  Z)
• .0398 .4500 .4898 49

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• z -1.00, Smaller Portion Red Blue
• z -1.645, Smaller Portion Blue
• (Red Blue) - Blue Red

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• z -1.645 z -1.00
• (z -1.00, Smaller Portion) (z -1.645,
  Smaller Portion)
• .1587 - .0500 .1087 11

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The Normal Distribution

• What are the scores that lie in the middle 50 of
  a distribution of scores with µ 50 and s 10?
• Look for Smaller Portion .2500 on Table E.10
• z .67
• Solve for X using z-score formula
• Scores 56.7 and 43.3

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The Normal Distribution
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The Normal Distribution

• Other uses for z-scores
• Converting two variable to a standard metric
• You took two exams, you got an 80 in Statistics
  and a 50 in Biology you cannot say which one
  you did better in without knowing about the
  variability in scores in each
• If the class average in Stats was a 90 and the
  s.d. 15, what would we conclude about your score
  now? How is it different than just using the
  score itself?
• If the mean in Bio was a 30 and the s.d. was a 5,
  you did 4 s.ds above the mean (a z-score of 4)
  or much better than everyone else

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The Normal Distribution

• Other uses for z-scores
• Converting variables to a standard metric
• This also allows us to compare two scores on
  different metrics
• i.e. two tests scored out of 100 same metric
• one test out of 50 vs. one out of 100 two
  different metrics
• Is 20/50 better than 40/100? Is it better when
  compared to the class average?
• Allows for quick comparisons between a score and
  the rest of the distribution it is a part of

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The Normal Distribution

• Standard Scores scores with a predetermined
  mean and standard deviation, i.e. a z-score
• Why convert to standard scores?
• You can compare performance on two different
  tests with two different metrics
• You can easily compute Percentile ranks
• but they are population-relative!
• Percentile the point below which a certain
  percent of scores fall
• i.e. If you are at the 75thile (percentile),
  then 75 of the scores are at or below your score

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The Normal Distribution

• How do you compute ile?
• Convert your raw score into a z-score
• Look at Table E.10, and find the Smaller
  Portion if your z-score is negative and the
  Larger Portion if it is positive
• Multiply by 100

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The Normal Distribution

• New Score New s.d. (z) New Mean
• New IQ Score 15 (2) 100 130
• T-Score commonly used standardized normal
  distribution w/ mean 50 and s.d. 10
• T-Score 10 (2) 50 70