15 September 2003

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15 September 2003
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15 September 2003

• 1.2 Describing Distributions with Numbers
• Five-number summaries and boxplots
• Changing the unit of measurement
• 1.3 The Normal Distributions
• Density curves
• The normal distribution
• Standardized scores
• Areas under a standard normal curve (Table A)
• Normal quantile plots

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FIVE-NUMBER SUMMARIES and BOXPLOTS
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Calculating the variance and the standard
deviation
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What if we add 10 to each number?
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What if we add 10 to each number?
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What if we multiply each number by 10?
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What if we multiply each number by 10?
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What if we multiply each number by 10?
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CHANGING THE UNIT OF MEASUREMENT

• If you add or subtract the same amount from each
  value in a distribution, then
• the mean is increased or decreased by that amount
• the spread is not changed
• If you multiply or divide each value in a
  distribution by the same amount, then
• the mean is multiplied or divided by that amount
• the variance is multiplied or divided by the
  square of that amount
• the standard deviation is multiplied or divided
  by that amount

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STANDARDIZED SCORES

• A standardized score (or z score) tells us far
  above or below the mean a given number falls, in
  standard deviation units.

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STANDARDIZED SCORES

• For example, suppose the grades on a quiz have
  mean of 85 and standard deviation of 5 points.
• If your grade is 95, your z score is 2.00,
  because you are two standard deviations above the
  mean.
• Your friend whose grade is 83 has what z score?
• Another friend tells you that his z score is
  -1.00. Whats his grade?
• 65 70 75 80 85 90 95
  100
• 
  
• -4 -3 -2 -1 0 1 2
  3

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Standardized scores
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What if we add 10 to each number?
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What if we multiply each number by 10?
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STANDARDIZED SCORES

• If you take a list of numbers and
• add the same amount to each number,
• subtract the same amount from each number,
• multiply each number by the same amount, or
• divide each number by the same amount,
• the z scores do not change.

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THE NORMAL CURVE
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THE NORMAL CURVE
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The normal curve for womens heights
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How many women are at least 67 inches tall?
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How many women are at least 67 inches tall?

• z (67 - 64.5) / 2.5 1.00
• The relative frequency of women at least 67
  inches tall is approximately equal to the
  fraction of a standard normal curve which lies to
  the right of 1.00.
• Table A says that 84 of the area lies to the
  left of 1.00, so 16 must lie to the right.
• So about 16 of women are at least 67 tall.

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How many women are at least 68 inches tall?

• z (68 - 64.5) / 2.5 1.40
• The relative frequency of women at least 68
  inches tall is approximately equal to the
  fraction of a standard normal curve which lies to
  the right of 1.40.
• Table A says that 91.92 of the area lies to the
  left of 1.40, so 8.08 must lie to the right.
• So about 8 of women are at least 68 tall.

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How many women are at least 68 inches tall?
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What SAT-verbal score is at the 90th percentile?
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What SAT-verbal score is at the 90th percentile?

• Table A says that about 90 of a normal histogram
  lies left of 1.28
• ( X - 505 ) / 110 1.28
• X 110 (1.28) 505
• X 645.8
• So approximately 90 of SAT-v scores are less
  than 645.8

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What fraction of SAT-v scores are less than 645.8?

• z ( 645.8 - 505 ) / 110
• ( 140.8 ) / 110
• 1.28
• According to Table A, 89.97 of a normal
  histogram lies to the left of 1.28, so about 90
  of the SAT scores will be less than 645.8.

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• 1.2 Describing Distributions with Numbers
• Five-number summaries and boxplots
• Changing the unit of measurement
• 1.3 The Normal Distributions
• Density curves
• The normal distribution
• Standardized scores
• Areas under a standard normal curve (Table A)
• Normal quantile plots

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Sections to skip

• 1.3 Normal Quantile Plots
• (pp 78-83)
• 2.1 Categorical Explanatory Variables
• (pp 113-114)