Chapter 13: Normal Distributions

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Chapter 13 Normal Distributions

• Exploring data for one quantitative variable
• Always plot the data Histogram or stemplot
• Look for an overall pattern and for striking
  deviations such as outliers.
• Describe center and spread with the five-number
  summary or the mean and standard deviation.
• Overall pattern of a large number of observations
  is regular enough to be described by a smooth
  curve.

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Density Curves

• Density curve A curve that is superimposed on a
  density histogram to outline the shape.
• The histogram shows the proportion in each class
  and the area under the curve is 1.
• Density curves offer an easy and quick way of
  describing the shape of a distribution.

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Figure 13.1, p. 243
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Figure 13.3, p. 245
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Using a Density Curve

• Histograms show either frequencies (counts) or
  relative frequencies (proportions) in each class
  interval.
• Density curves show the proportion of
  observations in any region by areas under the
  curve.

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Figure 13.4, p. 246
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Center and Spread of a Density Curve

• Center Three Measures
• Mode The most frequently occurring value(s). On
  a density curve, this is where highest point
  occurs.
• Median The point that divides the area under the
  curve in half. (p. 247)
• Mean The point at which the curve would balance
  if made out of solid material. (p. 247)

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Figure 13.5, p. 247Mean and Median on Density
Curves
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Figure 13.6, p. 247Mean as balancing point
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Symmetric and Skewed Curves

• For a symmetric density curve, the mean, median,
  and mode are all equal. They lie in the center
  of the curve.
• For a skewed density curve, the mean is pulled
  away from the mode and median in the direction of
  the long tail.

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Normal Density Curves (p. 249)

• The normal curves are symmetric, bell-shaped
  curves that have these properties
• A specific normal curve is described by its mean
  and standard deviation.
• The mean is the center of the distribution. It
  is located at the center of symmetry of the
  curve.
• The standard deviation gives the shape of the
  curve. It is the distance from the mean to the
  change-of-curvature points on the other side.

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Figure 13.7, p. 248 Mean and Standard Deviation
For Two Normal Curves
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Why Study Normal Curves?

• The normal curves are useful for describing many
  variables.
• Examples
• Health data Heights, weight, blood pressure
• Standardized test scores SAT, GRE, IQ
• Times in sporting events Running, swimming, etc.

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The 68-95-99.7 Rule or The Empirical Rule (p.
250)

• In any normal distribution, approximately
• 68 of the observations fall within one standard
  deviation of the mean.
• 95 of the observations fall within two standard
  deviations of the mean.
• 99.7 of the observations fall within three
  standard deviations of the mean

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Figure 13.8, p. 250
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Example IQ Scores for 12-year-olds

• IQ scores of 12-year-olds have normal
  distribution with a mean of 100 and a standard
  deviation of 16.
• Compute an interval in which the middle 68 of
  scores will fall. Interval for 95, 99.7.
• What percent of 12-year-olds will have IQ scores
  higher than 100? 116? 132?
• What percent will have IQ scores lower than 116?
  100? 84?

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Standard Score (p. 252)

• The standard score of an observation is the
  number of standard deviations away from the mean
  that the observation falls.
• The standard score for any observation is
• Standard score observation mean
  standard deviation
• z x -?x s

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Comparing Scores on SAT and ACT

• The SAT and the ACT measure the same kind of
  ability.
• SAT verbal scores have a normal distribution with
  mean 500 and standard deviation 100.
• ACT verbal scores are normally distributed with
  mean 18 and standard deviation 6.
• Ricky scores 450 on the SAT and Seth scores 16 on
  the ACT. Who has the higher score?
• Example 3 on p. 253 Similar example with two
  scores that are above the mean for each test.

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Percentiles and the Normal Curve

• Percentiles
• The cth percentile of a distribution is the
  value such that c percent of the observations lie
  below it and the rest of the observations lie
  above it. (p. 253)
• Table B, p. 547 gives standard scores and
  percentiles.

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Percentiles on the SAT

• Example Assume verbal SAT scores have an
  approximate normal distribution with a mean of
  500 and a standard deviation of 100.
• Suppose Fred receives a score of 400. What is
  Freds percentile rank?
• Suppose Emma receives a score of 600. What is
  Emmas percentile rank?