Chapter 2: The Normal Distribution

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Chapter 2 The Normal Distribution

• Section 1 Density Curves and the Normal
  Distribution

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Density Curves

• A density curve is similar to a histogram, but
  there are several important distinctions.
• 1. Obviously, a smooth curve is used to represent
  data rather than bars. However, a density curve
  describes the proportions of the observations
  that fall in each range rather than the actual
  number of observations.
• 2. The scale should be adjusted so that the total
  area under the curve is exactly 1. This
  represents the proportion 1 (or 100).

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Density Curves

• 3. While a histogram represents actual data
  (i.e., a sample set), a density curve represents
  an idealized sample or population distribution.

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Density Curves Mean Median

• Three points that have been previously made are
  especially relevant to density curves.
• 1. The median is the "equal areas" point.
  Likewise, the quartiles can be found by dividing
  the area under the curve into 4 equal parts.
• 2. The mean of the data is the "balancing" point.
• 3. The mean and median are the same for a
  symmetric density curve.

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Greek 101

• Since the density curve represents "idealized"
  data, we use Greek letters mu m for mean and
  sigma s for standard deviation.

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Shapes of Density Curves

• We have mostly discussed right skewed, left
  skewed, and roughly symmetric distributions that
  look like this

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Bimodal Distributions

• We could have a bi-modal distribution. For
  instance, think of counting the number of tires
  owned by a two-person family. Most two-person
  families probably have 1 or 2 vehicles, and
  therefore own 4 or 8 tires. Some, however, have
  a motorcycle, or maybe more than 2 cars. Yet,
  the distribution will most likely have a hump
  at 4 and at 8, making it bi-modal.

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Uniform Distributions

• We could have a uniform distribution. Consider
  the number of cans in all six packs. Each pack
  uniformly has 6 cans. Or, think of repeatedly
  drawing a card from a complete deck. One-fourth
  of the cards should be hearts, one-fourth of the
  cards should be diamonds, etc.

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Other Distributions

• Many other distributions exist, and some do not
  clearly fall under a certain label. Frequently
  these are the most interesting, and we will
  discuss many of them.

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Normal Curves

• Curves that are symmetric, single-peaked, and
  bell-shaped are often called normal curves and
  describe normal distributions.
• All normal distributions have the same overall
  shape. They may be "taller" or more spread out,
  but the idea is the same.

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What does it look like?
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Normal Curves µ and s

• The "control factors" are the mean µ and the
  standard deviation s.
• Changing only µ will move the curve along the
  horizontal axis.
• The standard deviation s controls the spread of
  the distribution. Remember that a large s implies
  that the data is spread out.

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Finding µ and s

• You can locate the mean µ by finding the middle
  of the distribution. Because it is symmetric, the
  mean is at the peak.
• The standard deviation s can be found by
  locating the points where the graph changes
  curvature (inflection points). These points are
  located a distance s from the mean.

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The 68-95-99.7 Rule

• In a normal distribution with mean µ and standard
  deviation s
• 68 of the observations are within s of the mean
  µ.
• 95 of the observations are within 2 s of the
  mean µ.
• 99.7 of the observations are within 3 s of the
  mean µ.

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The 68-95-99.7 Rule
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Why Use the Normal Distribution???

• 1. They are occur frequently in large data sets
  (all SAT scores), repeated measurements of the
  same quantity, and in biological populations
  (lengths of roaches).
• 2. They are often good approximations to chance
  outcomes (like coin flipping).
• 3. We can apply things we learn in studying
  normal distributions to other distributions.

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Heights of Young Women

• The distribution of heights of young women aged
  18 to 24 is approximately normally distributed
  with mean ? 64.5 inches and standard deviation
  ? 2.5 inches.

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The 68-95-99.7 Rule
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Use the previous chart...

• Where do the middle 95 of heights fall?
• What percent of the heights are above 69.5
  inches?
• A height of 62 inches is what percentile?
• What percent of the heights are between 62 and 67
  inches?
• What percent of heights are less than 57 in.?

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But...

• However, NOT ALL DATA are normal or even close to
  normal. Salaries, for instance, are generally
  right skewed. Nonnormal data are common and often
  interesting.