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Chapter 4: The Normal Distribution


Title: Chapter 2: The Normal Distribution Author: Stoney Pryor Last modified by: Belton ISD Created Date: 8/25/2005 1:14:28 AM Document presentation format – PowerPoint PPT presentation

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Title: Chapter 4: The Normal Distribution

Chapter 4 The Normal Distribution
  • Density Curves and the Normal Distribution

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).

Density Curves
  • 3. While a histogram represents actual data
    (i.e., a sample set), a density curve represents
    an idealized sample or population distribution.
    (describes the proportion of the observations)
  • 4. Always on or above the horizontal axis
  • 5. We will still utilize mu m for mean and sigma
    s for standard deviation.

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.

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

Other Shapes (previously discussed)
  • Uniform Distributions
  • Bi-modal Distributions
  • Multi-modal Distributions

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 them later.
  • It is the only way to see what is really going on!

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.

What does it look like?
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.

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.

The 68-95-99.7 (Empirical)Rule
  • In a NORMAL DISTRIBUTIONS with mean µ and
    standard deviation s
  • 68 of the observations are within s of the mean
  • 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.

The 68-95-99.7 Rule
Why Use the Normal Distribution???
  • 1. They 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.

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.

The 68-95-99.7 Rule
Use the previous chart...
  • Where do the middle 95 of heights fall?
  • What percent of the heights are above 69.5
  • A height of 62 inches is what percentile?
  • What percent of the heights are between 62 and 67
  • What percent of heights are less than 57 in.?

  • Suppose, on average, it takes you 20 minutes to
    drive to school, with a standard deviation of 2
    minutes. Suppose a normal model is appropriate
    for the distribution of drivers times.
  • How often will you arrive at school in less than
    20 minutes?
  • How often will it take you more than 24 minutes?