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SOAN 3120
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Michelle Dumas
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Chapter 3

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Sociology and Anthropology

SOAN 3120

Michelle Dumas

Fall

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Chapter 3: The Normal Distributions
Density Curves
Exploring a distribution:
1. Always plot data: make a graph, usually a histogram or a stemplot
2. Look for the overall pattern (shape, center, spread) and for striking
deviations such as outliers
3. Calculate a numerical summary to briefly describe center and the spread
4. Sometimes the overall pattern of a large number of observations is so regular
that we can describe it by a smooth curve
Our eyes respond to the areas of bars in a histogram, the bar areas represent
proportions of the observations
In moving from histogram bars to a smooth curve, we make a specific choice:
we adjust the scale of the graph so that the total area under the curve is
exactly 1
The total area represents the proportion 1, that is, all the observations
We can then interpret areas under the curve as proportions of the
observations
The curve is now a density curve
A density curve is a curve that:
o Is always on or above the horizontal axis, and
o Has area exactly 1 underneath it
o A density curve describes the overall pattern of distribution. The area
under the curve and above any range of values is the portion of all
observations that fall in that range
Density curves like distributions come in many shapes
o Both show the overall shape and the bumps in the long tail
A density curve is often a good description of the overall pattern of a
distribution
Outliers which are deviations from the overall pattern, are not described by
the curve
Of course no real set of data is exactly described y a density curve, the curve
is an idealized description that is easy to use and accurate enough for
practical use
Describing Density Curves
Our measures of center and spread apply to density curves as well as to
actual sets of observations
Areas under the curve show the proportions of the total number of
observations
So the median on the density curve is the equal areas point, the point with half
the area under the curve to its left and the reaming half the area to its right
You can roughly divide the area under the curve into four equal parts to get
the quartiles
o One fourth of the area under the curve is to the left of the first quartile
o Three fourths of the area is to the left of the third quartile The median of a symmetric density curve is exactly at its center
There are mathematical ways to find the median when the curve is skewed
o The median of a skewed right density curve show the mean is pulled
way from the median toward the long tail
The mean density curve is the balance point, at which the curve would balance
if made of solid material
The symmetric curve balances at its center because the two sides are
identical, the mean and median of a symmetric density curve are equal
The mean of a skewed distribution is pulled toward the long tail
The usual notion for the mean of a density curve is
We write the standard deviation of a density curve as
We can roughly located the by eye, as the balance point, but there is no
easy way to locate the standard deviation by eye for density curves
Normal Distributions
Normal curves describe distributions called normal distributions
Normal distributions are special and not all “normal” in the sense of being
average
We capitalize Normal to remind you that these curves are special
They illustrate important facts:
o All normal curves have the same overall shape: symmetric, single
peaked and bell shaped
o Any specific normal curve is completely described by giving its mean
and standard deviation
o The mean is located at the center of the symmetric curve and is the
same as the median. Changing mean without changing the standard
deviation moves the normal curve along the horizontal axis without
changing its spread
o The standard deviation controls the spread of a normal curve.
Curves with larger deviations are more spread out
The standard deviation is the natural measure of spread for the Normal
distributions
We can also locate by eye on a Normal curve
o The standard deviation is the distance from the center to the change
of curvature points on either side
Why are Normal distributions important in statistics?
1. Normal distributions are good descriptions for some distributions of
real data scores on takes taken by many people SAT exams, and
repeated careful measurements of the same quantity, and
characteristics of biologi

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