Transcript Normal Distributions PowerPoint

Normal Distributions
Fitting a Density Curve to a
Histogram
Skewed right
Symmetrical and Bell shaped :
Approximately normal
Density Curves
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A curve through the tops of the histogram bars that
smoothes out the irregular ups and downs
Histograms show counts of observations in each class,
while Density curves show the proportion of observations
in any region
The area under the curve represents this proportion, so the
total area under the curve is equal to 1, or 100%.
Means and Medians of Density
Curves
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The median is the equal areas point – 50% of the
observations are on either side of it
The mean is the balancing point:
 On a skewed density curve, the mean is pulled toward the tail
Skewed right: Mean is bigger than the median
► Skewed left: Mean is smaller than the median
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 On a symmetrical density curve, the mean and median are equal
Answer the questions as you watch the video
http://www.learner.org/courses/againstallodds/unitpages/unit08.html
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All normal curves have what characteristic shape?
Symmetric bell
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A normal distribution can be completely described by which two
numbers?
Mean and standard deviation
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A normal distribution follows the Empirical Rule. Fill in the percentages
below:
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68%
95%
99.7%
mean
of the values lie within 1 standard deviation of the mean
of the values lie within about 2 standard deviations of the mean
(or almost all) of the values lie within 3 standard deviations of the
95% of women are between what two heights?
60.5” and 70.5”
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How tall must a woman and man be to join the Boston Beanstalks
Club? Man: 6’2”
Woman: 5’10”
Normal Density Curves
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Symmetric
Bell shaped with only one peak
Tails fall off quickly, so no
outliers
Mean and median are equal
The point at which the
curvature changes is located
one standard deviation from the
mean
Normal curves are fully
described by the mean (center)
and standard deviation (spread)
Find the mean and standard
deviation of the normal curves
► There
are two normal
curves graphed at the
right, A and B.
► Which one has the
greater mean? What
is it? B, about 46, is a greater
mean than A, at about 13
► Which one has the
greater standard
deviation? What is it?
B, about 5, is a greater
st.dev. than A, at about 3
A
B
Barack Obama
47
Presidents’ Ages at Inauguration
The presidents’ ages at inauguration are approximately
normal (not exactly normal).
► Using the graph at the right, centered at mean 54.6 with
st.dev 6.2, what percent of the presidents ages were:
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 within 1 standard deviation of the mean? (15 + 13)/44 = 64%
 within 2 standard deviations of the mean? (28+6+7)/44 = 93%
 within 3 standard deviations of the mean? 100%
Empirical Rule (68-95-99.7) of
Normal Distributions
► In
any normal distribution,
approximately:
 68% of the observations fall
within 1 standard deviation of
the mean
 95% of the observations fall
within 2 standard deviations
of the mean
 99.7% of the observations
fall within 3 standard
deviations of the mean
SAT Example:
SAT scores are approximately normally distributed with a
mean of 500 and a standard deviation of 100
50% of scores lie below
500, and 50% lie above
500
► 68% of scores fall within
400-600, so 34% fall
within 500-600
► What percent of scores fall
within 500-700? ½(95) = 47.5%
► What percent of scores fall
within 600-700? 47.5-34 = 13.5%
► What percentile is a
student at who scores a
700? 50+47.5 = 97.5%tile
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Or (100-95)/2 = 2.5,
and 2.5 + 95 = 97.5%
CD Example:
The playing times of compact discs are approximately normally
distributed; the average time in a large collection is 35 minutes, and
the standard deviation is 5 minutes.
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The middle 68% of the playing
times falls between what two
times? 30-40 minutes
What percent of the cd’s playing
times are between 25 and 30
minutes? 13.5%
What percent of the cd’s playing
times are below 30? 16%
A playing time of 45 minutes
corresponds to what percentile
of cd playing times for this
collection? 97.5%
A playing time of 40 minutes
corresponds to what percentile
of cd playing times for this
collection? 84%
20 25 30 35 40 45 50
Playing times
Percents to Memorize!
0.15%
2.35%
13.5%
34%
34%
13.5%
2.35%
0.15%