Download Normal Curves

Mr. Markwalter
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People who keep
organized notebooks are
doing the best
People who copy down
my examples are doing
the best
People who ask questions
are doing the best
∴Take our a notebook.
No more loose leaf
I will start putting up
models for note-taking
We can make histograms of data.
 But sometimes we have a lot of data and:
THE OVERALL PATTERN OF A LARGE NUMBER
OF OBSERVATIONS IS SO REGULAR WE CAN
DESCRIBE IT BY A SMOOT CURVE!
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947 students tested
Distribution of scores is below
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We can look at it in the raw numbers OR
We can fit a curve (in red) that is a good
model
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If we shade all the scores less than 6, what
percentage of scores do you think we
shaded?
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30.3% or 287 people out of 947
That means the total area of the bars would
be 100% or a proportion of 1!
30.3%
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If we want to talk about the red curve, we
make the total area below the curve 1.
30.3%
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The area below the curve less than 6 is 0.293.
That is 29.3% of the area which is less than 6.
30.3%
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The curve is a pretty good model for the bars!
30.3%
29.3%
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Is always on or above the horizontal axis
Has an area of 1 underneath it
A density curve describes the overall pattern
of distribution.
The area under the curve and above any
interval is the proportion of observations that
fall in that interval.
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Mean is the balancing point of the curve
Median is the marker of equal areas; divides
the area under the curve in half.
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If the area to the left line in the density curve
shown below is 0.40, what is the area of the
other part?
0.40
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What percentage of observations are to the
left of the line in the curve below?
0.40
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There is one kind of curve that trumps them
all.
We see it more than anything else
It is the basis of 95% of statistics.
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Describe Normal Distributions
They are defined by two numbers
Mean: μ
Standard Deviation: σ (the average distance
from the mean)
Bell Shaped
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They are defined by two numbers
Mean: μ
Standard Deviation: σ
Bell Shaped
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As usual, the area under the curve is 1
Let’s take a look.
http://wwwstat.stanford.edu/~naras/jsm/NormalDensity
/NormalDensity.html
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In the Normal distribution with mean μ and
standard deviation σ:
About 68% of observations fall within σ of μ.
About 95% of observations fall within 2σ of μ.
About 99.7% of observations fall within 3σ of
μ.
Usually we define a Normal curve like this
 N(μ, σ)
 N(6, 1) means we have a curve with mean 6
and standard deviation 1.
 Using our 68-95-99.7 Rule…
6-1=5 and 6+1=7
68% of the observations are between 5 and 7.
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I make candies. The
mean mass of the
candy is 100g and
the standard
deviation is 5.
Draw a Normal
curve for the
situation.
What percentage of
candies is between
95g and 105g?
What percentage of
candies is less than
105g?
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I make throw
frisbees. My mean
throw is 75 yards
with a standard
deviation of 5 yards
Draw a Normal
curve for the
situation.
What percentage of
throws is between
65 and 85 yards?
What percentage of
throws is less than
above 70 yards?
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Spend 15 minutes doing this worksheet.
You may work with those around you.
If you do not finish it is homework.
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1, 4, 5, 5, 6, 9
Find the standard deviation