Download PS3.3Two

3.3 Day II: Measures of
Variance
By the end of class you will be able to make
estimates about standard deviations by using
the “rule of thumb”, Chebyshev’s Theorem,
and the Empirical Rule
Heights of Students (inches)
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Enter class data in a list
Calculate the mean
Calculate the standard deviation
Is there an easier way to describe how
spread out the data is?
Is there a way to predict the shortest and
tallest student?
Estimating Standard Deviation
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It takes some time to find the standard deviation.
We will estimate it using the “Rule of Thumb”:
range
s
4
Estimate:
Real St. Deviation: s =
Estimating Data (“rule of
thumb”)
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If the standard deviation and mean are known
we can estimate the lowest and highest data
value
Mean = _____, st. deviation = _____
Predicted minimum value=
X  2s
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Predicted maxiumum value =
X  2s
How many heights are 1 standard
deviation away from the mean?
How many are two standard
deviations away from the mean?
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Counting = tedious!
Empirical Rule (for symmetric/bell shaped
distributions)
Chebyshev’s Theorem (for any
distribution)
Empirical Rule
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About 68% of data are within 1 standard
deviation of the mean.
About 95% of the data are within 2
standard deviations of the mean.
About 99.7% of the data are within 3
standard deviations of the mean.
Empirical Rule: SAT Scores
Chebyshev’s Theorem
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The proportion of data values from a data set
that will fall within k standard deviations of the
mean will be at least
 1 
1  2 
k 
This means that if k=2, at least 75% of all data
values are within 2 standard deviations of the
mean.
This means that if k = 3….
Chebychev’s Theorem
Estimate the standard deviation of
wait times at the bank:
6.5, 6.6, 6.7,6.8, 7.1, 7.3, 7.4, 7.7,
7.7, 7.7
The mean of heights of men is 69in
with a standard deviation of 2.8in.
Use the range of thumb rule to find
the minimum and maximum
heights.
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Minimum: 63.4in
Maximum: 74.6in
The mean of heights of men is 69in
with a standard deviation of 2.8in.
What percentage of men have
heights between 60.6 in and
77.4in? (assume bell shape)
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99.7% using the empirical rule
The mean of heights of men is 69in
with a standard deviation of 2.8in.
What percentage of men have
heights between 60.6 in and
77.4in? (assume no bell shape)
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89% using Chebyshev’s Theorem
How many heights would be in this range
of 50 men were sampled?
44.5
Remember: You must always add
and subtract standard deviations
from the mean to determine how
many standard deviations you are
away from the mean!
Summarizer
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3 Facts about data variation
2 Facts about estimating variation
1 Question/Concern you still have