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The Normal distribution
Chapter 5
Assessing Normality (5.4)
• You have a bunch of data.
• Question: Is it Normal?
Assessing Normality (5.4)
• You have a bunch of data.
• Question: Is it Normal?
– Check if your data has the same properties that
Normal data does
1. Construct a histogram or stem and leaf plot.
Does it look Normal-ish? (moundshaped and
symmetric)
1. Construct a histogram or stem and leaf plot.
Does it look Normal-ish? (moundshaped and
symmetric)
2. Find the % of data within 1 standard
deviation of the mean, 2 standard deviations
and 3 standard deviations (should be 68%,
95%, 99.6%)
1. Construct a histogram or stem and leaf plot.
Does it look Normal-ish? (moundshaped and
symmetric)
2. Find the % of data within 1 standard
deviation of the mean, 2 standard deviations
and 3 standard deviations (should be 68%,
95%, 99.6%)
3. IQR/s=1.3
1. Construct a histogram or stem and leaf plot.
Does it look Normal-ish? (moundshaped and
symmetric)
2. Find the % of data within 1 standard
deviation of the mean, 2 standard deviations
and 3 standard deviations (should be 68%,
95%, 99.6%)
3. IQR/s=1.3
4. Construct a Normal Probability Plot
1. Construct a histogram or stem and leaf plot.
Does it look Normal-ish? (moundshaped and
symmetric)
2. Find the % of data within 1 standard
deviation of the mean, 2 standard deviations
and 3 standard deviations (should be 68%,
95%, 99.6%)
3. IQR/s=1.3
4. Construct a Normal Probability Plot
Note: 3 and 4 are true and good checks for Normality,
but we will not be covering them in this class.
Toilet Flush Example
Is this data normal??
Toilet Flush Example
Is this data normal??
1. Construct a histogram or stem and
leaf plot. Does it look Normal-ish?
(moundshaped and symmetric)
Housefly Wings Example
Is this data set normal??
Housefly Wings Example
Is this data normal??
1. Construct a
histogram or stem and
leaf plot. Does it look
Normal-ish?
(moundshaped and
symmetric)
Housefly Wings Example
Is this data normal??
Stem Leaf Plot
3|678899
4|000011111122222223333333344444444455555555556666
666666777777777888888889999999
5|0000001111223345
n = 100
Mean = 45.5
St. Dev = 3.92
Housefly Wings Example
Is this data normal??
Stem Leaf Plot
3|678899
4|000011111122222223333333344444444455555555556666
666666777777777888888889999999
5|0000001111223345
n = 100
Mean = 45.5
St. Dev = 3.92
2. Find % of data within 1,2,3
standard deviations of the
mean
Housefly Wings Example
Stem Leaf Plot
3|678899
4|000011111122222223333333344444444455555555556666
666666777777777888888889999999
5|0000001111223345
n = 100
Mean = 45.5
St. Dev = 3.92
2. Find % of data within 1,2,3
standard deviations of the
mean
(mean-s,mean+s)=(41.6,49.4)
(mean-2s,mean+2s)=(37.7,53.3)
(mean-3s,mean+3s)=(?,?)
What percent of
the fly data is in
these intervals?
Approximating a Binomial with a
Normal
• Flip a fair coin 100 times. How many heads do
we get?
Approximating a Binomial with a
Normal
• Flip a fair coin 100 times. How many heads do
we get?
– Repeat this process 1000 times. What do we
expect our results to look like?
Approximating a Binomial with a
Normal
• Flip a fair coin 100 times. How many heads do
we get?
– Repeat this process 1000 times. What do we
expect our results to look like?
Does this shape remind you of
anything??
We can (often) approximate a
Binomial Distribution with a Normal!
• Ex: Given the above flips, let X be the number
of Heads.
– Find the P(X<=45)=
Could do the binomial probabilities. But we’d have
to find: P(X=0)+P(X=1)+…+P(X=45)
Or we could convert it to a normal. Find z-score and
use a table!!
Example of Normal Approximation of
Binomial Random Variable
Do on chalkboard!
Is this approximation valid?
• Note that the interval (x-3s,x+3s) contains nearly
all the data (see Empirical Rule of Thumb or
Chebyshev’s Rule)
• For the approximation to be valid we need that
interval to be in the middle of our binomial
possibilities. The range of binomial possibilities is
(0,n)
• Therefore approximation is valid if:
(x-3s) > 0
(x+3s) < n