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Plan for today:
Chapter 13: Normal distribution
Normal Distribution
Normal Distribution:
Two variables we usually use to describe a sample.
(Although five-number summary is also a good choice.)
Measure of central tendency:
sample mean:
x1  x2 
X
n
xn
Measure of variability or “spread” :
sample std dev:
s
1 N
2
( xi  x )

N  1 n1
Normal Distribution:
So we want to find a distribution which is fully determined
by those two numbers. For a general distribution, it’s not
true.
Normal Distribution:
Normal density curve is symmetric, bell-shaped.
Normal Distribution:
(pic form: Wikipedia)
A specific Normal curve is completely described by
giving its mean and its standard deviation.
Normal Distribution:
(pic form: Wikipedia)
The mean determines the center of the distribution. It is
located at the center of symmetry of the curve.
Thus mean = median for Normal Distribution
Normal Distribution:
(pic form: Wikipedia)
The standard deviation determines the shape of the
curve.
Why Normal Distribution:
You may wonder: Why we call the distribution normal
distribution?
For example, flip a fair coin 100 times, the proportion of
getting head is p̂ .
What if we repeat the procedure m times. how those p̂
distributed.
Each day, I flipped a fair coin 100 times and got the proportion
of getting a head.
Day 1 Day 2
Day 3 Day 4
Day 5
…
0.49
0.43
0.56
0.55
0.57
…
68-95-99.7 Rule (or Three-sigma Rule):
68% of the observations fall within one standard
deviation of the mean.
95% of the observations fall within two standard
deviations of the mean.
99.7% of the observations fall within three standard
deviations of the mean.
68-95-99.7 Rule (or Three-sigma Rule):
A generalization of 68-95-99.7 rule is standard scores.
Standard Scores:
observation  mean
standard score 
standard deviation
A standard score of 1 says that the observation in
question lies one standard deviation above the mean.
Standard scores can be used to compare values in
different distributions.
Example in the Textbook:
Jennie scored 600 on the SAT and her friend Gerald scored
21 on the math part of ACT. Assuming that both tests
measure the same kind of ability, who has the higher score?
The performance depends on where those two scores lie in
their distribution (percentile). Because the percentage
makes more sense.
SAT: mean = 500
ACT: mean = 18
standard deviation = 100
standard deviation = 6
600  500
z1 
1
100
21  18
z1 
 0.5
6
Normal Distribution:
The area under any density curve is always 1 which
corresponding to 100 percent.
Calculation the Chance (a.k.a. Probability):
So The area is designed to express the corresponding
chance/probability a observation could be.
Calculation the Chance (a.k.a. Probability):
93.32 % of the area is shaded. So the
chance that an observation is less
than 1.5 is 93.32%.
Calculation the Chance (a.k.a. Probability):
86.64% area is shaded. So the chance
that an observation is between -1.5
and 1.5 is 86.64%.
Calculation the Chance (a.k.a. Probability):
6.68% area is shaded. So the chance
that an observation is larger than 1.5
is 6.68%.
Percentiles:
The cth percentile is a value such that c percent of
the observation lie below it.
93.32 % of the area is shaded.
1.5 is the 93.32th percentile of
this particular distribution.
Forward/Backward Problems:
How can we find the corresponding area at test and
HW?
Answer: using Table B on the textbook.
Caution: Table B only provides percentiles for the
standard scores.
Forward Problem
Observation
Standard
Score
Backward Problem
Percentile
Normal Table (Forward):
Once you obtained the standard score and
know the area you should work on.
Standard
Percentile
score
65.54 % of the area is under 0.4.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
50.00
53.98
57.93
61.79
65.54
69.15
72.58
75.80
78.81
81.59
84.13
86.43
88.49
90.32
91.92
93.32
Normal Table (Forward):
Once you obtained the standard score and
know the area you should work on.
The area above 0.4 is 100-65.54.
Standard
Percentile
score
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
50.00
53.98
57.93
61.79
65.54
69.15
72.58
75.80
78.81
81.59
84.13
86.43
88.49
90.32
91.92
93.32
Normal Table (Forward):
Once you obtained the standard score and
know the area you should work on.
The area between 0.2 and 1.5 is
93.32-57.93.
Standard
Percentile
score
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
50.00
53.98
57.93
61.79
65.54
69.15
72.58
75.80
78.81
81.59
84.13
86.43
88.49
90.32
91.92
93.32