Download § 13.1 - 13.3 Populations, Surveys and Random Sampling

§ 16.1 - 16.2
Approximately Normal Distributions;
Normal Curves
Approximately Normal
Distributions of Data
 Suppose the following is a bar graph for the
height distribution of 205 randomly chosen
men.
Approximately Normal
Distributions of Data
Notice that the graph is roughly ‘Bell-Shpaed’
Approximately Normal
Distributions of Data
Now look at the case with a sample size of
968 men:
70
60
50
40
30
20
10
87
84
81
78
75
72
69
66
63
60
57
54
51
48
0
Approximately Normal
Distributions of Data
Here the ‘Bell’ behaviour is more apparent:
70
60
50
40
30
20
10
87
84
81
78
75
72
69
66
63
60
57
54
51
48
0
Approximately Normal
Distributions of Data
 Data that is distributed like the last
two examples is said to be in an
approximately normal distribution.
 If the ‘bell-shape’ in question were
perfect then the data would be said
to be a normal distribution. The
bell-shaped curves are called
normal curves.
Normal Distributions
 Normal curves are all bell-shaped.
However they can look different
from one another:
Normal Distributions:
Properties
 Symmetry: Every normal curve is
symmetric about a vertical axis.
This axis is the line x =  where  is the
mean/average of the data.
 Mean = Median
Normal Distributions:
Properties
 Symmetry:
Every
normal
curve
is
Left-Half
Right-Half
symmetric
about
a
vertical
axis.
50% of data
data  is the
This axis is the line 50%
x= of
where
mean/average of the data.
 Mean = Median
 = mean = median
Normal Distributions:
Properties
 Standard Deviation: The data’s
standard deviation, , is the distance
between the curve’s points of inflection
and the mean.
(Inflection points are where a curve
changes from ‘opening-up’ to ‘openingdown’ and vice-versa.)
Normal Distributions:
Properties
 Standard Deviation: The data’s
standard deviation, , is thePoints
distance
between the curve’s points ofofinflection
Inflection
and the mean.
(Inflection points are where a curve
changes from ‘opening-up’ to ‘openingdown’ and vice-versa.)
-

+
Normal Distributions:
Properties
 Quartiles: The first and third quartiles
for a normally distributed data set can
be estimated by
Q3 ≈  + (0.675)
Q1 ≈  - (0.675) 
Normal Distributions:
Properties
 Quartiles: The first and third quartiles
for a normally distributed data set can
be estimated by
50%
Q3 ≈  + (0.675)
25%
Q1 ≈  - (0.675) 
Q1

Q3
25%
Example: Find the mean, median,
standard deviation and the first and
third quartiles.
Point
of
Inflection
43
50
Example: Find the mean, median,
standard deviation and the first and
third quartiles.
Points
of
Inflection
36
39
Example: Find the mean, median
and standard deviation.
25%
64.6125
73.875
§ 16.4
The 68-95-99.7 Rule
The 68-95-99.7 Rule
(For normal distributions)
1) (Roughly) 68% of all data is within one
standard deviation of the mean, .
(I.e. - 68% of the data lies between
 -  and  + )
The 68-95-99.7 Rule
(For normal distributions)
1) (Roughly) 68% of all data is within one
standard deviation of the mean, .
(I.e. - 68% of the data lies between
68%
 -  and  + )
of
Data
16%
of
Data
16%
of
Data
-

+
The 68-95-99.7 Rule
(For normal distributions)
1) 68% of all data is within one standard
deviation of the mean, .
2) 95% of data is within two standard
deviations of the mean.
(I.e. - between  -  and  + )
The 68-95-99.7 Rule
(For normal distributions)
1) (Roughly) 68% of all data is within one
standard deviation of the mean, .
95%
2) 95% of data is within two standard
of
deviations of the mean.
Data
(I.e. - between
2.5%
of
Data
2.5%
of
Data
 - 2

 + 2
The 68-95-99.7 Rule
(For normal distributions)
1) 68% of all data is within one standard
deviation of the mean, .
2) 95% of data is within two standard
deviations of the mean.
3) 99.7% of data is within three standard
deviations of the mean.
The 68-95-99.7 Rule
(For normal distributions)
1) 68% of all data is within one standard
deviation of the mean, .
2) 95% of data is within two
deviations of the mean.
3)
0.15%
of
99.7%
Data
99.7%
standard
of
Data
of data is within three standard
0.15%
deviations of the mean.
of
Data
 - 3

 + 3
The 68-95-99.7 Rule
(For normal distributions)
4) The range of the data R is
estimated by
R ≈ 6
Example: Find the mean, median,
standard deviation and the first and
third quartiles.
68%
36
52
Example: Find the standard
deviation and the first and third
quartiles.
84%
6.22
10.35
Example: Find the mean and
standard deviation.
2.5%
25
125