§ 16.1 - 16.2 § 16.1 - 16.2 Approximately Normal Approximately Normal Distributions; Distributions; Normal Curves Normal Curves
§ 16.1 - 16.2§ 16.1 - 16.2Approximately Normal Distributions; Approximately Normal Distributions;
Normal CurvesNormal Curves
Approximately Normal Approximately Normal Distributions of DataDistributions of Data
Suppose the following is a bar graph for the height distribution of 205 randomly chosen men.
0
2
4
6
8
10
12
14
16
Frequency
48 51 54 57 60 63 66 69 72 75 78 81 84 87
Height (inches)
Heights of N=205 Men
Approximately Normal Approximately Normal Distributions of DataDistributions of Data
Notice that the graph is roughly ‘Bell-Shpaed’
0
2
4
6
8
10
12
14
16
Frequency
48 51 54 57 60 63 66 69 72 75 78 81 84 87
Height (inches)
Heights of N=205 Men
Approximately Normal Approximately Normal Distributions of DataDistributions of Data
Now look at the case with a sample size of 968 men:
0
10
20
30
40
50
60
70
48 51 54 57 60 63 66 69 72 75 78 81 84 87
Approximately Normal Approximately Normal Distributions of DataDistributions of Data
Here the ‘Bell’ behaviour is more apparent:
0
10
20
30
40
50
60
70
48 51 54 57 60 63 66 69 72 75 78 81 84 87
Approximately Normal Approximately Normal Distributions of DataDistributions 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 DistributionsNormal Distributions
Normal curves are all bell-shaped. However they can look different from one another:
Normal Distributions: Normal Distributions: PropertiesProperties
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: Normal Distributions: PropertiesProperties
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
= mean = median = mean = median
Right-Half50% of dataRight-Half50% of data
Left-Half50% of dataLeft-Half50% of data
Normal Distributions: Normal Distributions: PropertiesProperties
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 ‘opening-down’ and vice-versa.)
Normal Distributions: Normal Distributions: PropertiesProperties
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 ‘opening-down’ and vice-versa.)
+ + - -
Pointsof
Inflection
Pointsof
Inflection
Normal Distributions: Normal Distributions: PropertiesProperties
Quartiles: The first and third quartiles for a normally distributed data set can be estimated by
Q3 ≈ + (0.675)
Q1 ≈ - (0.675)
Normal Distributions: Normal Distributions: PropertiesProperties
Quartiles: The first and third quartiles for a normally distributed data set can be estimated by
Q3 ≈ + (0.675)
Q1 ≈ - (0.675)
Q3Q3Q1Q1
50%50%
25%25%25%25%
Example: Example: Find the mean, Find the mean, median, standard deviation and median, standard deviation and
the first and third quartiles.the first and third quartiles.
4343 5050
Pointof
Inflection
Pointof
Inflection
Example: Example: Find the mean, Find the mean, median, standard deviation and median, standard deviation and
the first and third quartiles.the first and third quartiles.
3939
Pointsof
Inflection
Pointsof
Inflection
3636
Example: Example: Find the mean, Find the mean, median and standard deviation.median and standard deviation.
73.87573.87564.612564.6125
25%25%
§ 16.4§ 16.4 The 68-95-99.7 Rule The 68-95-99.7 Rule
The 68-95-99.7 Rule(For normal distributions)
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)
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 + )
+ + - -
68%of
Data
68%of
Data
16%of
Data
16%of
Data
16%of
Data
16%of
Data
The 68-95-99.7 Rule(For normal distributions)
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)
The 68-95-99.7 Rule(For normal distributions)
1) (Roughly) 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
+ 2 + 2 - 2 - 2
95%of
Data
95%of
Data
2.5%of
Data
2.5%of
Data
2.5%of
Data
2.5%of
Data
The 68-95-99.7 Rule(For normal distributions)
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)
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.
+ 3 + 3 - 3 - 3
99.7%of
Data
99.7%of
Data
0.15%of
Data
0.15%of
Data
0.15%of
Data
0.15%of
Data
The 68-95-99.7 Rule(For normal distributions)
The 68-95-99.7 Rule(For normal distributions)
4) The range of the data R is estimated by
R ≈ 6
Example:Example: Find the mean, Find the mean, median, standard deviation and median, standard deviation and
the first and third quartiles.the first and third quartiles.
3636 5252
68%68%
Example:Example: Find the standard Find the standard deviation and the first and third deviation and the first and third
quartiles.quartiles.
10.3510.35
84%84%
6.226.22
Example:Example: Find the mean and Find the mean and standard deviation.standard deviation.
125125
2.5%2.5%
2525