§ 16.1 - 16.2 approximately normal distributions; 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