§ 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:

Approximately Normal Distributions of Data  Here the ‘Bell’ behaviour is more apparent:

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 symmetric about a vertical axis. This axis is the line x=  where  is the mean/average of the data.  Mean = Median  = mean = median Right-Half 50% of data Left-Half 50% of data

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 ‘opening- down’ and vice-versa.)

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 ‘opening- down’ and vice-versa.)    +   -  Points of Inflection

Normal Distributions: Properties  Quartiles: The first and third quartiles for a normally distributed data set can be estimated by Q 3 ≈  + (0.675)  Q 1 ≈  - (0.675) 

Normal Distributions: Properties  Quartiles: The first and third quartiles for a normally distributed data set can be estimated by Q 3 ≈  + (0.675)  Q 1 ≈  - (0.675)    Q3Q3 Q3Q3 Q1Q1 Q1Q1 50% 25%

Example: Find the mean, median, standard deviation and the first and third quartiles Point of Inflection

Example: Find the mean, median, standard deviation and the first and third quartiles. 39 Points of Inflection 36

Example: Find the mean, median and standard deviation %

§ 16.4 The Rule

The 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 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 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 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  95% of Data 95% of Data 2.5% of Data 2.5% of Data 2.5% of Data 2.5% of Data

The 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 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  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 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 %

Example: Find the standard deviation and the first and third quartiles % 6.22

Example: Find the mean and standard deviation % 25