The Normal Distribution

The idealized binomial distribution

 

                                                

 

is called the normal distribution

 

                                                  

 

 

There are many normal distributions with different means and different standard deviations.

 

Comparison of Binomial and Normal Distributions

 

Using Microsoft Excel, we calculated the binomial distribution

 

and the normal distribution

 

 

The results of this comparison can be visualized as
 

In the above figure, the binomial distribution is in blue color and the normal distribution is in red.

 

 

Standard Normal Distribution

The standard normal distribution has the mean equal to zero. Its standard deviation and variance are equal to one. The abscissa of the standard normal distribution is in the standard scores z, and its ordinate y equals

 

 

                                                       

 

 

The above formula can be also written as the special case of the Euler's Gamma function

 

                                                        

 

where the constant a equals

 

                                                          

 

i.e., .3989, the standard scores z are denoted by x, the constant b equals zero, the constant c equals .5, and the constant d is equal to 2. Thus

 

                                                     

 

Visualizing Standard Normal Distribution by using the Cruise Scientific Visual Statistics Studio

To visualize ordinates of the normal distribution, select (Arguments, -3 z, t - +3 z, t ), mark the Standard Normal Distribution n 61, click the Generate command, and the Display the Sequence command. Next, select (Distributions, Ordinates of Normal Distribution) to generate values for the ordinates of the standard normal distribution as

Clicking on the Stereographs and Define Stereograph commands,

 

 

Standard Normal Distribution within the Microsoft Excel Framework

 

Using Microsoft Excel, the abscissa and ordinate for the standard normal distribution can be generated as

 

 

Microsoft Excel

Fx = NORMDIST(A1:A31,0,1,FALSE)

 

Comparison of Binomial and Normal Distributions

For n greater than 30, there is not too much difference between the binomial and normal distributions, as shown in the figure below.

However, the distributions are noticeably shifted. This shift is due to the fact that the abscissa for the normal distribution is a continuous variable whereas the abscissa for the binomial distribution is a discrete variable, a point series. Consider that it is not possible to have any values between, say, 2 and 3 heads of a coin. For the binomial distribution a value of +.5 is sometimes applied to the abscissa of the binomial distribution. This is called a correction for continuity.

 

The binomial distribution was generated as

 

Microsoft Excel

Fx = BINOMDIST(A1,31,.5,FALSE)

Binomial Distribution

 

 

Microsoft Excel

Fx = NORMDIST(A1:A31,15,2.7386,FALSE)

Normal Distribution