Normal Distribution

Functionalities

Draw a Mathematical-based Normal Distribution

  • Draw a Normal Distribution with N(μ, σ); μ indicates the mean (location), and σ indicates its standard deviation (shape).
  • Calculate the position x0 of a user-defined probability Pr(X ≤ x0) that is the possibility of a variable X being in an interval (-∞, x0] from the probability distribution.
    In the curve, the left area to the red-line indicates this possibility value, and the intersection of the red line and horizontal axis (X-axis) is the x0.
  • Calculate the probability Pr(μ – n × σ < X ≤ μ + n × σ) that is the possibility of a variable X being in an interval (μ – n × σ, μ + n × σ]
    In the curve, the blue area indicates this possibility value.

Draw a Simulated-based Normal Distribution

  • Generate and download random numbers of normal distribution using a user-defined sample size.
  • Draw histogram of the generated random numbers.
  • Calculate the Mean(μ) and Standard Deviation(σ) of the generated random numbers.
  • Calculate the position x0 of a user-defined probability Pr(X ≤ x0) that is the possibility of a variable X being in an interval (-∞, x0] from the probability distribution of the generated random numbers.

Draw a User Data-based Normal Distribution

  • Upload your data using Manual Input or from CSV/TXT files.
  • Draw histogram and density plots of your data.
  • Calculate the Mean(μ) and Standard Deviation(σ) of your data.
  • Calculate the position x0 of a user-defined probability Pr(X ≤ x0) that is the possibility of a variable X being in an interval (-∞, x0] from the probability distribution of your data.

Case Example

Suppose we wanted to see the shape of N(0, 1) and wanted to know 1. at which point x0 when Pr(X < x0) = 0.025, and 2. what is the probability between means +/- 1SD

Please follow the Steps, and Outputs will give real-time analytical results.

Outputs

Mathematical-based Plot


                

                  

                

                

                  The position of the red line and the blue ares
                

                

                

                

              

              

                
Simulation-based Plot

                Histogram from random numbers
                

                
                  
                  Download Random Numbers
                
                

                  Sample descriptive statistics
                

                

              

              

                Data preview
                

                
Distribution of Your Data

                Density from upload data
                

                Histogram from upload data
                

                CDF from upload data
                

                

                  Sample descriptive statistics
                

                

              

            

          

          

        

        

          

            
Exponential Distribution

          

          


 Functionalities


    

  •  Draw an Exponential Distribution with E(Rate); Rate indicates the rate of change
    • 

  •  Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line
    • 

  •  Get the probability distribution from simulation numbers in Simulation-based tab
    • 

  •  Download the random number in Simulation-based tab
    • 

  •  Get the mean, SD, and Pr(X ≤ x0) of simulated numbers
    • 

  •  Get the probability distribution of your data which can be roughly compared to E(Rate)
    • 




Case Example

Suppose we wanted to see the shape of E(2), and wanted to know at which point x0 when Pr(X < x0)= 0.05 

 Please follow the Steps, and Outputs will give real-time analytical results.




          

          

            

              
            

            

              

                Outputs
              

              

                
Mathematical-based Plot

                Exponential distribution plot
                

                

                

                  

                

                

                  The position of red line
                

                

                

              

              

                
Simulation-based Plot

                Histogram from random numbers
                

                
                  
                  Download Random Numbers
                
                

                  Sample descriptive statistics
                

                

              

              

                Data preview
                

                
Distribution of Your Data

                Density from upload data
                

                Histogram from upload data
                

                CDF from upload data
                

                

                  Sample descriptive statistics
                

                

              

            

          

          

        

        

          

            
Gamma Distribution

          

          


 Functionalities


    

  •  Draw a Gamma Distribution with Gamma(α, θ); α controls the shape, 1/θ controls the change of rate
    • 

  •  Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line
    • 

  •  Get the probability distribution from simulation numbers in Simulation-based tab
    • 

  •  Download the random number in Simulation-based tab
    • 

  •  Get the mean, SD, and Pr(X ≤ x0) of simulated numbers
    • 

  •  Get the probability distribution of your data which can be roughly compared to  Gamma(α, θ)
    • 




Case Example

Suppose we wanted to see the shape of Gamma(9,0.5), and wanted to know at which point x0 when Pr(X < x0)= 0.05 


 Please follow the Steps, and Outputs will give real-time analytical results.



          

          

            

              
            

            

              

                Outputs
              

              

                
Mathematical-based Plot

                Gamma distribution plot
                

                

                

                

              

              

                
Simulation-based Plot

                Histogram from random numbers
                

                
                  
                  Download Random Numbers
                
                

                  Sample descriptive statistics
                

                

              

              

                Data preview
                

                
Distribution of Your Data

                Density from upload data
                

                Histogram from upload data
                

                CDF from upload data
                

                

                  Sample descriptive statistics
                

                

              

            

          

          

        

        

          

            
Beta Distribution

          

          


 Functionalities


    

  •  Draw a Beta Distribution with Beta(α, β); α, β controls the shape

  •  Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line
    • 

  •  Get the probability distribution from simulation numbers in Simulation-based tab
    • 

  •  Download the random number in Simulation-based tab
    • 

  •  Get the mean, SD, and Pr(X ≤ x0) of simulated numbers
    • 

  •  Get the probability distribution of your data which can be roughly compared to Beta(α, β)
    • 




Case Example

Suppose we wanted to see the shape of Beta(12, 12), and wanted to know at which point x0 when Pr(X < x0)= 0.05 


 Please follow the Steps, and Outputs will give real-time analytical results.



          

          

            

              
            

            

              

                Outputs
              

              

                
Mathematical-based Plot

                Beta distribution plot
                

                

                

                

              

              

                
Simulation-based Plot

                Histogram from random numbers
                

                
                  
                  Download Random Numbers
                
                

                  Sample descriptive statistics
                

                

                
     Explanation 
   
    
    
  •   Mean = α/(α+β) 
    • 
    
  •   SD = sqrt(α*β/(α+β)^2(α+β+1)) 
    • 
   

    
              

              

                Data preview
                

                
Distribution of Your Data

                Density from upload data
                

                Histogram from upload data
                

                CDF from upload data
                

                

                  Sample descriptive statistics
                

                

              

            

          

          

        

        

          

            
Student's T Distribution

          

          


 Functionalities


    

  •  Draw a T Distribution with T(v); v is the degree of freedom related to your sample size and control the shape
    • 

  •  Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line
    • 

  •  Get the probability distribution from simulation numbers in Simulation-based tab
    • 

  •  Download the random number in Simulation-based tab
    • 

  •  Get the mean, SD, and Pr(X ≤ x0) of simulated numbers
    • 

  •  Get the probability distribution of your data which can be roughly compared to T(v)
    • 




Case Example

Suppose we wanted to see the shape of T(4) and wanted to know at which point x0 when Pr(X < x0)= 0.025 


 Please follow the Steps, and Outputs will give real-time analytical results.



          

          

            

              
            

            

              

                Outputs
              

              

                
Mathematical-based Plot

                T distribution plot
                
The blue curve is the standard normal distribution

                

                

                

                

              

              

                
Simulation-based Plot

                Histogram from random numbers
                

                
                  
                  Download Random Numbers
                
                

                  Sample descriptive statistics
                

                

              

              

                Data preview
                

                
Distribution of Your Data

                Density from upload data
                

                Histogram from upload data
                

                CDF from upload data
                

                

                  Sample descriptive statistics
                

                

              

            

          

          

        

        

          

            
Chi-Squared Distribution

          

          


 Functionalities


    

  •  Draw a Chi-Squared Distribution with Chi(v); v is the degree of freedom related to your sample size and control the shape
    • 

  •  Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line
    • 

  •  Get the probability distribution from simulation numbers in Simulation-based tab
    • 

  •  Download the random number in Simulation-based tab
    • 

  •  Get the mean, SD, and Pr(X ≤ x0) of simulated numbers
    • 

  •  Get the probability distribution of your data which can be roughly compared to Chi(v)
    • 




Case Example

Suppose we wanted to see the shape of Chi(4), and wanted to know at which point x0 when Pr(X < x0)= 0.05 


 Please follow the Steps, and Outputs will give real-time analytical results.



          

          

            

              
            

            

              

                Outputs
              

              

                
Mathematical-based Plot

                Chi-square distribution plot
                

                

                

                

              

              

                
Simulation-based Plot

                Histogram from random numbers
                

                
When the number of bins is 0, plot will use the default number of bins

                
                  
                  Download Random Numbers
                
                

                  Sample descriptive statistics
                

                

                
     Explanation 
   
    
    
  •   Mean = v 
    • 
    
  •   SD = sqrt(2v) 
    • 
   

    
              

              

                Data preview
                

                
Distribution of Your Data

                Density from upload data
                

                Histogram from upload data
                

                CDF from upload data
                

                

                  Sample descriptive statistics
                

                

              

            

          

          

        

        

          

            
F Distribution

          

          


 Functionalities


    

  •  Draw a F Distribution with F(df1, df2) ; df1 and df2 are the degree of freedom related to your sample size and control the shape

  •  Get the probability distribution of x0 that Pr(X ≤ x0) = left to the red-line

  •  Get the probability distribution from simulation numbers in Simulation-based tab

  •  Download the random number in Simulation-based tab

  •  Get the mean, SD, and Pr(X ≤ x0) of simulated numbers

  •  Get the probability distribution of your data which can be roughly compared to F(df1, df2)




Case Example

Suppose we wanted to see the shape of F(100, 10), and wanted to know at which point x0 when Pr(X < x0)= 0.05 


 Please follow the Steps, and Outputs will give real-time analytical results.



          

          

            

              
            

            

              

                Outputs
              

              

                
Mathematical-based Plot

                F distribution plot
                

                

                

                

              

              

                
Simulation-based Plot

                Histogram from random numbers
                

                
                  
                  Download Random Numbers
                
                

                  Sample descriptive statistics
                

                

              

              

                Data preview
                

                
Distribution of Your Data

                Density from upload data
                

                Histogram from upload data
                

                CDF from upload data
                

                

                  Sample descriptive statistics