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