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.


Step 1. Select the data source

Mathematical-based, simulated-data-based, or user data-based


Step 2. Set parameters

Set Parameters for N(μ, σ)


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


Data point can be separated by , ; /Enter /Tab /Space

Data be copied from CSV (one column) and pasted in the box

Missing value is input as NA


Upload data will cover the example data

2. Show 1st row as column names?

3. Use 1st column as row names? (No duplicates)

Correct separator and quote ensure the successful data input

Find some example data here

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


Step 3. Show Probability


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.


Step 1. Select the data source

Mathematical-based, simulated-data-based, or user data-based


Step 2. Set parameters

Set Parameters for E(Rate)


                

Mean = SD = 1/Rate


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


Data point can be separated by , ; /Enter /Tab /Space

Data be copied from CSV (one column) and pasted in the box

Missing value is input as NA


Upload data will cover the example data

2. Show 1st row as column names?

3. Use 1st column as row names? (No duplicates)

Correct separator and quote ensure the successful data input

Find some example data here

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


Step 3. Show Probability


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.


Step 1. Select the data source

Mathematical-based, simulated-data-based, or user data-based


Step 2. Set parameters

1. Set Parameters for Gamma(k, θ)


                
  • Mean = kθ
  • Variance = kθ2
  • When the number of bins is 0, plot will use the default number of bins


    Data point can be separated by , ; /Enter /Tab /Space

    Data be copied from CSV (one column) and pasted in the box

    Missing value is input as NA


    Upload data will cover the example data

    2. Show 1st row as column names?

    3. Use 1st column as row names? (No duplicates)

    Correct separator and quote ensure the successful data input

    Find some example data here

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


    Step 3. Show Probability


    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.


    Step 1. Select the data source

    Mathematical-based, simulated-data-based, or user data-based


    Step 2. Set parameters

    1. Set Parameters for Beta(α, β)

    
                    
  • Mean = α / (α + β)
  • Variance = αβ/[(α + β)^2(α + β+1)]
  • When the number of bins is 0, plot will use the default number of bins


    Data point can be separated by , ; /Enter /Tab /Space

    Data be copied from CSV (one column) and pasted in the box

    Missing value is input as NA


    Upload data will cover the example data

    2. Show 1st row as column names?

    3. Use 1st column as row names? (No duplicates)

    Correct separator and quote ensure the successful data input

    Find some example data here

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


    Step 3. Change Probability


    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.


    Step 1. Select the data source

    Mathematical-based, simulated-data-based, or user data-based


    Step 2. Set parameters

    1. Set Parameters for T(v)

    
                    

    SD = v/(v-2)


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


    Data point can be separated by , ; /Enter /Tab /Space

    Data be copied from CSV (one column) and pasted in the box

    Missing value is input as NA


    Upload data will cover the example data

    2. Show 1st row as column names?

    3. Use 1st column as row names? (No duplicates)

    Correct separator and quote ensure the successful data input

    Find some example data here

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


    Step 2. Show Probability


    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.


    Step 1. Select the data source

    Mathematical-based, simulated-data-based, or user data-based


    Step 2. Set parameters

    1. Set Parameters for Chi(v)

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


    Data point can be separated by , ; /Enter /Tab /Space

    Data be copied from CSV (one column) and pasted in the box

    Missing value is input as NA


    Upload data will cover the example data

    2. Show 1st row as column names?

    3. Use 1st column as row names? (No duplicates)

    Correct separator and quote ensure the successful data input

    Find some example data here

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


    Step 2. Show Probability


    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.


    Step 1. Select the data source

    Mathematical-based, simulated-data-based, or user data-based


    Step 2. Set parameters

    1. Set Parameters
  • Mean = v2 / (v2 - 2), for v2 > 2
  • Variance = [ 2 * v2^2 * ( v1 + v2 - 2 ) ] / [ v1 * ( v2 - 2 )^2 * ( v2 - 4 ) ] for v2 > 4

    You can adjust x-axes range

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


    Data point can be separated by , ; /Enter /Tab /Space

    Data be copied from CSV (one column) and pasted in the box

    Missing value is input as NA


    Upload data will cover the example data

    2. Show 1st row as column names?

    3. Use 1st column as row names? (No duplicates)

    Correct separator and quote ensure the successful data input

    Find some example data here

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


    Step 3. Show Probability


    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