# 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

• 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

#### 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

#### Mathematical-based Plot

The position of the red line and the blue ares

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

Data preview

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
• 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

#### 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

#### Mathematical-based Plot

Exponential distribution plot

The position of red line

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

Data preview

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
• 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

#### 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

#### Mathematical-based Plot

Gamma distribution plot

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

Data preview

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
• 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

#### 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

#### Mathematical-based Plot

Beta distribution plot

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

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

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
• 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

#### 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

#### Mathematical-based Plot

T distribution plot

The blue curve is the standard normal distribution

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

Data preview

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
• 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

#### 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

#### 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

Sample descriptive statistics

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

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
• 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

#### 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

• 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

#### Mathematical-based Plot

F distribution plot

#### Simulation-based Plot

Histogram from random numbers

Sample descriptive statistics

Data preview