**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 x
_{0}of a user-defined probability Pr(X ≤ x_{0}) that is the possibility of a variable X being in an interval (-∞, x_{0}] 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 x_{0}. - 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 x
_{0}of a user-defined probability Pr(X ≤ x_{0}) that is the possibility of a variable X being in an interval (-∞, x_{0}] 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 x
_{0}of a user-defined probability Pr(X ≤ x_{0}) that is the possibility of a variable X being in an interval (-∞, x_{0}] from the probability distribution of your data.

**The position of the red line and the blue ares**

**Sample descriptive statistics**

**Sample descriptive statistics**

- Draw an Exponential Distribution with E(Rate); Rate indicates the rate of change
- Get the probability distribution of x
_{0}that Pr(X ≤ x_{0}) = 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 ≤ x
_{0}) of simulated numbers - Get the probability distribution of your data which can be roughly compared to E(Rate)

**The position of red line**

**Sample descriptive statistics**

**Sample descriptive statistics**

- Draw a Gamma Distribution with Gamma(α, θ); α controls the shape, 1/θ controls the change of rate
- Get the probability distribution of x
_{0}that Pr(X ≤ x_{0}) = 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 ≤ x
_{0}) of simulated numbers - Get the probability distribution of your data which can be roughly compared to Gamma(α, θ)

**Sample descriptive statistics**

**Sample descriptive statistics**

- Draw a Beta Distribution with Beta(α, β); α, β controls the shape
- Get the probability distribution of x
_{0}that Pr(X ≤ x_{0}) = 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 ≤ x
_{0}) of simulated numbers - Get the probability distribution of your data which can be roughly compared to Beta(α, β)

**Sample descriptive statistics**

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

**Sample descriptive statistics**

- 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 x
_{0}that Pr(X ≤ x_{0}) = 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 ≤ x
_{0}) of simulated numbers - Get the probability distribution of your data which can be roughly compared to T(v)

The blue curve is the standard normal distribution

**Sample descriptive statistics**

**Sample descriptive statistics**

- 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 x
_{0}that Pr(X ≤ x_{0}) = 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 ≤ x
_{0}) of simulated numbers - Get the probability distribution of your data which can be roughly compared to Chi(v)

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

Download Random Numbers
**Sample descriptive statistics**

- Mean = v
- SD = sqrt(2v)

**Sample descriptive statistics**

- Draw a F Distribution with F(df
_{1}, df_{2}) ; df_{1}and df_{2}are the degree of freedom related to your sample size and control the shape - Get the probability distribution of x
_{0}that Pr(X ≤ x_{0}) = 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 ≤ x
_{0}) of simulated numbers - Get the probability distribution of your data which can be roughly compared to F(df
_{1}, df_{2})

**Sample descriptive statistics**

**Sample descriptive statistics**