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

Select plot

Step 2. Set parameters

Set Parameters for N(μ, σ)

1. Mean(μ), the dashed line, indicates the location

2. Standard Deviation(σ), indicates the shape

3. Blue Area = Pr(Mean-n*SD < X < Mean+n*SD)

4. Change range of x-axis, symmetric to 0

Sample size of simulated numbers

The number of bins in histogram

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

Manual Input
Upload Data

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

1. Choose CSV/TXT file

Browse...

2. Show 1st row as column names?

Yes

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

Yes

4. Which separator for data?

Comma (,): CSV often uses this

One Tab (->|): TXT often uses this

Semicolon (;)

One Space (_)

Correct separator and quote ensure the successful data input

Find some example data here

The number of bins in histogram

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

Step 3. Show Probability

Area Proportion Left to Red line = Pr(X < x₀), x₀ is the position of Red-line

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

Select plot

Step 2. Set parameters

Set Parameters for E(Rate)

Rate (> 0) indicates the rate of change, input rate

Or. Calculate Rate from Mean and SD (Mean = SD), input mean

Mean = SD = 1/Rate

2. Change the range of x-axis > 0

Sample size of simulated numbers

The number of bins in histogram

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

Manual Input
Upload Data

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

1. Choose CSV/TXT file

Browse...

2. Show 1st row as column names?

Yes

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

Yes

4. Which separator for data?

Comma (,): CSV often uses this

One Tab (->|): TXT often uses this

Semicolon (;)

One Space (_)

Correct separator and quote ensure the successful data input

Find some example data here

The number of bins in histogram

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

Step 3. Show Probability

Area Proportion Left to Red-line = Pr(X < x₀), x₀ is the position of Red-line

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

Select plot

Step 2. Set parameters

1. Set Parameters for Gamma(k, θ)

k > 0, Shape parameter

θ > 0, Scale parameter

Or. Calculate k and θ from Mean and SD (Mean = SD), input mean

Input SD

Mean = kθ

Variance = kθ²

Change the range of x-axis, > 0

Sample size of simulated numbers

The number of bins in histogram

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

Manual Input
Upload Data

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

1. Choose CSV/TXT file

Browse...

2. Show 1st row as column names?

Yes

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

Yes

4. Which separator for data?

Comma (,): CSV often uses this

One Tab (->|): TXT often uses this

Semicolon (;)

One Space (_)

Correct separator and quote ensure the successful data input

Find some example data here

The number of bins in histogram

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

Step 3. Show Probability

Area Proportion Left to Red-line = Pr(X < x₀), x₀ is the position of Red-line

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

Select plot

Step 2. Set parameters

1. Set Parameters for Beta(α, β)

α > 0, Shape parameter

β > 0, Shape parameter

Or. Calculate k and θ from Mean and SD (Mean = SD), input mean

Input SD

Mean = α / (α + β)

Variance = αβ/[(α + β)^2(α + β+1)]

Change the range of x-axis, > 0

Sample size of simulated numbers

The number of bins in histogram

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

Manual Input
Upload Data

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

1. Choose CSV/TXT file

Browse...

2. Show 1st row as column names?

Yes

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

Yes

4. Which separator for data?

Comma (,): CSV often uses this

One Tab (->|): TXT often uses this

Semicolon (;)

One Space (_)

Correct separator and quote ensure the successful data input

Find some example data here

The number of bins in histogram

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

Step 3. Change Probability

Area Proportion Left to Red-line = Pr(X < x₀), x₀ is the position of Red-line

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

Select plot

Step 2. Set parameters

1. Set Parameters for T(v)

v > 0, Degree of Freedom, related to the shape

Or. Calculate Rate from SD, input SD

SD = v/(v-2)

Change the range of x-asis

Sample size of simulated numbers

The number of bins in histogram

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

Manual Input
Upload Data

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

1. Choose CSV/TXT file

Browse...

2. Show 1st row as column names?

Yes

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

Yes

4. Which separator for data?

Comma (,): CSV often uses this

One Tab (->|): TXT often uses this

Semicolon (;)

One Space (_)

Correct separator and quote ensure the successful data input

Find some example data here

The number of bins in histogram

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

Step 2. Show Probability

Area Proportion Left to Red-line = Pr(X < x₀), x₀ is the position of Red-line

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

Select plot

Step 2. Set parameters

1. Set Parameters for Chi(v)

v > 0, Degree of Freedom = Mean = Variance/2

Change the range of x-axis, > 0

Sample size of simulated numbers

The number of bins in histogram

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

Manual Input
Upload Data

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

1. Choose CSV/TXT file

Browse...

2. Show 1st row as column names?

Yes

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

Yes

4. Which separator for data?

Comma (,): CSV often uses this

One Tab (->|): TXT often uses this

Semicolon (;)

One Space (_)

Correct separator and quote ensure the successful data input

Find some example data here

The number of bins in histogram

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

Step 2. Show Probability

Area Proportion Left to Red-line = Pr(X < x₀), x₀ is the position of Red-line

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

Select plot

Step 2. Set parameters

1. Set Parameters

v1 > 0, Degree of Freedom 1

v2 > 0, Degree of Freedom 2

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

Range of x-axis, > 0

Sample size of simulated numbers

The number of bins in histogram

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

Manual Input
Upload Data

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

1. Choose CSV/TXT file

Browse...

2. Show 1st row as column names?

Yes

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

Yes

4. Which separator for data?

Comma (,): CSV often uses this

One Tab (->|): TXT often uses this

Semicolon (;)

One Space (_)

Correct separator and quote ensure the successful data input

Find some example data here

The number of bins in histogram

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

Step 3. Show Probability

Area Proportion Left to Red-line = Pr(X < x₀), x₀ is the position of Red-line

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