shape

Normal Distribution Calculator

Our Normal Distribution Calculator is a free online tool that displays the probability distribution for the given data set

Normal Distribution Calculator

Normal Distribution Calculator

Enter the parameters to compute the CDF, PDF, and generate a random sample.

Results

CDF at \( x \):

PDF at \( x \):

Random Sample:

Normal Distribution: \(f(x;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\exp\Bigl(-\frac{(x-\mu)^2}{2\sigma^2}\Bigr)\), CDF: \(\Phi(x)=0.5\Bigl[1+\operatorname{erf}\Bigl(\frac{x-\mu}{\sigma\sqrt{2}}\Bigr)\Bigr]\).

Normal distribution function

The normal distribution, also known as the Gaussian distribution, is a probability distribution characterized by its bell-shaped curve that is symmetric about the mean, and the probability density function is defined by f(x) = (1 / (σ * √(2π))) * exp(-(x – μ)² / (2σ²)), where μ is the mean and σ is the standard deviation. 

Here’s a more detailed explanation:

Key Concepts:

  • Probability Distribution: The normal distribution describes the probability of a random variable taking on a certain value within a given range. 
  • Symmetry: The distribution is symmetrical around its mean, meaning values equidistant from the mean have equal probabilities. 
  • Bell-Shaped Curve: The graph of a normal distribution is often referred to as a “bell curve” because of its shape. 
  • Parameters: The normal distribution is characterized by two parameters: the mean (μ) and the standard deviation (σ). 
  • Mean (μ): The average or center of the distribution. 
  • Standard Deviation (σ): A measure of the spread or dispersion of the data around the mean. 

The Probability Density Function (PDF):

  • Formula:The formula for the PDF of a normal distribution is: f(x) = (1 / (σ * √(2π))) * exp(-(x – μ)² / (2σ²)).
    • f(x) is the probability density at a specific value x. 
    • e is the base of the natural logarithm (approximately 2.71828). 
    • μ is the mean. 
    • σ is the standard deviation. 
    • π is the mathematical constant pi (approximately 3.14159). 
  • Interpretation:The PDF describes the relative likelihood of a random variable taking on a specific value. 
  • Area Under the Curve:The area under the curve of the PDF between two values corresponds to the probability of the random variable falling within that range. 

The Standard Normal Distribution:

  • A special case of the normal distribution where the mean (μ) is 0 and the standard deviation (σ) is 1.
  • It is often used as a reference for making calculations and comparisons with other normal distributions. 

Why is the Normal Distribution Important?

  • Central Limit Theorem:Many real-world phenomena are approximately normally distributed, and the Central Limit Theorem states that the distribution of sample means tends toward a normal distribution as the sample size increases. 
  • Real-World Applications:The normal distribution is used in many fields, such as statistics, economics, finance, and engineering. 
  • Examples:
    • Measurements of heights or weights of individuals. 
    • Errors in scientific experiments. 
    • Test scores or examination grades. 

Related Calculators

Previous Post
CASC –
Next Post
Binomial Confidence

Comments are closed