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