Explore Normal Distribution Probabilities

Visualize and calculate probabilities for normal distributions with ease. Understand the bell curve and its applications in statistics.

Probability Statistics Distributions

Distribution Parameters

Mean (

\mu

)

Average value

Standard Deviation (

\sigma

)

Spread of data

Lower Bound (a)

Upper Bound (b)

Probability & Visualization

Probability P(a ≤ X ≤ b):

Visualize the normal distribution by entering parameters and calculating the probability.

Understanding Normal Distribution

The normal distribution, often called the Gaussian distribution or bell curve, is fundamental in statistics. It describes how the values of a variable are distributed. In a normal distribution, most values cluster around the mean, forming a symmetrical bell shape.

It's defined by two key parameters:

Mean ( $\mu$ ): The average value, determining the center of the distribution.
Standard Deviation ( $\sigma$ ): Measures the spread or variability of the distribution. A larger standard deviation means a wider curve.

The probability density function, which describes the shape of the normal distribution, is given by:

f(x; \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}

This calculator helps you find the probability that a random variable $$ X $$ falls within a specific range $$ [a, b] $$ for a given normal distribution. This is crucial in many fields for risk assessment, quality control, and making predictions based on data.