Normal Distribution Probability Density Function (PDF) Calculator
The probability density function of a normal distribution describes the likelihood of a continuous random variable taking on a specific value based on its mean and variance.
Formula first
Overview
This formula represents the classic bell-shaped Gaussian curve, where the peak is defined by the mean (μ) and the spread or width is controlled by the variance (σ²). It is the cornerstone of inferential statistics, as the Central Limit Theorem dictates that the sums of many independent random variables tend toward this distribution. The integral of this function over any interval represents the probability that the random variable falls within that range.
Symbols
Variables
x = Random Variable, = Mean, = Variance
Apply it well
When To Use
When to use: Use this to model physical, biological, or social phenomena where data points cluster around a central average with symmetric deviations.
Why it matters: It allows for the calculation of probabilities, hypothesis testing, and the estimation of parameters in nearly all scientific and engineering fields.
Avoid these traps
Common Mistakes
- Confusing standard deviation (σ) with variance (σ²).
- Assuming the PDF value is a probability itself, rather than a density (the probability of an exact point is 0).
One free problem
Practice Problem
For a normal distribution with a mean (μ) of 0 and a variance (σ²) of 1, calculate the density f(x) at x = 0.
Solve for:
Hint: Recall that = 1 and the expression simplifies to 1/sqrt(2π).
The full worked solution stays in the interactive walkthrough.
References
Sources
- Feller, W. (1968). An Introduction to Probability Theory and Its Applications.
- Ross, S. M. (2014). A First Course in Probability.
- Gauss, C. F. (1809). Theoria motus corporum coelestium.