Normal PDF Calculator
Probability Density Function of the Normal Distribution, defined by mean (μ) and standard deviation (σ).
Formula first
Overview
The Normal PDF defines the probability density of a continuous random variable that follows a symmetric, bell-shaped distribution. It is uniquely determined by the mean, which sets the location of the peak, and the standard deviation, which controls the spread or width of the curve.
Symbols
Variables
f(x) = Probability Density, x = Value, \mu = Mean, \sigma = Std Deviation
Apply it well
When To Use
When to use: Apply this equation when modeling natural phenomena like heights, weights, or test scores that cluster around a central average. It is also the appropriate choice when the Central Limit Theorem suggests that the sum of independent random variables tends toward a normal distribution.
Why it matters: This distribution is the foundation of frequentist statistics, enabling the calculation of P-values, Z-scores, and confidence intervals. Its mathematical properties allow scientists to quantify uncertainty and predict the likelihood of outcomes in everything from engineering to psychology.
Avoid these traps
Common Mistakes
- Evaluating PDF instead of CDF for probability.
- Sigma in denominator.
One free problem
Practice Problem
In a distribution with a mean (μ) of 50 and a standard deviation (σ) of 10, calculate the probability density (y) at the point where x = 50.
Solve for:
Hint: When x equals the mean, the exponent term becomes zero, making the entire exponential part of the equation equal to 1.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Wikipedia: Normal distribution
- Probability and Statistics for Engineers and Scientists by Walpole, Myers, Myers, Ye
- Mathematical Statistics with Applications by Wackerly, Mendenhall, Scheaffer
- Normal distribution (Wikipedia article)
- Probability and Statistics for Engineering and the Sciences by Jay L. Devore
- Edexcel A-Level Mathematics — Statistics (Normal Distribution)