Normal Distribution Probability Density Function Calculator

Formula first

Overview

The Normal Distribution Probability Density Function (PDF) describes a continuous random variable with a symmetric bell-shaped curve. Its height at a particular x-value depends on the mean and standard deviation, and the total area under the curve is 1.

Symbols

Variables

f(x | $\mu$, ${\sigma }^2$) = Probability density, x = Value of x, $\mu$ = Mean, ${\sigma }^2$ = Variance, e = Euler's number

Apply it well

When To Use

When to use: Use this formula when modeling continuous data that clusters around a mean and forms a bell-shaped distribution.

Why it matters: It is one of the foundational distributions in statistics and appears in confidence intervals, hypothesis tests, and many real-world measurement processes.

Avoid these traps

Common Mistakes

• Treating the PDF height as a probability.
• Forgetting the 1/(sigma * sqrt(2pi)) normalizing factor.
• Confusing variance with standard deviation.
• Using the formula for non-continuous outcomes.

One free problem

Practice Problem

For a standard normal distribution with mean 0 and variance 1, find the probability density at x = 0.

Solve for: $f(x|\mu ,{\sigma }^2)$

Hint: At the mean, the exponential term becomes 1.

The full worked solution stays in the interactive walkthrough.

Keep going

Related Formulas

References

Sources

1. Britannica Editors (2026) 'Standard normal distribution' Encyclopaedia Britannica.
2. NIST/SEMATECH e-Handbook of Statistical Methods, normal distribution section.
3. NIST Special Publication 1017, standard normal distribution references.
4. Standard probability and statistics textbooks
5. Any standard textbook on probability and statistics or advanced calculus.