Normal Distribution PDF Calculator

Formula first

Overview

The Normal Distribution Probability Density Function (PDF) describes the distribution of a continuous random variable characterized by a symmetric, bell-shaped curve. It defines the relative likelihood of a variable taking on a specific value based on the mean (center) and standard deviation (spread) of the dataset.

Symbols

Variables

x = x Value, $\mu$ = Mean, $\sigma$ = Std Dev, f(x) = Density

Apply it well

When To Use

When to use: Apply this equation when modeling natural phenomena like human height, blood pressure, or measurement errors that tend to cluster around a central average. It is essential when data follows the Central Limit Theorem, indicating that the sum of independent factors leads to a normal distribution.

Why it matters: It is the foundational distribution in statistics, enabling the calculation of z-scores, confidence intervals, and p-values. Understanding this curve allows researchers to predict the probability of specific outcomes in fields ranging from quality control to financial risk management.

Avoid these traps

Common Mistakes

• Thinking this gives P(X<x) (CDF).
• Forgetting the 1/sqrt(2pi) term.

One free problem

Practice Problem

Calculate the probability density (y) at the mean (u) of 10, assuming a standard deviation (s) of 2.

Solve for: $y$

Hint: When x equals u, the exponent term e to the power of zero becomes 1.

The full worked solution stays in the interactive walkthrough.

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Related Formulas

References

Sources

1. Wikipedia: Normal distribution
2. A First Course in Probability by Sheldon Ross
3. Wikipedia: Probability density function
4. Casella and Berger, Statistical Inference
5. Wikipedia: Central Limit Theorem
6. Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability and Statistics for Engineers and Scientists (9th ed.). Pearson.
7. Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). W. W. Norton & Company.
8. Standard curriculum — Mathematical Statistics