Normal Distribution Calculator

Summary

Enter mean, standard deviation, and x to get the PDF, CDF (P(X ≤ x)), and z-score instantly.

How it works

1. Enter the population mean (μ) — the center of the distribution.
2. Enter the standard deviation (σ) — controls the spread; must be greater than zero.
3. Enter your x value — the point you want to evaluate.
4. The calculator returns the PDF (height of the curve at x), the CDF P(X ≤ x), and the z-score.
5. The tail probabilities P(X > x), P(X < x), and the two-tailed P(|X - μ| ≤ |x - μ|) are also shown.
6. Use the chart to see the shaded area representing the cumulative probability.

Use cases

• Statistics courses: check homework answers for normal probability problems.
• Quality control: find the probability a measurement falls within spec limits.
• Finance: estimate the probability of a return exceeding a threshold.
• Psychology/research: convert raw scores to percentile ranks.
• Machine learning: evaluate how far a data point is from the mean in standard deviations.
• Standardized testing: convert raw scores to z-scores and percentiles.

Frequently Asked Questions