# Normal Distribution Calculator: Probabilities in Real Time
The Normal Distribution Calculator computes any normal distribution probability instantly: left tail, right tail, central interval, and inverse. Simply enter the mean and standard deviation to get the result and visualize it on the Gaussian bell curve. 4 Calculation Modes
Live Real-Time Results
Free No Registration
# Calculation Modes
| Mode | Description | Example Use |
|---|---|---|
| P(X ≤ a) | Cumulative probability up to value a (left tail). | Percentage of students below a grade threshold. |
| P(X ≥ a) | Right-tail probability from value a onwards. | Probability of exceeding a quality threshold. |
| P(a ≤ X ≤ b) | Probability within a central or asymmetric interval. | Proportion of parts within tolerance. |
| Inverse | Find X such that P(X ≤ x) = p (the p-th percentile). | Critical value in a hypothesis test. |
The 68 95 99.7 Empirical Rule
In any normal distribution: 68% of data falls within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. Verify this by calculating P(-1 ≤ z ≤ 1) with μ=0 and σ=1.# Quick Reference Glossary
- Mean (μ)
- Location parameter. Determines the center of the Gaussian bell curve.
- Standard Deviation (σ)
- Scale parameter. Controls the width of the bell curve. Must be strictly positive.
- Z-score
- Standardized value: z = (X - μ) / σ. Indicates how many standard deviations X is from the mean.
- Probability Density Function. The area under the PDF over an interval equals the probability of that interval.