Normal Distribution Calculator
Evaluate Gaussian normal distribution metrics for continuous variables. Computes probability density function (PDF), cumulative distribution function (CDF), Z-score standardization, and upper-tail probabilities.
The specific observation or boundary point
Center of the bell curve
Spread parameter (strictly positive)
Area under the bell curve from −∞ up to x
How to Use This Calculator
- Enter the value x to evaluate.
- Enter the mean (μ) of your normal distribution (0 for standard normal).
- Enter the standard deviation (σ) (1 for standard normal).
- Inspect the cumulative percentile probability and bell curve density height.
Formula & Calculation Methodology
The Gaussian probability density function describes the symmetrical bell curve. The area under the entire curve equals 1.0 (100%).
Worked Example
Standard normal curve at x = 0 (μ = 0, σ = 1)
Evaluating x = 1.96 standard deviations
Frequently Asked Questions
Why does the normal distribution peak at x = μ?↓
The exponential term e^[−(x − μ)² / 2σ²] achieves its maximum value of 1.0 when (x − μ) = 0, producing peak height 1 / (σ√(2π)).
What is the 68–95–99.7 rule in a normal distribution?↓
Approximately 68.27% of observations lie within 1 standard deviation of the mean, 95.45% within 2 standard deviations, and 99.73% within 3 standard deviations.
Can the PDF density value exceed 1.0?↓
Yes. Probability density is not a probability; when standard deviation is very small (e.g. σ = 0.1), the curve is tall and narrow, exceeding 1.0 at its apex.
What is a limitation of the normal distribution model?↓
Real-world data often exhibits fat tails (excess kurtosis) or skewness, meaning extreme events occur more frequently than Gaussian models predict.
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Source Citation & Disclaimer
Informational tool only; not licensed tax, legal, or financial advice.