# Normal Distribution Calculator

Free normal distribution calculator. Calculate probabilities, z-scores, and PDF values for the normal (Gaussian) bell curve distribution.

## What this calculates

Calculate probabilities for the normal (Gaussian) distribution. Find the probability that a value falls below, above, or between two points on the bell curve.

## Inputs

- **Mean (μ)** — The mean of the normal distribution.
- **Standard Deviation (σ)** — min 0.0001 — The standard deviation (must be positive).
- **X Value** — The value to calculate the probability for.
- **Lower Bound (optional)** — Lower bound for P(a < X < b) calculation.
- **Upper Bound (optional)** — Upper bound for P(a < X < b) calculation.

## Outputs

- **Z-Score** — The standardized z-score for the given x value.
- **P(X < x)** — Probability that X is less than the given value.
- **P(X > x)** — Probability that X is greater than the given value.
- **P(a < X < b)** — Probability that X falls between the lower and upper bounds.
- **PDF f(x)** — The probability density function value at x.
- **Interpretation** — formatted as text — Plain language interpretation.

## Details

The normal distribution (bell curve) is the most important distribution in statistics.

PDF (Probability Density Function):
f(x) = (1 / (σ√(2π))) × e^(-(x-μ)²/(2σ²))

CDF (Cumulative Distribution Function):
P(X ≤ x) = Φ((x - μ) / σ)

where Φ is the standard normal CDF.

68-95-99.7 Rule

- 68% of data falls within 1σ of the mean

- 95% within 2σ

- 99.7% within 3σ

## Frequently Asked Questions

**Q: What is the standard normal distribution?**

A: The standard normal distribution has mean = 0 and standard deviation = 1. Any normal distribution can be converted to the standard form using z-scores: z = (x - μ) / σ. This allows you to use standard normal tables for any normal distribution.

**Q: What is the 68-95-99.7 rule?**

A: Also called the empirical rule, it states that for a normal distribution: approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations.

**Q: How do I know if my data is normally distributed?**

A: Use a histogram or Q-Q plot to visually check. Formal tests include the Shapiro-Wilk test and the Kolmogorov-Smirnov test. Many statistical methods are robust to moderate deviations from normality, especially with large sample sizes (Central Limit Theorem).

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Source: https://vastcalc.com/calculators/statistics/normal-distribution
Category: Statistics
Last updated: 2026-04-21