Normal Distribution Calculator

The Normal Distribution Calculator computes the z-score, probability density function (PDF), and cumulative distribution function (CDF) for any value $$x$$ from a normal distribution with specified mean $$\mu$$ and standard deviation $$\sigma$$.

The normal (Gaussian) distribution is the most important probability distribution in statistics, described by the PDF:

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \, e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

The z-score standardizes any normally distributed value to the standard normal distribution ($$\mu = 0, \sigma = 1$$):

$$z = \frac{x - \mu}{\sigma}$$

The CDF gives the probability that a random variable $$X$$ takes a value less than or equal to $$x$$:

$$\Phi(x) = P(X \le x) = \int_{-\infty}^{x} f(t) \, dt$$

Since the normal CDF has no closed-form expression, this calculator uses the Abramowitz and Stegun rational approximation with five polynomial terms, achieving accuracy to approximately 6 decimal places.

The Standard Normal Distribution

When $$\mu = 0$$ and $$\sigma = 1$$, we obtain the standard normal distribution. Any normal distribution can be converted to standard normal via the z-score transformation. This is the foundation of z-tables used throughout statistics.

The Empirical Rule (68-95-99.7)

For any normal distribution:

• About 68.27% of values fall within $$\mu \pm \sigma$$
• About 95.45% fall within $$\mu \pm 2\sigma$$
• About 99.73% fall within $$\mu \pm 3\sigma$$

This rule provides quick mental estimates before computing exact probabilities.

Properties

The normal distribution is symmetric about its mean, with mean = median = mode. It is completely determined by two parameters: $$\mu$$ (location) and $$\sigma$$ (scale). The total area under the curve equals 1. The inflection points occur at $$x = \mu \pm \sigma$$.

Applications

The normal distribution appears throughout science and statistics due to the Central Limit Theorem: the sum (or average) of many independent random variables tends toward a normal distribution regardless of the underlying distribution. This makes it fundamental for hypothesis testing, confidence intervals, quality control, financial modeling, and measurement error analysis.