Normal Distribution Calculator

The normal distribution (also called the Gaussian distribution or bell curve) is the most important probability distribution in statistics. It describes the distribution of many natural phenomena — heights, test scores, measurement errors — and is the foundation of most parametric inference.

Notation

$X \sim N(\mu, \sigma^2)$

where $\mu$ is the mean (center of the bell) and $\sigma^2$ is the variance ($\sigma$ is the standard deviation, the spread).

Probability Density Function (PDF)

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$

Key Properties

• Symmetric about $\mu$: mean = median = mode
• Bell-shaped: highest at $\mu$, tails extend infinitely
• Total area under the curve = 1 (it's a probability distribution)
• Empirical rule: ~68% of data within $\pm 1\sigma$, ~95% within $\pm 2\sigma$, ~99.7% within $\pm 3\sigma$

Interval	Approx. probability
$\mu \pm \sigma$	68.27%
$\mu \pm 2\sigma$	95.45%
$\mu \pm 3\sigma$	99.73%

The standard normal distribution is the special case $Z \sim N(0,1)$ — mean 0, standard deviation 1. Its CDF is written $\Phi(z)$.

Standardizing (Computing Z-Scores)

Any normal random variable $X \sim N(\mu, \sigma^2)$ can be converted to a standard normal variable:

$Z = \frac{X - \mu}{\sigma}$

The z-score tells you how many standard deviations $x$ is above or below the mean.

Computing Probabilities

All normal probability problems reduce to looking up $\Phi(z)$ (the area to the left of $z$):

$P(X < x) = P\!\left(Z < \frac{x-\mu}{\sigma}\right) = \Phi\!\left(\frac{x-\mu}{\sigma}\right)$

$P(X > x) = 1 - \Phi\!\left(\frac{x-\mu}{\sigma}\right)$

$P(a < X < b) = \Phi\!\left(\frac{b-\mu}{\sigma}\right) - \Phi\!\left(\frac{a-\mu}{\sigma}\right)$

Finding Percentiles (Inverse Normal)

To find the value $x$ such that $P(X \leq x) = p$:

$x = \mu + \sigma \cdot z_p$

where $z_p = \Phi^{-1}(p)$ is the $p$-th percentile of the standard normal.

Common percentiles:

Percentile	$z$
90th	1.282
95th	1.645
97.5th	1.960
99th	2.326

The normal distribution is central to statistics because of the Central Limit Theorem (CLT): the sampling distribution of the sample mean $\bar{X}$ approaches $N(\mu, \sigma^2/n)$ as $n$ grows, regardless of the original population's shape.

This means:

• Confidence intervals and hypothesis tests for means use normal (or t) critical values
• Many test statistics (t, z, F, $\chi^2$) are derived from normal distributions
• Even non-normal data can be analyzed using normal approximations when $n$ is large enough

Normal Approximation to the Binomial

When $n$ is large and $p$ is not too extreme, $X \sim \text{Binomial}(n, p)$ is approximately $N(np,\, np(1-p))$:

$P(X \leq k) \approx \Phi\!\left(\frac{k + 0.5 - np}{\sqrt{np(1-p)}}\right)$

(The $+0.5$ is the continuity correction for the discrete-to-continuous approximation.)

Validity: use this approximation when $np \geq 10$ and $n(1-p) \geq 10$.