Normal Distribution Calculator
Calculate Gaussian normal distribution probabilities $P(X \le x)$, $P(X \ge x)$, $P(x_1 \le X \le x_2)$, standard $Z$-scores, percentiles, and inverse quantiles with live interactive SVG bell curve charts.
Normal Distribution Definition & Key Formulas
A normal (Gaussian) distribution is a symmetric, continuous probability distribution characterized by its mean μ (center) and standard deviation σ (spread). The total area under its bell-shaped probability density curve equals exactly 1.
Gaussian Probability Density (PDF) & Cumulative Distribution (CDF)
The normal distribution $\mathcal{N}(\mu, \sigma^2)$ is the foundation of inferential statistics. Its smooth, bell-shaped probability density function $f(x)$ is defined across all real numbers $x \in (-\infty, \infty)$:
The cumulative distribution function (CDF) represents the integrated area under the curve from $-\infty$ up to a given cutoff $x$, representing the probability $P(X \le x)$:
The Empirical 68–95–99.7 Rule & Standard Deviation Intervals
Approximately 68.27% of all observations lie within $\pm 1$ standard deviation of the mean. Roughly 31.73% fall in the outside tails.
Approximately 95.45% of observations lie within $\pm 2\sigma$. The classic 95% confidence interval threshold corresponds to $\pm 1.96\sigma$.
Approximately 99.73% of observations lie within $\pm 3\sigma$. Observations beyond $\pm 3\sigma$ occur only $0.27\%$ of the time (1 in 370).
Z-Score Standardization & Reading Standard Normal Tables
Any normal random variable $X \sim \mathcal{N}(\mu, \sigma^2)$ can be linearly transformed into the standard normal distribution $Z \sim \mathcal{N}(0, 1)$ with mean $0$ and variance $1$:
Indicates the value $x$ is above the population average. Cumulative percentile $P(Z \le z) > 0.50$.
Indicates the value $x$ is below the population average. Cumulative percentile $P(Z \le z) < 0.50$.
The Central Limit Theorem (CLT) & Sampling Distributions
The Central Limit Theorem states that if you take sufficiently large random samples ($n \ge 30$) from any population with mean $\mu$ and standard deviation $\sigma$, the distribution of the sample means $\bar{X}$ will be approximately normal:
Here, $\sigma / \sqrt{n}$ is the Standard Error of the Mean (SEM), explaining why larger sample sizes yield increasingly narrow and precise estimates of the true population parameter.
Engineering & Industrial Applications (Six Sigma & Quality Control)
Six Sigma Quality Control
Industrial manufacturing targets process capability tolerances of $6\sigma$, allowing fewer than 3.4 defect parts per million opportunities (DPMO).
Standardized Test Scoring (SAT/GRE)
Educational testing services scale raw exam scores onto Gaussian normal distributions with calibrated mean and standard deviation benchmarks.
Financial Risk Modeling (VaR)
Value at Risk (VaR) models portfolio market risk by calculating the 1st or 5th percentile lower tail loss limits under normal market volatility assumptions.
Graded Step-by-Step Numerical Solutions
Find the probability that a random observation falls between 85 and 115 in a distribution with $\mu = 100$ and $\sigma = 15$.
1. Compute $Z_1$: $Z_1 = (85 - 100) / 15 = -15 / 15 = -1.00$.
2. Compute $Z_2$: $Z_2 = (115 - 100) / 15 = +15 / 15 = +1.00$.
3. Evaluate CDFs: $\Phi(+1.00) \approx 0.84134$, $\Phi(-1.00) \approx 0.15866$.
4. Subtract cumulative bounds: $P(85 \le X \le 115) = 0.84134 - 0.15866 = 0.68268$.
Probability $P(85 \le X \le 115) \approx 68.27\%$.
Common Pitfalls & Non-Normal Data Traps
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