A normal distribution (Gaussian distribution) is a symmetric, bell-shaped probability distribution defined by two parameters: the mean (μ) which determines the center, and the standard deviation (σ) which determines the spread. It is the most important distribution in statistics, appearing naturally in countless real-world phenomena — from heights and IQ scores to measurement errors and financial returns. The total area under the curve always equals 1.

The PDF (Probability Density Function) gives the height of the curve at a specific x-value — it shows relative likelihood but is not a probability itself. The CDF (Cumulative Distribution Function) gives the actual probability that a random value falls ≤ x, represented by the area under the curve to the left of x. For example, if CDF(x)=0.84, there's an 84% chance a randomly drawn value will be ≤ x.

A Z-score (standard score) measures how many standard deviations a value is from the mean: z = (x − μ) / σ. Z-scores allow you to compare values from different normal distributions on a common scale. They're essential for hypothesis testing, confidence intervals, and understanding where a particular value stands relative to the population. A Z-score of +2 means the value is 2 standard deviations above the mean — roughly in the 97.7th percentile.

The Empirical Rule states that for a normal distribution: ~68.27% of data falls within ±1σ of the mean, ~95.45% falls within ±2σ, and ~99.73% falls within ±3σ. This rule provides quick, intuitive probability estimates without needing exact calculations. Values beyond ±3σ are considered outliers, occurring less than 0.3% of the time.

This calculator uses the Abramowitz and Stegun approximation for the error function (erf), which provides CDF values accurate to about 1.5 × 10⁻⁷ (7 decimal places). This is more than sufficient for all practical applications including academic research, quality control, and financial modeling. The PDF is computed directly from the exact formula.

This calculator is specifically designed for normal (Gaussian) distributions. If your data is skewed, heavy-tailed, or follows a different distribution (e.g., exponential, Poisson, binomial), the results will not be accurate. Consider using distribution-specific tools or performing a normality test (like Shapiro-Wilk or Kolmogorov-Smirnov) before applying normal distribution calculations to your data.

Probability density is the rate of change of probability at a given point. For continuous distributions, the probability of any single exact value is technically 0 — instead, probabilities are defined over intervals (areas under the curve). The PDF value tells you how dense the probability is near that point. Higher PDF values indicate regions where observations are more likely to cluster. The PDF at the mean is always 1/(σ√(2π)).

Finding the x-value corresponding to a given cumulative probability requires the inverse CDF (quantile function). While this tool focuses on forward calculations, you can approximate it by adjusting the x-value until the CDF matches your target probability. For precise inverse calculations, many statistical software packages (R, Python SciPy, Excel's NORM.INV) offer built-in quantile functions. The inverse CDF is also key for generating random normal variates and constructing confidence intervals.