A random number distribution describes how the values of a random variable are spread or arranged. Different distributions (Normal, Uniform, Exponential, Binomial, Poisson) model different real-world phenomena — from natural measurements like heights (Normal) to counting rare events (Poisson) or modeling waiting times (Exponential).

A histogram is a graphical representation that organizes data into bins (intervals) and shows the frequency of values within each bin as bars. The number of bins significantly affects interpretation: too few bins hide detail, while too many create noise. Common rules include Sturges' formula (k = ⌈log₂(n) + 1⌉), Scott's rule, and the Freedman-Diaconis rule. For most datasets, 20-50 bins work well.

The Normal distribution, also known as the Gaussian distribution or "bell curve," is the most important probability distribution in statistics. It's characterized by its symmetric bell shape, defined by mean (μ) and standard deviation (σ). The Central Limit Theorem states that the sum of many independent random variables tends toward a Normal distribution, making it fundamental in statistical inference, quality control, and natural phenomena modeling.

The Box-Muller transform converts two independent uniform random numbers (U₁, U₂) into two independent standard Normal random numbers. The formula: Z₁ = √(-2·ln(U₁)) · cos(2π·U₂) and Z₂ = √(-2·ln(U₁)) · sin(2π·U₂). To get a Normal distribution with mean μ and standard deviation σ, we then compute X = μ + σ·Z. This elegant method is widely used in Monte Carlo simulations.

Skewness measures the asymmetry of a distribution. A skewness of 0 indicates a perfectly symmetric distribution (like the Normal distribution). Positive skewness means the right tail is longer (data clusters to the left), common in income distributions. Negative skewness means the left tail is longer (data clusters to the right). Skewness helps identify whether data is normally distributed or requires transformation before analysis.

Continuous distributions (Normal, Uniform, Exponential) can take any value within a range and are described by probability density functions (PDFs). Discrete distributions (Binomial, Poisson) can only take specific, countable values (often integers) and are described by probability mass functions (PMFs). In this tool, discrete distributions produce integer-only outputs, while continuous ones produce decimal values.

For a histogram to reliably represent the underlying distribution, a larger sample size is always better. With fewer than 100 samples, the histogram shape can be quite erratic. Around 1,000-5,000 samples, the shape becomes reasonably clear. For 10,000+ samples, the histogram closely approximates the theoretical distribution. This tool allows up to 200,000 samples for highly accurate visualization.