Probability Distribution Calculator - Normal, Binomial & Poisson
Math Calculators
Calculate PDF, PMF, and CDF values for the Normal, Binomial, and Poisson probability distributions. Visualize the distribution curve instantly in your browser.
Paste your raw observations into the Statistics Calculator, then copy its Mean result into this field. Keep the observations in the same units used here.
Paste your raw observations into the Statistics Calculator and copy Std Dev. Choose sample when the data is a sample, or population when it is the complete population.
Common probability distributions
| Distribution | Use case | Key parameters |
|---|---|---|
| Normal (Gaussian) | Heights, test scores, measurement errors | Mean (μ), standard deviation (σ) |
| Binomial | Coin flips, pass/fail outcomes in n trials | n (trials), p (probability of success) |
| Poisson | Rare events in fixed time/space (calls per hour) | λ (average rate) |
| Uniform | Equal probability outcomes (dice roll) | a (min), b (max) |
| Exponential | Time between events (wait time) | λ (rate) |
The 68-95-99.7 rule
For a normal distribution: 68% of values fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ. This is also called the empirical rule.
How discrete endpoint values are handled
Binomial and Poisson distributions count whole-number outcomes, so this tool requires whole numbers for n and k. Their boundary values are valid: with a binomial probability of 0, exactly zero successes has probability 1; with probability 1, exactly n successes has probability 1. Likewise, a Poisson process with rate λ = 0 has probability 1 of zero events. These cases are evaluated directly instead of relying on logarithms such as log(0), which are undefined.
Choosing and interpreting a distribution
Use the binomial model only when there is a fixed number of independent trials with the same success probability. Use Poisson for a count over a fixed interval when its rate is reasonably constant. The normal curve is continuous, so its PDF is a density rather than the probability of one exact value; probabilities over ranges come from differences between CDF values. The chart and all calculations run locally in your browser.
Related calculators
- StatisticsEstimate mean and standard deviation from observations before modeling them.
- Probability CalculatorCombine a model probability with complements, intersections, or unions.
- Z-ScoreStandardize a normal observation and see its percentile position.
- Confidence IntervalBuild an interval estimate from sample statistics instead of a point probability.