standarddeviationcalculator.net

Updated

Probability distribution calculators

Each calculator here takes the parameters of one probability distribution and returns probabilities, percentiles, the mean and the standard deviation. Discrete distributions count things; continuous ones model measurements, waiting times and proportions. Use the chooser to match your situation to the right model.

Which calculator do I need?

You have or wantUse
Successes in a fixed number of independent trials with the same pBinomial distribution calculator
Successes in a sample drawn without replacement from a small populationHypergeometric distribution calculator
The trial on which the first success happensGeometric distribution calculator
Counts of events per hour, per page or per square metre at an average ratePoisson distribution calculator
A bell-shaped measurement and the area beyond a valueNormal distribution calculator
An area or percentile, and want the cut-off valueInverse normal distribution calculator
The waiting time until the next event at a constant rateExponential distribution calculator
Time to failure of a component, with a wear-out or early-failure patternWeibull distribution calculator

Discrete distributions: counting successes and events

For outcomes you count: successes in trials, trials until a success, or events in an interval.

The normal distribution

Areas, cut-offs and the 68–95–99.7 rule for bell-shaped data.

Continuous distributions for times, sizes and proportions

Skewed or bounded models for waiting times, lifetimes, incomes, distances and proportions.

Choosing a distribution

Start with what you are measuring. If it is a count, you want a discrete distribution. The binomial counts successes in a fixed number of independent trials; the geometric and negative binomial fix the number of successes and count the trials instead; the hypergeometric is the binomial’s counterpart when you sample without replacement; and the Poisson counts events with no fixed number of trials.

If it is a measurement, you want a continuous distribution. Symmetric, bell-shaped data suits the normal. Waiting times at a constant rate are exponential; lifetimes whose failure rate rises or falls with age are Weibull. Positive, right-skewed quantities such as incomes or particle sizes are often lognormal, and proportions between 0 and 1 are modelled with the beta.

Worked comparison: with or without replacement

Deal 5 cards from a standard deck. What is the chance of exactly 2 hearts? The cards are not replaced, so the right model is hypergeometric with N = 52, K = 13, n = 5: P(X = 2) = C(13, 2) × C(39, 3) / C(52, 5) = 0.2743. Treating each card as an independent 1-in-4 draw (binomial with n = 5, p = 0.25) gives 0.2637, which is about 4% too low. The binomial approximation only works when the sample is a small fraction of the population, say under 5%.

Common mix-ups

  • Two geometric conventions. Some books count trials up to and including the first success (starting at 1), others count failures before it (starting at 0). The calculator shows both.
  • P(X < k) vs P(X ≤ k). For discrete distributions they differ by P(X = k); for continuous ones they are equal.
  • Lognormal parameters. μ and σ describe ln X, not X itself. Enter the mean and SD of X if that is what you have.
  • Empirical rule vs exact areas. The empirical rule rounds to 68–95–99.7; the exact figures are 68.27%, 95.45% and 99.73%.

Guides to read alongside

Common questions

When can I use the Poisson instead of the binomial?
When n is large and p is small, the binomial with parameters n and p is close to a Poisson with λ = np. A common rule of thumb is n ≥ 20 and p ≤ 0.05; for n = 100 and p = 0.03 the two give P(X ≤ 3) = 0.6472 to four places.
What is the difference between the normal and inverse normal calculators?
The normal calculator goes from a value to an area: given x, it returns the probability below, above or between. The inverse normal goes the other way, from an area to the value that cuts it off, such as the score at the 90th percentile.
Which distribution fits waiting times?
If events occur at a constant average rate and independently, the time until the next one is exponential. If the rate changes with age, as with parts that wear out, the Weibull fits better; with shape k = 1 it reduces to the exponential.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.