StatReason

Discrete Distribution Calculator

Enter any set of outcomes and their probabilities to check the distribution is valid and summarize its center and spread.

What this answers

This calculator answers two things at once: "does this set of outcomes and probabilities form a valid probability mass function?" and, if so, "what are its mean, variance, and standard deviation?" A valid PMF requires every probability to be nonnegative and the whole set to sum to exactly 1.

How it is calculated

The mean is each outcome weighted by its probability, summed across all outcomes. The variance is the probability-weighted average squared distance of each outcome from that mean, and the standard deviation is its square root, the same definitions used throughout this project's descriptive statistics engines, just weighted by stated probabilities instead of observed sample frequencies.

Worked example

For a fair six-sided die (outcomes 1 through 6, each with probability 1/6): the total probability is exactly 1, the mean is 3.5, and the variance works out to approximately 2.917, giving a standard deviation of about 1.708.

Assumption audit

Calculated from your data: whether your probabilities are nonnegative and sum to 1, and if so, the mean, variance, and SD of the resulting distribution.
Evidence to review: where your stated probabilities came from; this calculator validates internal consistency, not whether the probabilities themselves are a good model of the real-world process.
You must verify: that every possible outcome is actually included in your list; a PMF that sums to 1 but omits a real possible outcome will still pass this calculator's validation while misrepresenting the actual process.

Common mistakes

A common mistake is rounding individual probabilities and then finding the total does not sum to exactly 1, triggering this calculator's validation error; adjust one value slightly or use more decimal places to fix genuine rounding artifacts. Another is listing the same outcome twice with a split probability instead of merging it into one row with the combined probability, which does not break the math but makes the distribution harder to read. A third is assuming a small variance means the distribution is "more certain" in some absolute sense, when variance is only meaningful relative to the specific outcome scale being used.

Limitations

This calculator handles discrete distributions with a finite, explicitly listed set of outcomes; it does not model continuous distributions or infinite discrete distributions like the Poisson, which have their own dedicated calculators.