Probability · Distributions
Probability Distribution Calculator
Density, cumulative probability, tail areas, interval probabilities and quantiles for the six distributions you actually use — normal, t, chi-square, F, binomial and Poisson — each with a shaded plot and the exact value.
Result
In plain English
A probability distribution is just a description of how likely different outcomes are. The curve (or the bars, for counts) shows where values tend to fall; the whole area underneath adds up to 1 — that is, 100% of the possibilities. Whatever you ask it to compute, the shaded region in the plot is the probability you're after.
- PDF / PMF
- How concentrated the chances are at a particular value. For measured things (height, weight) it's a density — the height of the curve. For counts (heads in 10 flips) it's the actual probability of exactly that value.
- CDF — P(X ≤ x)
- The chance of getting your value or anything below it: the area under the curve up to that point.
- Upper tail — P(X ≥ x)
- The chance of your value or anything above it — the area in the right-hand tail.
- Interval — P(a ≤ X ≤ b)
- The chance of landing somewhere between two values.
- Quantile (inverse)
- The reverse question: you give a probability, it hands back the cut-off value — e.g. “the score that 95% of people fall below.”
- mean
- The balancing point of the distribution — its long-run average value.
- std deviation
- The typical distance of values from the mean. Bigger = more spread out.
Frequently asked
What's the difference between the PDF/PMF and the CDF?
The PDF (continuous) or PMF (discrete) gives the height or probability at a single point; the CDF gives the cumulative probability up to a point, P(X ≤ x). Tail and interval probabilities are read off the CDF.
For a discrete distribution, is P(X ≥ k) the same as P(X > k)?
No — discrete distributions place real probability on each integer, so P(X ≥ k) = 1 − P(X ≤ k−1), which includes k itself. This calculator and its exported code handle the ±1 correctly.
Which distribution should I choose?
Normal for measurements that cluster symmetrically; t for small-sample means; chi-square and F for variance and ANOVA-type tests; binomial for the number of successes in n trials; Poisson for counts of rare events over time or space.
What is the difference between a discrete and a continuous distribution?
A discrete distribution (binomial, Poisson) places probability on separate whole-number outcomes, so P(X = k) is a real, non-zero probability you can read off directly. A continuous distribution (normal, t, chi-square, F) spreads probability over a range, where any exact value has probability zero and only intervals — P(a ≤ X ≤ b) — carry weight. That is why the discrete tools report a PMF and the continuous ones a PDF.