367
Normal distribution
Bell-curve probabilities — below, above, or between, with the z-score shown.
CALCULATOR · STATISTICS · PROBABILITY
Probability Distribution Calculator
“What are the odds?” is a question with three classic answers. The normal distribution handles bell-curved quantities — heights, test scores, measurement error. The binomial handles yes/no trials — coin flips, free throws, pass/fail batches. The Poisson handles rare events over time — calls per hour, typos per page. Pick your distribution below: a score below 70 with mean 100 and SD 15 happens 2.28% of the time; exactly 3 heads in 10 flips happens 11.72%; exactly 2 arrivals when 4 are expected happens 14.65%. Every answer ships with its formula.
368
Binomial distribution
Yes/no trials — exactly, at most, or at least k successes in n tries.
369
Poisson distribution
Rare events over time or space — exactly, at most, or at least k events.
Which distribution is yours?
| Distribution | Use it when | Needs | Example |
|---|---|---|---|
| Normal | Continuous, bell-curved data | mean μ, SD σ | Test scores, heights, errors |
| Binomial | Fixed yes/no trials | n trials, p per trial | Coin flips, free throws, defect counts |
| Poisson | Rare events per interval | rate λ | Calls/hour, typos/page, goals/match |
The worked examples
Normal: scores with μ = 100, σ = 15. What fraction scores below 70? z = (70 − 100) / 15 = −2.00, and Φ(−2) = 2.28% — about 1 in 44. Between 85 and 115 (z = ±1) is 68.27% — the famous 68% rule, computed not quoted.
Binomial: 10 fair coin flips — exactly 3 heads? C(10,3) × 0.510 = 120/1024 = 11.72%. At most 3 heads sums k = 0…3 → 17.19%. Note the asymmetry: “exactly” is always smaller than the cumulative versions.
Poisson: a shop averages λ = 4 customers an hour — exactly 2 next hour? e−4 × 42 / 2! = 14.65%. Six or more arrivals: 21.49% — rare events still happen about one hour in five.
Output · formula · meaning
| Output | Formula | What it means |
|---|---|---|
| Normal probability | Φ((x−μ)/σ), the standard normal CDF | 2.28% below 70 — the share of the bell curve under the cutoff. |
| Z-score | z = (x − μ) / σ | −2.00 — the value is two standard deviations below the mean. |
| Binomial P(X = k) | C(n,k) pk (1−p)n−k | 11.72% — exactly 3 heads in 10 flips. |
| Poisson P(X = k) | e−λ λk / k! | 14.65% — exactly 2 arrivals at rate 4. |
Who uses this
- Students: checking homework on z-scores, binomial tables, and Poisson problems — with the formula visible.
- Quality engineers: defect probabilities — at most k bad parts in a batch of n.
- Ops planners: staffing for rare-event demand — the Poisson “at least” answers “how often are we swamped?”
- Sports bettors-turned-students: honest odds for streaks and counts — no, the coin has no memory.
Accuracy & limitations
The normal CDF uses the Abramowitz–Stegun erf approximation (error under 10−7) — plenty for any practical question. Binomial and Poisson sums are computed exactly by recurrence, not approximated, for n up to 1,000 and k up to 200. The models assume their textbook conditions: normal needs roughly bell-shaped data, binomial needs independent trials with fixed p, Poisson needs a constant rate with independent events. Break the assumptions and the numbers are precise but wrong. Educational tool, not a substitute for a statistics course.
Cite this calculatorBright Side Kit. “Probability Distribution Calculator.” brightoolbox.com, 2026.
Bright Side Kit. "Probability Distribution Calculator." https://brightoolbox.com/probability-distribution-calculator/ Accessed October 23, 2026.Frequently asked questions
What is the difference between PDF and CDF?
The PDF (density) is the curve’s height — not a probability. The CDF (cumulative) is the area under the curve up to a point — that is a probability. This calculator reports CDF-style answers (“below”, “at most”) because those are the questions people actually ask.
When do I use normal vs binomial vs Poisson?
Normal: continuous measurements that cluster around a middle (heights, scores). Binomial: a fixed number of yes/no trials (flips, free throws). Poisson: events happening at some average rate over time or space (calls per hour). If you are counting successes in n tries, it is binomial; if there is no fixed n, think Poisson.
What is a z-score?
How many standard deviations a value sits from the mean: z = (x − μ) / σ. A z of −2.00 means “two SDs below average” — about 2.28% of a normal distribution falls below it. Z-scores let you compare values from completely different distributions on one scale.
Why is P(X = exactly 70) basically zero for the normal curve?
Because the normal distribution is continuous — the probability of hitting any single exact point is zero; only intervals have probability. That is why the normal panel offers below / above / between instead of “exactly”. For exact counts, use binomial or Poisson.
How accurate are these calculations?
Very. The normal CDF uses a classic approximation accurate to about 1 in 10 million, and the binomial/Poisson values are computed exactly by recurrence — no normal approximation, no sampling. The remaining error is your model choice, not the arithmetic.
What does λ (lambda) mean in the Poisson panel?
The average number of events per interval — 4 customers per hour, 0.5 typos per page. The distribution then spreads probability across 0, 1, 2, … events around that average. Bigger λ shifts the whole hump right; the calculator shows exactly how.