Continuous vs. discrete: the first fork in the road
Normal is continuous — it models measurements that can take any value within a range (height, measurement error, test scores approximately). Binomial and Poisson are both discrete — they model counting outcomes (how many successes, how many events), never fractional values. This single distinction — "am I measuring a continuous quantity or counting discrete occurrences" — immediately rules out normal for anything that's fundamentally a count.
Normal distribution: the bell curve, defined by two numbers
Fully described by its mean (μ, where the peak sits) and standard deviation (σ, how spread out it is). It shows up constantly because of the Central Limit Theorem — the sum or average of many independent random factors tends toward a normal distribution regardless of the individual factors' own distribution, which is why so many real-world measurements (subject to many small, independent influences) end up approximately bell-shaped.
Use normal when: modeling a continuous measurement influenced by many small independent factors.
Example: heights of adults in a population, measurement error in a sensor reading.
Binomial: counting successes in a fixed number of independent trials
Applies when you have a fixed number of trials (n), each trial has exactly two possible outcomes (success/failure), the trials are independent of each other, and the success probability (p) is the same for every trial. The binomial distribution answers: "what's the probability of exactly k successes out of n trials?"
Use binomial when: n is fixed and known in advance, and p stays constant across trials.
Example: probability of exactly 7 heads in 10 coin flips; probability of exactly 3 defective
items in a batch of 50, given a known defect rate.
Poisson: counting events in a fixed interval, with no fixed "n"
This is the one people most often misapply. Poisson models the number of events occurring in a fixed interval (of time, area, or volume) when those events happen independently and at a constant average rate (λ) — critically, there's no fixed "number of trials" the way binomial requires; events could in principle happen any number of times.
Use Poisson when: there's no natural "n" — you're counting occurrences of something
that could happen any number of times in a fixed window, at a known average rate.
Example: number of customer arrivals at a store per hour; number of typos per page
of a manuscript; number of network requests per second to a server.
The practical test: does "n" and "p" actually exist?
If you can clearly name a fixed number of independent trials (n) and a constant per-trial success probability (p), you're looking at binomial. If instead you're counting occurrences with no natural upper bound and no meaningful "per-trial probability" — just an average rate over an interval — you're looking at Poisson. This single question resolves the vast majority of binomial-vs-Poisson confusion.
| Question | Binomial | Poisson |
|---|---|---|
| Fixed number of trials (n)? | Yes | No — no natural "n" |
| Constant per-trial probability (p)? | Yes | N/A — rate (λ) instead |
| Example | Coin flips, defect counts in a fixed batch | Arrivals per hour, typos per page |
Common mistakes
- Using binomial when there's no actual fixed "n." Counting rare events over a time period (customer arrivals, server errors per minute) usually has no natural trial count — that's a signal to reach for Poisson instead.
- Applying the normal distribution to a small discrete count. The normal distribution is a reasonable approximation to binomial or Poisson only under certain conditions (large n, moderate p) — using it directly on small counts without justification can be inaccurate.
- Forgetting Poisson requires a constant rate. If the underlying rate genuinely changes over the interval (rush hour vs. midnight arrivals lumped into one "per hour" average), a single Poisson parameter misrepresents the real pattern.
FAQ
How do I decide between binomial and Poisson for a counting problem?
Check whether there's a fixed, known number of independent trials with a constant success probability (binomial) — if instead you're counting occurrences with no natural upper bound, just a constant average rate over an interval, use Poisson.
Why does the normal distribution show up so often in real data?
The Central Limit Theorem — sums or averages of many independent small factors tend toward a normal distribution regardless of those factors' own individual distributions, which is why so many naturally-occurring continuous measurements approximate a bell curve.
Can the normal distribution ever substitute for binomial or Poisson?
As an approximation, under certain conditions (binomial with large n and moderate p; Poisson with a sufficiently large λ) — but it's an approximation with its own accuracy conditions, not a universal substitute for the exact discrete calculation.
Compute exact PDF/PMF and CDF values for all three distributions with the Probability Distribution Calculator — entirely client-side.