Choosing an appropriate distribution

Matching the situation to a distribution

The key Excellence skill is choosing the right model by checking which distribution's conditions fit the situation:
- Binomial — a count of successes in a fixed number of independent trials with constant $p$ (e.g. heads in 20 tosses).
- Poisson — a count of events over an interval at a constant average rate (e.g. calls per hour).
- Normal — a continuous measurement clustering symmetrically around a mean (e.g. heights).
- Uniform (rectangular) — outcomes equally likely across a range (e.g. a random time in an hour).
- Triangular — a continuous variable with a known minimum, most likely value (mode), and maximum, used when little data is available.

Checking a model fits

After choosing, check the conditions actually hold — e.g. for a binomial, are the trials really independent with a constant probability?
If the conditions are broken (like sampling without replacement for a binomial), the model is not appropriate and its answers are unreliable.

Worked ExampleChoosing the distribution

For each situation, name the most appropriate distribution and justify it: (a) the number of sixes in 12 rolls of a die; (b) the number of texts a phone receives per minute; (c) the time a randomly chosen commuter waits for a bus that comes exactly every 10 minutes.

Part (a) — binomial

There is a fixed number of trials (12 rolls), two outcomes (six or not), constant $p = \tfrac{1}{6}$ , and the rolls are independent — so it is binomial.

Part (b) — Poisson

It is a count of events over an interval (per minute) at an average rate, with no fixed number of trials — so it is Poisson.

Part (c) — uniform

With a bus exactly every 10 minutes, the wait is equally likely to be anywhere from 0 to 10 minutes — a continuous uniform distribution.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Which distribution models the number of heads in 30 coin tosses?

Merit

A shop records the number of customers arriving each 10 minutes. State the appropriate distribution and justify it.

Excellence

A researcher wants to model the lengths of adult eels, which cluster around an average length. They plan to use a binomial distribution. Explain why this is inappropriate and state a better choice.