Stop 02 · Math FoundationsTopics · 6/9

Distributions (Normal, Binomial, Poisson)

In short

Most real data follows a few familiar shapes. Spot the shape and you instantly know which formulas, tests and models fit your problem.

The normal distribution is the classic bell curve many metrics approximate, binomial models counts of successes across repeated yes/no trials, and Poisson models the count of events in a fixed interval.

The mascot arranges a row of glass jars of marbles, the fullest jar in the middle.

Think of it like this

Marbles in jars

Drop a thousand marbles through a board of pins and most pile up in the middle jars, with fewer at the edges. That bell shape is the normal distribution. Other kinds of counting, like right answers or calls per hour, make their own shapes.

Watch it happen

Which distribution fits?

Your data

Measured

Counted

Normal

Binomial

Poisson

Heights

MCQs guessed right

Calls per hour

1/4First ask: is it measured on a scale, or counted in whole numbers?

The big ideas

What you need to know

Normal: the bell curve

About 68% of values fall within one standard deviation of the mean, and about 95% within two.

Binomial: n tries, chance p

Fixed number of yes/no trials, same chance each time. Mean is n × p.

Poisson: counts in a window

Events happening at an average rate, independently. Its mean and variance are equal.

Plot before you assume

Draw a histogram first. Income and wait times are often skewed, not normal.

See it in code

Walk through the code

python

1from scipy import stats2 3# Normal: heights, mean 165 cm, sd 74print(stats.norm(165, 7).cdf(180))       # share shorter than 1805 6# Binomial: guess 10 MCQs, 4 options each7print(stats.binom(10, 0.25).pmf(5))      # exactly 5 right8 9# Poisson: 3 failed payments an hour on average10print(stats.poisson(3).pmf(0))           # an hour with zero

1/3Normal: what share of people are shorter than 180 cm.

scipy.stats has every common distribution. cdf gives 'this value or less', pmf gives 'exactly this count'.

Save yourself the pain

Common mistakes

Oops

Assuming every column is normally distributed.

Do this instead

Plot a histogram. Skewed data like income may need a log transform or a different model.

Oops

Using binomial when there's no fixed number of trials.

Do this instead

If you're counting events in time with no upper limit, Poisson usually fits better.

Your mission

Guess the shape

~20 min

Quick check

You're modelling the number of orders a small café gets every 10 minutes. Which distribution fits best?

Go deeper

1Sketch each distribution's shape and note where it typically shows up in real data.
2Learn the 68-95-99.7 rule for the normal distribution.
3Identify a real-world example (in a domain you know) that would fit each distribution.