Data Scientist · #045 · September 6, 2026 · 2 min read
Which probability distribution fits your data?
Normal, binomial, Poisson, log-normal, uniform and exponential: what each shape models, the tell that identifies it, and the plot-it-first rule that prevents the classic mistakes.
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Every model and every rule of thumb assumes a shape, and most data people never check which one their data actually has. Six shapes explain most of it. The print-ready A4 PDF is at the bottom.
Normal
±1σ ≈ 68%,±2σ ≈ 95%: the quick math.- Heights, measurement errors: sums of small causes.
- CLT: means become normal,
n ≥ 30.
Binomial
- n trials, probability p each: the count of successes.
mean = np,var = np(1-p).- Conversions, churn: a yes/no repeated.
Poisson
- Events per window, rate λ.
mean = var = λ: the tell.- Tickets, arrivals: rare and independent.
Long tails
- Log-normal: salaries, revenue.
mean >> median: the skew tell.log(x)first: then it behaves.
Two more
- Uniform: anything equally likely.
- Exponential: time between events.
- Memoryless: its famous property.
Match the data to the shape
- Conversion rate: binomial. n visitors, each converts with probability p.
- Tickets per hour: Poisson. Rare independent events in a fixed window.
- Revenue per user: log-normal. Most pay little, a few whales pay a lot.
- Time between failures: exponential. The waiting-time twin of Poisson.
- Mean of samples: normal, by the CLT, almost whatever the source shape.
That is the interview version: name the variable, name the distribution, name why.
The trap: treating everything as normal
| You assumed | Reality | Do instead |
|---|---|---|
| revenue is normal | log-normal: whales rule the mean | median, or log(x) first |
| ±3σ finds outliers | long tails live past 3σ legally | plot, then pick fences |
| mean = typical user | mean can beat the 90th pctile | report percentiles |
The 68/95 shortcuts and z-score fences are only true for the bell shape. One histogram costs ten seconds and prevents every mistake in this table.
Frequently asked questions
How do I know which distribution my data follows?
Plot it first: one histogram costs ten seconds and beats any assumption. Then use the tells. Sums of many small causes look normal; counts of successes over n yes/no trials are binomial; counts of rare independent events per window are Poisson, where mean equals variance; and mean far above median says long tail, usually log-normal.
What does the central limit theorem actually say?
That the mean of enough independent samples is approximately normally distributed, almost regardless of the source distribution's shape, with n of 30 or more as the usual working threshold. It is why the 68/95 shortcuts apply to sample means and why so much of testing leans on the normal curve, even when the raw data is skewed.
Why is revenue per user not normally distributed?
Because most users pay little and a few whales pay a lot, which makes it log-normal: a long right tail where the mean sits far above the median and can even beat the 90th percentile. Report the median or percentiles, or take log(x) first, after which the data behaves.
What is the difference between Poisson and exponential distributions?
Two views of the same process. Poisson counts how many rare independent events land in a fixed window, at rate lambda, with mean equal to variance as the tell. Exponential measures the waiting time between those events, and its famous property is being memoryless: having waited already changes nothing about the wait ahead.
Get the free PDF
One page, print-ready, free to share. No signup needed.