Probability Distributions, Explained
A plain-English tour of the distributions behind everyday data
A probability distribution is just a list of everything that could happen, with a weight on each possibility. That's it. The trick — and the reason statisticians seem to have a favorite distribution for every occasion — is that different kinds of randomness have different shapes, and once you recognize the shape of yours, an enormous amount of ready-made math comes free.
This page walks through the seven distributions you will meet most often, in plain English: what each one describes, what its parameters (the dials you can turn) actually do, and when you would reach for it — with one everyday example and one business example each. Every chart below is interactive: drag the sliders and watch the shape respond.
One piece of vocabulary before we start: distributions come in two kinds. Discrete distributions describe things you count (die rolls, conversions, customers per hour) and are drawn as bars. Continuous distributions describe things you measure (heights, waiting times, percentages) and are drawn as smooth curves.
Uniform (Discrete)
Discrete (bars)Possible values: Whole numbers between a and b · Mean: (a + b) / 2
The "every option is equally likely" distribution for things you can count. If the outcome must be one of a handful of whole numbers and no outcome is favored over any other, this is your distribution. Its picture is the flattest possible: a row of equal-height bars.
The dials you can turn
- a (lower end point): The smallest value the outcome can take. Moving a to the right slides the whole row of bars right and, by shrinking the number of options, makes each remaining bar taller.
- b (upper end point, greater than a): The largest value the outcome can take. Widening the gap between a and b spreads the same total probability over more options, so every bar gets shorter — more possibilities means each one is individually less likely.
When to reach for it
Choose it when outcomes are countable, bounded, and you have no reason to believe any one is more likely than another — pure symmetric randomness, like a fair draw. It is also the honest starting point when you genuinely know nothing except the range.
In everyday life
Rolling a fair six-sided die is the textbook case: a = 1, b = 6, and every face has exactly a 1-in-6 chance. The same goes for drawing a raffle ticket numbered 1 to 100 out of a hat.
In business
A call center that routes each incoming call to one of 8 agents completely at random is assigning calls with a discrete uniform distribution — every agent should get about 1 call in 8, which is exactly the "expected split" idea behind the SRM check in A/B testing.
Mean = (a + b) / 2 = 3.5 — the long-run average outcome if you repeated this over and over.
Binomial
Discrete (bars)Possible values: Whole numbers between 0 and n · Mean: n × p
The "how many successes out of n tries" distribution. Repeat the same yes/no experiment n times — each try succeeding with the same probability p, independently — and the binomial tells you how likely each possible success count is.
The dials you can turn
- n (number of trials, 0 or more): How many times you run the yes/no experiment. More trials stretches the distribution over more possible counts and (in absolute terms) widens it — though relative to n, results actually get more predictable as n grows.
- p (probability of success, between 0 and 1): The chance any single try succeeds. It positions the bulk of the distribution: the peak sits near n × p. At p = 0.5 the shape is symmetric; pushing p toward 0 or 1 crowds the bars against that end.
When to reach for it
Choose it whenever your data is a count of successes from a fixed number of independent, identical yes/no trials: heads or tails, converted or not, clicked or ignored. It is the distribution underneath every conversion-rate A/B test on this site.
In everyday life
Flip a fair coin 10 times: the number of heads follows a Binomial(n = 10, p = 0.5). Five heads is the most likely single outcome, but you should be entirely unsurprised by 4 or 6.
In business
Email 500 customers a promotion that historically gets a 4% click rate. The number of clicks follows a Binomial(n = 500, p = 0.04): expect about 20, and the chart shows how much a "good day" (28 clicks) or "bad day" (12) is just noise.
Mean = n × p = 6 — the long-run average outcome if you repeated this over and over.
Poisson
Discrete (bars)Possible values: Whole numbers 0 and up · Mean: m
The "how many events in a window" distribution. When independent events arrive at a steady average rate — with no fixed cap on how many can occur — the Poisson tells you how likely you are to see 0, 1, 2, 3, … of them in a given stretch of time or space.
The dials you can turn
- m (mean, greater than 0): The average number of events per window — its one and only dial. Small m piles probability on 0 and 1 with a long right tail; as m grows, the peak moves right and the shape becomes more symmetric and bell-like. Uniquely, the variance equals the mean, so busier processes are also noisier in absolute terms.
When to reach for it
Choose it for counts of arrivals or occurrences with no natural maximum: events happen independently, at a steady average rate, and two never occur at exactly the same instant. If "how many could there possibly be?" has no hard answer, think Poisson rather than binomial.
In everyday life
A dedicated birdwatcher spots on average 3 rare birds per outing. Poisson(m = 3) says a birdless outing still happens about 5% of the time, and a spectacular 6-bird day about 5% too — streaks and droughts are built into steady randomness.
In business
A coffee shop gets an average of 12 customers per hour in the mid-afternoon. Poisson(m = 12) tells the owner how often to expect a 20-customer rush — about 2% of hours — which is exactly the kind of number you need for staffing decisions.
Mean = m = 4 — the long-run average outcome if you repeated this over and over.
Uniform (Continuous)
Continuous (curve)Possible values: Any value between a and b · Mean: (a + b) / 2
The "anywhere in the range, equally likely" distribution for measurements rather than counts. Instead of equal bars over whole numbers, it is one flat rectangle: every value between a and b is equally plausible, and nothing outside the range can happen.
The dials you can turn
- a (lower end point): The smallest possible value. Moving it slides the left wall of the rectangle.
- b (upper end point, greater than a): The largest possible value. Because total probability must equal 1, the rectangle’s height is exactly 1 / (b − a): widen the range and the rectangle gets lower, narrow it and it gets taller.
When to reach for it
Choose it when a measurement is guaranteed to fall in a known range, with no part of that range more likely than any other. It is also the workhorse behind simulations: random number generators produce Uniform(0, 1) draws, which get transformed into every other distribution.
In everyday life
Spin a fair prize wheel and measure the angle where it stops: any angle from 0° to 360° is equally likely — Uniform(a = 0, b = 360). The same logic covers where a dropped ball lands along the edge of a roulette wheel.
In business
A ferry departs every 30 minutes and a traveler arrives at the dock at a random moment. Their wait time is Uniform(0, 30): an average wait of 15 minutes, with a 28-minute wait exactly as likely as a 2-minute one.
Mean = (a + b) / 2 = 5 — the balance point of the curve.
Normal
Continuous (curve)Possible values: All numbers · Mean: m
The famous bell curve — the distribution of quantities that are the sum of many small, independent influences. Values near the middle are common, values far from it are increasingly rare, and the fall-off is perfectly symmetric on both sides.
The dials you can turn
- m (mean): The center of the bell — simultaneously the average, the most likely value, and the point of symmetry. Changing m slides the entire curve left or right without changing its shape.
- s (standard deviation, greater than 0): The width of the bell — how far typical values stray from the center. Small s means a tall, narrow spike (very consistent data); large s means a low, wide dome (very variable data). About 68% of values fall within one s of the mean, and 95% within two.
When to reach for it
Choose it for measurements shaped by many small additive effects: heights, measurement errors, daily averages. It also earns a special role from the central limit theorem — averages of almost anything become normal as samples grow — which is why the A/B test calculators on this site can use it for conversion-rate comparisons.
In everyday life
Adult heights are the classic example: many genes and environmental factors each nudge height a little, so it comes out normal — most people near the average, and 7-footers vanishingly rare. A golfer’s drive distances behave the same way.
In business
Daily revenue for a stable store tends to be roughly normal around its typical day: m = $12,000 and s = $1,500 means about 95% of days land between $9,000 and $15,000 — so a $8,000 day is a genuine signal worth investigating, not routine noise.
Mean = m = 0 — the balance point of the curve.
Gamma
Continuous (curve)Possible values: Positive numbers · Mean: a × b
The "waiting time" distribution. Where the Poisson counts how many events happen in a window, the gamma measures how long you wait for a given number of them. It lives only on positive numbers and is typically lopsided: a hump on the left with a long tail stretching right.
The dials you can turn
- a (shape, greater than 0): Roughly, "how many events are you waiting for". At a = 1 the curve starts high at zero and decays (short waits dominate). As a grows, the hump moves away from zero and the curve becomes more symmetric — waiting for many events averages out the randomness of each one.
- b (scale, greater than 0): The typical spacing between events — the stretch factor. Doubling b doubles every wait: the whole curve stretches to the right and flattens, without changing its fundamental shape. The mean is simply shape × scale.
When to reach for it
Choose it for positive, right-skewed quantities — durations, sizes, amounts — where values can’t go below zero, most observations are moderate, and a long tail of large values is real: time to finish a repair, insurance claim sizes, rainfall totals.
In everyday life
Waiting for your third bus of the morning: if buses arrive randomly about every 10 minutes, your total wait follows a Gamma(shape = 3, scale = 10) — 30 minutes on average, sometimes lucky, occasionally much longer. Fishing works the same way: time to catch your third fish.
In business
A support team resolves tickets in three stages — triage, fix, and verify — each taking about 2 hours of random effort. Total resolution time is roughly Gamma(shape = 3, scale = 2): 6 hours on average, with the long right tail explaining those occasional tickets that take all day.
Mean = a × b = 2 — the balance point of the curve.
Beta
Continuous (curve)Possible values: Between 0 and 1 · Mean: a / (a + b)
The distribution for percentages, proportions, and probabilities themselves. It lives entirely between 0 and 1, and its two parameters act like a running tally of evidence: a counts the "yes" outcomes you’ve seen, b counts the "no" outcomes.
The dials you can turn
- a ("yes" count, greater than 0): Evidence for the high side. Bigger a pulls the curve toward 1. Think of it as the number of successes you’ve observed (plus one, in the standard uninformed starting point).
- b ("no" count, greater than 0): Evidence for the low side. Bigger b pulls the curve toward 0. Together, the total a + b controls confidence: Beta(2, 8) and Beta(20, 80) both center near 20%, but the second — built on ten times the evidence — is far narrower. At a = b = 1 the curve is perfectly flat: total ignorance.
When to reach for it
Choose it whenever the quantity you’re uncertain about is itself a rate or percentage — a conversion rate, a free-throw percentage, a defect rate. It is the engine inside Bayesian A/B testing: the posterior curves drawn by our A/B test calculator are exactly Beta distributions updated with your data.
In everyday life
A basketball player has made 7 of 10 free throws this season. Your belief about their true skill is a Beta(a = 8, b = 4): probably around 65–70%, but the wide curve admits anywhere from 45% to 90%. After 100 more shots the curve will narrow dramatically.
In business
Your new landing page has converted 30 of 1,000 visitors. Your uncertainty about the true conversion rate is Beta(a = 31, b = 971) — centered near 3.1%, plausibly 2.1% to 4.2%. This is precisely the posterior distribution chart shown by the A/B test calculator on this site.
Mean = a / (a + b) = 0.29 — the balance point of the curve.
Where these show up in A/B testing
If you run experiments, you are already using three of these without necessarily naming them. Each visitor either converts or doesn't — a binomial process. At real traffic volumes, differences between conversion rates follow a normal distribution, which is what powers the frequentist significance test. And your uncertainty about the conversion rate itself is a beta distribution — the posterior curves drawn by our A/B test calculator are exactly the beta distributions on this page, updated with your data. Not sure which tool you need? Answer two quick questions.
Frequently Asked Questions
What is a probability distribution, in plain English?
It is a complete list of everything that could happen, with a weight on each possibility saying how likely it is. A die roll has six possibilities at 1/6 each; a person’s height has infinitely many possibilities concentrated near the average. The distribution is the shape of that uncertainty — and picking the right shape for your data is what lets you calculate anything useful from it.
What is the difference between discrete and continuous distributions?
Discrete distributions describe things you count — 0, 1, 2, 3 conversions, die faces, customers per hour — so their chart is a set of bars, one per possible outcome. Continuous distributions describe things you measure — heights, waiting times, percentages — where any value in a range is possible, so their chart is a smooth curve, and probability corresponds to area under the curve rather than the height at a single point.
How do I pick which distribution fits my data?
Ask three questions. Counting or measuring? Counting points to discrete (uniform, binomial, Poisson), measuring to continuous. Is there a natural limit? A fixed number of tries points to binomial; no cap on the count points to Poisson; values trapped between 0 and 1 point to beta. What shape do you expect? Equally likely everywhere is uniform, symmetric around a typical value is normal, and positive with a long right tail is gamma.
Which of these distributions power the A/B testing calculators on this site?
Three of them do the heavy lifting. Each visitor converting or not is a binomial process. The normal distribution approximates conversion-rate differences at scale, which is what the frequentist significance test relies on. And the beta distribution represents uncertainty about the conversion rate itself — the posterior curves in the Bayesian panel of the A/B test calculator are beta distributions updated with your data.
What does the mean of a distribution actually tell me?
It is the long-run average: the value you would get by repeating the random process endlessly and averaging the results. It is the balance point of the chart, not necessarily the most likely value — for skewed shapes like the gamma, the mean sits to the right of the peak because the long tail of large values drags the average up.