Probability
Common Continuous Distributions
In Random Variables & Probability Distributions we saw that a continuous variable can take any value in a range, and that its PDF gives probabilities as areas under a curve. As with discrete variables, a small set of well-known patterns covers most real situations. This post covers four of them: uniform, normal, exponential and log-normal. For the discrete counterparts, see Four Common Discrete Distributions.
Uniform distribution
The continuous uniform distribution describes a situation where every value in an interval is equally likely. The curve is a flat rectangle, and the probability of a range is simply its width as a share of the whole interval.
Imagine a customer who calls at a random moment during a 10-minute window. The probability that the call comes between minute 2 and minute 5 is 3 out of 10, or 30%.
Where it is used
As a building block for simulations and random assignment, and whenever you know a value lies within a range but have no reason to favor any part of it. It is the continuous counterpart of the die roll.
Normal distribution
The normal distribution is bell-shaped and symmetric around the mean. Most values sit close to the center, and the further out you go, the rarer they become. It describes many natural and business measurements, such as heights, test scores or delivery times, and it is also the model behind many statistical methods.
Learn more
The 68-95-99.7 rule, z-scores and how to spot unusual values are covered in Normal Distribution.
Exponential distribution
The exponential distribution models the waiting time until the next event, when events happen independently at a known average rate. Short waits are the most likely, and long waits get steadily rarer, so the curve starts high and falls away to the right.
Suppose a call center receives 2 calls per hour on average. The time between two calls follows an exponential distribution with an average wait of 30 minutes. About 63% of the gaps are shorter than 30 minutes, and only about 14% are longer than an hour.
Poisson and exponential
They describe the same process from two sides. The Poisson distribution counts how many events happen in a fixed period, the exponential measures the time between two events.
Log-normal distribution
The log-normal distribution describes values that cannot be negative and are skewed to the right: most values are small, and a few are very large. If you take the logarithm of such values, they follow a normal distribution.
Order values are a typical example. Most customers spend a modest amount, and a few place very large orders. In the chart, the typical (median) order is about 20 euros, but the large orders pull the average up to about 28 euros.
Mean vs median
With right-skewed data such as order values, incomes or response times, the median describes a typical case better than the mean. Report both.
Which distribution should I use?
A quick way to choose is to ask what you are measuring.
| Distribution | What it models | Business example |
|---|---|---|
| Uniform | All values in a range equally likely | A call at a random moment in a time window |
| Normal | Symmetric values around an average | Delivery times, test scores |
| Exponential | Waiting time until the next event | Time between support calls |
| Log-normal | Positive, right-skewed values | Order values, incomes |
How to explain it to stakeholders
“Our order values are not symmetric. Most customers spend a small amount and a few spend a lot, so the average is pulled up by those big orders. The typical order is about 20 euros, while the average is about 28.”
“That is why we report the median next to the average. If we only looked at the average, we would overestimate what a typical customer spends, and we would plan promotions and shipping thresholds for a customer who barely exists.”
“The question for us is therefore not ‘What is the average?’ but ‘Which number describes the customer we actually have?’”
Key takeaway
Choosing a distribution starts with a simple question: what are you measuring? Equal chances in a range point to the uniform, symmetric values around an average to the normal, waiting times to the exponential, and positive, right-skewed values to the log-normal. The right choice tells you which summary numbers to trust and which statistical method to use next.