# Choosing an appropriate distribution

You can express uncertainty about any variable by using a probability distribution. You can select a distribution based on available data, on the judgment of a knowledgeable expert, or on a combination of data and judgment. These questions about the uncertain quantity can help you select the most appropriate distribution:

## Is the quantity discrete or continuous?

When trying to express uncertainty about a quantity, the first technical question is whether the quantity is discrete or continuous.

A discrete quantity has a finite or countable number of possible values — for example, the gender of a person or the country of a person’s birth. Logical or Boolean variables are a type of discrete variable with only two values, true or false, sometimes coded as yes or no, present or absent, or 1 or 0 — for example, whether a person was born before January 1, 1950, or whether a person has ever resided in California.

A continuous quantity can be represented by a real number, and has infinitely many possible values between any two values in its domain. Examples are the quantity of an air pollutant released during a given period of time, the distance in miles of a residence from a source of air pollution, and the volume of air breathed by a specified individual during one year.

For a large discrete quantity, such as the number of humans residing within 50 miles of Disneyland on December 25, 1980, it is often convenient to treat it as continuous. Even though you know that the number of live people must be an integer, you might want to represent uncertainty about the number with a continuous probability distribution.

Conversely, it is often convenient to treat continuous quantities as discrete by partitioning the set of possible values into a small finite set of partitions. For example, instead of modeling human age by a continuous quantity between 0 and 120, it is often convenient to partition people into infants (age < 2 years), children (3 to 12), teenagers (13 to 19), young adults (20 to 40), middleaged (41 to 65), and seniors (over 65 years). This process is termed discretizing. It is often convenient to discretize continuous quantities before assessing probability distributions.

## Does the quantity have bounds?

If the quantity is continuous, it is useful to know if it is bounded before choosing a distribution — that is, does it have a minimum and maximum value?

Some continuous quantities have exact lower bounds. For example, a river flow cannot be less than zero (assuming the river cannot reverse direction). Some quantities also have exact upper bounds. For example, the percentage of a population that is exposed to an air pollutant cannot be greater than 100%.

Most real world quantities have de facto bounds — that is, you can comfortably assert that there is zero probability that the quantity would be smaller than some lower bound, or larger than some upper bound, even though there is no precise way to determine the bound. For example, you can be sure that no human could weigh more than 5000 pounds; you might be less sure whether 500 pounds is an absolute upper bound.

Many standard continuous probability distributions, such as the normal distribution, are unbounded. In other words, there is some probability that a normally distributed quantity is below any finite value, no matter how small, and above any finite value, no matter how large.

Nevertheless, the probability density drops off quite rapidly for extreme values, with near exponential decay, in fact, for the normal distribution. Accordingly, people often use such unbounded distributions to represent real world quantities that actually have finite bounds. For example, the normal distribution generally provides a good fit for the distribution of heights in a human population, even though you might be certain that no person’s height is less than zero or greater than 12 feet.

## How many modes does it have?

The mode of a distribution is its most probable value. The mode of an uncertain quantity is the value at the highest peak of the density function, or, equivalently, at the steepest slope on the cumulative probability distribution.

Important questions to ask about a distribution are how many modes it has, and approximately where it, or they, are? Most distributions have a single mode, but some have several and are known as multimodal distributions.

If a quantity has two or more modes, you can usually view it as a combination of two or more populations. For example, the distribution of ages in a daycare center at leaving time might include one mode at age 3 for the children and another mode at age 27 for the parents and caretakers. There is obviously a population of children and a population of parents. It is generally easier to decompose a multimodal quantity into its separate components and assess them separately than to assess a multimodal distribution. You can then assess a unimodal (single mode) probability distribution for each component, and combine them to get the aggregate distribution. This approach is often more convenient, because it lets you assess single-mode distributions, which are easier to understand and evaluate than multimodal distributions.

## Is it symmetric or skewed?

A symmetrical distribution is symmetrical about its mean. A skewed distribution is asymmetric. A positively skewed distribution has a thicker upper tail than lower tail; and vice versa, for a negatively skewed distribution.

Probability distributions in environmental risk analysis are often positively skewed. Quantities such as source terms, transfer factors, and dose-response factors, are typically bounded below by zero. There is more uncertainty about how large they might be than about how small they might be.

## Should you use a standard or a custom distribution?

The next question is whether to use a standard parametric distribution — for example, normal, lognormal, or beta — or a custom distribution, where the assessor specifies points on the cumulative probability or density function.

Considering the physical processes that generate the uncertainty in the quantity might suggest that a particular standard distribution is appropriate. More often, however, there is no obvious standard distribution to apply.

It is generally much faster to assess a standard distribution than a full custom distribution, because standard distributions have fewer parameters, typically from two to four. You should usually start by assigning a simple standard distribution to each uncertain quantity using a quick judgment based on a brief perusal of the literature or telephone conversation with a knowledgeable person. You should assess a custom distribution only for those few uncertain inputs that turn out to be critical to the results. Therefore, it is important to be able to select an appropriate standard distribution quickly for each quantity.