# Distributions

*… a core concept used in Quantitative Methods and Atlas104*

### Concept description

OnlineStatBook (link on right) distinguishes two categories of statistical distributions:

**frequency distribution**: A distribution of empirical data (as in Figure 1 to the right)**probability distribution**: A distribution of the probabilities of each possible outcome (as in Figure 2 to the right)

Investopedia (reference below) defines **probability distribution** as “a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.”

OnlineStatBook uses the example of the distribution of M&M candies in a bag of 55 M&M’s:

“The M&M’s were in six different colors. A quick count showed that there were 55 M&M’s: 17 brown, 18 red, 7 yellow, 7 green, 2 blue, and 4 orange [see Figure 1]… The distribution shown in Figure 1 concerns just my one bag of M&M’s. You might be wondering about the distribution of colors for all M&M’s. The manufacturer of M&M’s provides some information about this matter, but they do not tell us exactly how many M&M’s of each color they have ever produced. Instead, they report proportions rather than frequencies. Figure 2 shows these proportions. Since every M&M is one of the six familiar colors, the six proportions shown in the figure add to one. We call Figure 2 a probability distribution because if you choose an M&M at random, the probability of getting, say, a brown M&M is equal to the proportion of M&M’s that are brown (0.30).

“Notice that the distributions in Figures 1 and 2 are not identical. Figure 1 portrays the distribution in a sample of 55 M&M’s. Figure 2 shows the proportions for all M&M’s. Chance factors involving the machines used by the manufacturer introduce random variation into the different bags produced. Some bags will have a distribution of colors that is close to Figure 2; others will be further away.”

###### Types of probability distributions

Examples of probability distributions include Normal Distributions and Power Law and Pareto Distributions.

Investopedia notes that:

“There are many different classifications of probability distributions. Some of them include the **normal distribution**, **chi square distribution**, **binomial distribution**, and **Poisson distribution**. The different probability distributions serve different purposes. The binomial distribution, for example, evaluates the probability of an event occurring several times over a given number of trials and given the event’s probability in each trial. The usual example would use a fair coin and figuring the probability of that coin coming up heads in ten straight flips.

“The most commonly used distribution is the normal distribution and it is used frequently in finance, investing, science, and engineering. The normal distribution is fully characterized by its mean and standard deviation, meaning the distribution is not skewed and does exhibit kurtosis. This makes the distribution symmetric and it is depicted as a bell-shaped curve when plotted.”

**Atlas topic, subject, and course**

The Study of Quantitative Methods (core topic) in Quantitative Methods and Atlas104 Quantitative Methods.

###### Sources

David M. Lane and Heidi Ziemer, OnlineStatBook, at http://onlinestatbook.com/2/introduction/distributions.html,

Investopedia, Probability Distribution, at http://www.investopedia.com/terms/p/probabilitydistribution.asp, accessed 10 June 2017.

**Page created by:** Ian Clark, last modified 10 June 2017.

**Image:** Created from images on OnlineStatBook, at http://onlinestatbook.com/2/introduction/distributions.html, accessed 10 June 2017.