# 5.2: Theoretical Models for the Distribution of Data

- Page ID
- 219088

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

There are four important types of distributions that we will consider in this chapter: the uniform distribution, the binomial distribution, the Poisson distribution, and the normal, or Gaussian, distribution. In Chapter 3 and Chapter 4 we used the analysis of bags of M&Ms to explore ways to visualize data and to summarize data. Here we will use the same data set to explore the distribution of data.

## Uniform Distribution

In a uniform distribution, all outcomes are equally probable. Suppose the population of M&Ms has a uniform distribution. If this is the case, then, with six colors, we expect each color to appear with a probability of 1/6 or 16.7%. Figure \(\PageIndex{1}\) shows a comparison of the theoretical results if we draw 1699 M&Ms—the total number of M&Ms in our sample of 30 bags—from a population with a uniform distribution (on the left) to the actual distribution of the 1699 M&Ms in our sample (on the right). It seems unlikely that the population of M&Ms has a uniform distribution of colors!

## Binomial Distribution

A binomial distribution shows the probability of obtaining a particular result in a fixed number of trials, where the odds of that result happening in a single trial are known. Mathematically, a binomial distribution is defined by the equation

\[P(X, N) = \frac {N!} {X! (N - X)!} \times p^{X} \times (1 - p)^{N - X} \nonumber\]

where *P*(*X*,*N*) is the probability that the event happens *X *times in *N* trials, and where *p* is the probability that the event happens in a single trial. The binomial distribution has a theoretical mean, \(\mu\), and a theoretical variance, \(\sigma^2\), of

\[\mu = Np \quad \quad \quad \sigma^2 = Np(1 - p) \nonumber\]

Figure \(\PageIndex{2}\) compares the expected binomial distribution for drawing 0, 1, 2, 3, 4, or 5 yellow M&Ms in the first five M&Ms—assuming that the probability of drawing a yellow M&M is 435/1699, the ratio of the number of yellow M&Ms and the total number of M&Ms—to the actual distribution of results. The similarity between the theoretical and the actual results seems evident; in Chapter 6 we will consider ways to test this claim.

## Poisson Distribution

The binomial distribution is useful if we wish to model the probability of finding a fixed number of yellow M&Ms in a sample of M&Ms of fixed size—such as the first five M&Ms that we draw from a bag—but not the probability of finding a fixed number of yellow M&Ms in a single bag because there is some variability in the total number of M&Ms per bag.

A Poisson distribution gives the probability that a given number of events will occur in a fixed interval in time or space if the event has a known average rate and if each new event is independent of the preceding event. Mathematically a Poisson distribution is defined by the equation

\[P(X, \lambda) = \frac {e^{-\lambda} \lambda^X} {X !} \nonumber\]

where \(P(X, \lambda)\) is the probability that an event happens *X* times given the event’s average rate, \(\lambda\). The Poisson distribution has a theoretical mean, \(\mu\), and a theoretical variance, \(\sigma^2\), that are each equal to \(\lambda\).

The bar plot in Figure \(\PageIndex{3}\) shows the actual distribution of green M&Ms in 35 small bags of M&Ms (as reported by M. A. Xu-Friedman “Illustrating concepts of quantal analysis with an intuitive classroom model,” Adv. Physiol. Educ. **2013**, *37*, 112–116). Superimposed on the bar plot is the theoretical Poisson distribution based on their reported average rate of 3.4 green M&Ms per bag. The similarity between the theoretical and the actual results seems evident; in Chapter 6 we will consider ways to test this claim.

## Normal Distribution

A uniform distribution, a binomial distribution, and a Poisson distribution predict the probability of a discrete event, such as the probability of finding exactly two green M&Ms in the next bag of M&Ms that we open. Not all of the data we collect is discrete. The net weights of bags of M&Ms is an example of continuous data as the mass of an individual bag is not restricted to a discrete set of allowed values. In many cases we can model continuous data using a normal (or Gaussian) distribution, which gives the probability of obtaining a particular outcome, *P*(*x*), from a population with a known mean, \(\mu\), and a known variance, \(\sigma^2\). Mathematically a normal distribution is defined by the equation

\[P(x) = \frac {1} {\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2/(2 \sigma^2)} \nonumber\]

Figure \(\PageIndex{4}\) shows the expected normal distribution for the net weights of our sample of 30 bags of M&Ms if we assume that their mean, \(\overline{X}\), of 48.98 g and standard deviation, *s*, of 1.433 g are good predictors of the population’s mean, \(\mu\), and standard deviation, \(\sigma\). Given the small sample of 30 bags, the agreement between the model and the data seems reasonable.