Summarizing Data for Two Variables Using Graphical Displays

In the previous section we showed how a crosstabulation can be used to summarize the data for two variables and help reveal the relationship between the variables. In most cases, a graphical display is more useful for recognizing patterns and trends in the data. In this section, we introduce a variety of graphical displays

Data Visualization: Best Practices in Creating Effective Graphical Displays

Data visualization is a term used to describe the use of graphical displays to summarize and present information about a data set. The goal of data visualization is to communicate as effectively and clearly as possible, the key information about the data. In this section, we provide guidelines for creating an effective graphical display,

Measures of Location

1. Mean Perhaps the most important measure of location is the mean, or average value, for a variable. The mean provides a measure of central location for the data. If the data are for a sample, the mean is denoted by x; if the data are for a population, the mean is denoted by

Measures of Variability

In addition to measures of location, it is often desirable to consider measures of variability, or dispersion. For example, suppose that you are a purchasing agent for a large manufacturing firm and that you regularly place orders with two different suppliers. After several months of operation, you find that the mean number of days

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Measures of Distribution Shape, Relative Location, and Detecting Outliers

We have described several measures of location and variability for data. In addition, it is often important to have a measure of the shape of a distribution. In Chapter 2 we noted that a histogram provides a graphical display showing the shape of a distribution. An important numerical measure of the shape of a

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Five-Number Summaries and Boxplots

Summary statistics and easy-to-draw graphs based on summary statistics can be used to quickly summarize large quantities of data. In this section we show how five-number sum­maries and boxplots can be developed to identify several characteristics of a data set. 1. Five-Number Summary In a five-number summary, five numbers are used to summarize the

Measures of Association Between Two Variables

Thus far we have examined numerical methods used to summarize the data for one vari­able at a time. Often a manager or decision maker is interested in the relationship between two variables. In this section we present covariance and correlation as descriptive measures of the relationship between two variables. We begin by reconsidering the

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Data Dashboards: Adding Numerical Measures to Improve Effectiveness

In Section 2.5 we provided an introduction to data visualization, a term used to describe the use of graphical displays to summarize and present information about a data set. The goal of data visualization is to communicate key information about the data as effectively and clearly as possible. One of the most widely used

Random Experiments, Counting Rules, and Assigning Probabilities

In discussing probability, we deal with experiments that have the following characteristics: The experimental outcomes are well defined, and in many cases can even be listed prior to conducting the experiment. On any single repetition or trial of the experiment, one and only one of the possible experimental outcomes will occur. The experimental outcome

Events and Their Probabilities

In the introduction to this chapter we used the term event much as it would be used in everyday language. Then, in Section 4.1 we introduced the concept of an experiment and its associated experimental outcomes or sample points. Sample points and events provide the foundation for the study of probability. As a result,

Some Basic Relationships of Probability

Given an event A, the complement of A is defined to be the event consisting of all sample points that are not in A. The complement of A is denoted by Ac. Figure 4.4 is a diagram, known as a Venn diagram, which illustrates the concept of a complement. The rectangular area represents the

Conditional Probability

Often, the probability of an event is influenced by whether a related event already occurred. Suppose we have an event A with probability P(A). If we obtain new informa­tion and learn that a related event, denoted by B, already occurred, we will want to take advantage of this information by calculating a new probability

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Bayes’ Theorem

In the discussion of conditional probability, we indicated that revising probabilities when new information is obtained is an important phase of probability analysis. Often, we begin the analysis with initial or prior probability estimates for specific events of interest. Then, from sources such as a sample, a special report, or a product test, we

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Random Variables

A random variable provides a means for describing experimental outcomes using numer- and its associated experimen- ical values. Random variables must assume numerical values. In effect, a random variable associates a numerical value with each possible experimental outcome. The particular numerical value of the random variable depends on the outcome of the experiment. A

Developing Discrete Probability Distributions

The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. For a discrete random variable x, a probability function, denoted by f(x), provides the probability for each value of the random variable. The classical, subjective, and relative frequency methods of assign­ing probabilities can be used

Expected Value and Variance

The expected value, or mean, of a random variable is a measure of the central location for the random variable. The formula for the expected value of a discrete random variable x follows. Both the notations E(x) and m are used to denote the expected value of a random variable. Equation (5.4) shows that

Bivariate Distributions, Covariance, and Financial Portfolios

A probability distribution involving two random variables is called a bivariate probability distribution. In discussing bivariate probability distributions, it is useful to think of a bivariate experiment. Each outcome for a bivariate experiment consists of two values, one for each random variable. For example, consider the bivariate experiment of rolling a pair of dice.

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Binomial Probability Distribution

The binomial probability distribution is a discrete probability distribution that has many applications. It is associated with a multiple-step experiment that we call the binomial experiment. 1. A Binomial Experiment A binomial experiment exhibits the following four properties. PROPERTIES OF A BINOMIAL EXPERIMENT The experiment consists of a sequence of n identical trials. Two

Poisson Probability Distribution

In this section we consider a discrete random variable that is often useful in estimating the number of occurrences over a specified interval of time or space. For example, the random variable of interest might be the number of arrivals at a car wash in one hour, the number of repairs needed in 10

Hypergeometric Probability Distribution

The hypergeometric probability distribution is closely related to the binomial distribution. The two probability distributions differ in two key ways. With the hypergeometric distribu­tion, the trials are not independent; and the probability of success changes from trial to trial. In the usual notation for the hypergeometric distribution, r denotes the number of elements in