The “Elementary Statistics” Book by Larson and Farber

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Table of Contents

Descriptive Statistics
Probability
Discrete Random Variables
References

It is hard to disagree that statistics play an essential role in modern people’s everyday lives. Even ordinary persons may need to use statistics for some activities. Researchers need numbers and data to prove their point and provide evidence for some conclusions they make in their papers. While working with statistics may seem challenging, a book titled Elementary Statistics: Picturing the World by Larson and Farber (2012) explains its concepts in detail and offer numerous examples. What is more, an open-access resource developed by subject matter experts and experienced teachers also provides many insights and important information (Illowsky et al., 2022). This summer, we studied several chapters from these textbooks, and the following paragraphs contain their summary.

Descriptive Statistics

The primary objective of this chapter is to focus on specific ways to describe and organize data sets. The chapter provides a detailed overview of variations, averages, and trends, making the data easier to understand and work with. As noticed by the authors, during a data set organization and description, it is crucial to look for “its center, its variability (or spread), and its shape” (Larson & Farber, 2012, p. 38). One way to organize data sets is to form a frequency distribution, which is “a table that shows classes or intervals of data entries with a count of the number of entries in each class” (Larson & Farber, 2012, p. 38). The chapter teaches certain steps to create a frequency distribution and then differentiates between its graphs: frequency histogram, polygon, relative frequency histogram, and ogive.

Further, a newer way to display quantitative data is a stem-and-leaf plot. It is similar to histograms but “has the advantage that the graph still contains the original data values” and also “provides an easy way to sort data” (Larson & Farber, 2012, p. 53). Other graphing methods explained in the chapter are pie charts, Pareto charts, scatter plots, and others. The following two sections of Chapter 2 discuss measures of central tendency, which are values representing a central data set entry. Overall, the mode, the median, and the mean, are the most commonly used measures (Larson & Farber, 2012, p. 65). In section 2.4, it is discussed that the Empirical Rule applies only to bell-shaped distributions, while Chebychev’s Theorem applies to all distributions.

Probability

Further, Chapter 3 focuses on probability and covers the following topics: basic probability and counting concepts, conditional probability, the Multiplication Rule and the Addition Rule, and some other topics related to counting and probability. Overall, many people encounter probabilities, or likelihoods, of specific events on a daily basis, and many decisions are based on this concept. When an analyzed event is simple and contains only one outcome, a probability experiment may be conducted. When there are more possible outcomes, it is better to use the Fundamental Counting Principle (Larson & Farber, 2012, p. 130). The textbook’s authors also explore the three types of probability, namely, classical, empirical, and subjective (Larson & Farber, 2012). Section 3.2 discusses conditional probabilities, which are related to cases when one even’s the likelihood is explored considering that another event has already happened.

Then, dependent and independent events are mentioned, and to determine the likelihood of two events occurring in sequence, the Multiplication Rule can be used. Interestingly, while one event may or may not cause the other one, it is also possible that two events are mutually exclusive and cannot happen simultaneously (Larson & Farber, 2012, p. 156). The Addition Rule can be utilized to define the probability of mutually exclusive or other events. In the section that contains some additional concepts, the authors mention that several other techniques can be used to count the number of ways an event can occur. They are related to permutations and combinations, which are concepts essential to be considered when utilizing the Multiplication Rule.

Discrete Random Variables

Chapter 4 of the Introductory Statistics textbook outlines how to understand and recognize discrete probability distribution functions, interpret and calculate expected values, and recognize and apply the binomial, Poisson, geometric, and hypergeometric probability distributions. Overall, random variables describe the results of statistical experiments in words. They are divided into two groups: discrete and continuous ones, having countable and uncountable values, respectively (Illowsky et al., 2022). The two primary characteristics of discrete probability distribution functions are that “each probability is between zero and one, inclusive,” and “the sum of the probabilities is one” (Illowsky et al., 2022, para. 4.1). Next, the concept of an expected value is outlined: Illowsky et al. (2022) define this concept as an average expected after a long-term experiment.

Several sections of the paper are devoted to differentiating between the four probability distributions. First, a binomial experiment has a fixed number of trials, two potential outcomes, and independent trials that are repeated using identical conditions. Second, a geometric experiment has “repeating independent Bernoulli trials until a success is obtained,” an unlimited number of trials, and stable probabilities of success and failure (Illowsky et al., 2022, para. 4.4). Next, a hypergeometric experiment’s five characteristics are the following: samples are taken from two groups without replacement, there is a concern with the first group, each pick is not independent, and Bernoulli trials are not included. Finally, a Poisson probability distribution is relevant when there is a fixed space or period of time, the success likelihood is small, and the number of trials is large.

References

Illowsky, B., Dean, S., Birmajer, D., Blount, B., Boyd, S., Einsohn, M., Helmreich, J., Kenyon, L., Lee, S., & Taub, J. (2022). Introductory statistics.

Larson, R., & Farber, B. (2012). Elementary statistics: Picturing the world (5^th ed.). Pearson.

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The “Elementary Statistics” Book by Larson and Farber

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