2.2: Introduction to Counting
In the last section, we saw how to use the theorems we developed to answer some questions concerning the probability of an event. In this section (and in the following), we will learn how to count. After doing so, we will then connect counting to probability. We first start off with the following definition:
Combinatorial analysis refers to the mathematical theory of counting.
For many students, the topic of counting is often met with much dismay. Our goal here is to make counting as intuitive as possible. In order to do so, we will take a motivated approach to mostly every topic.
We first begin with a simple question that many students have heard of before. (However, if you did not hear of this question, then still try to answer it!).
Motivating Question: A man has 3 different shirts and 4 different pants. How many outfits does the man have? (For us, an outfit is a specification of the shirt and pants).
Most students correctly answer that the man has a total of 12 different shirts which is indeed correct. But if you are unsure as to why the answer is 12 or if you obtained a different answer, then let us explain where the 12 comes from. There are a few ways to prove your answer is correct. You may generate a chart or a tree diagram. The method of using a chart is discussed below:
A man has 3 different shirts and 4 different pants. How many outfits does the man have?
Solution
Let us call the three shirts \( S_1, S_2 \) and \( S_3 \) and the four pants \(P_1, P_2, P_3 \) and \( P_4 \). We may think of an outfit as being described by an order pair \( (x,y) \) where \(x \) represents the shirt and \( y \) represents the pants. To keep track of all the outfits, we make a table as shown below:
| \( S_1 \) | \(S_2 \) | \( S_3 \) | |
|---|---|---|---|
| \(P_1 \) | \( (S_1, P_1)\) | \((S_2, P_1)\) | \((S_3, P_1)\) |
| \( P_2 \) | \((S_1, P_2)\) | \((S_2, P_2)\) | \((S_3, P_2)\) |
| \( P_3 \) | \((S_1, P_3)\) | \((S_2, P_3)\) | \((S_3, P_3)\) |
| \( P_4 \) | \((S_1, P_4)\) | \((S_2, P_4)\) | \((S_3, P_4)\) |
We now must count the number of ordered pairs which appears in our chart. We see that there are \( 3 \times 4 = 12 \) entries, and hence 12 total possibilities.
For us, we would like to have a theorem which justifies the above multiplication. That theorem is what we will call The Basic Principle of Counting.
Suppose an experiment satisfies the following two criteria:
- The experiment is performed in two distinct parts or steps.
- The first part of the experiment has \( m \) possible outcomes and regardless of which of these outcomes may occur, the second part of the experiment has \( n \) possible outcomes.
Then the sample space, \( S \), has a total of \( m \times n \) possible outcomes.
Proof: We use a similar technique as presented in the above solution. Let the \( m \) possible outcomes in the first part be denoted by $\(s_1, s_2, \ldots, s_m \) and let the \( n \) possible outcomes in the second part be denoted by \( w_1, w_2, \ldots, w_n \). We now generate a chart to keep track of all possible outcomes:
| \( s_1 \) | \(s_2 \) | \( \ldots \) | \( s_m \) | |
|---|---|---|---|---|
| \(w_1 \) | \( (s_1, w_1)\) | \((s_2, w_1)\) | \( \ldots \) | \((s_m, w_1)\) |
| \( w_2 \) | \((s_1, w_2)\) | \((s_2, w_2)\) | \( \ldots \) | \((s_m, w_2)\) |
| \( \ldots \) | \((s_1, w_3)\) | \((s_2, w_3)\) | \( \ldots \) | \((s_m, w_3)\) |
| \( w_n \) | \((s_1, w_n)\) | \((s_2, w_n)\) | \( \ldots \) | \((s_m, w_n)\) |
Counting the number of ordered pairs, we see that there are \( \underbrace{m+m+\ldots + m}_{n ~ \text{times}} = mn \) possible outcomes.
We now have a more concise way of answering Example 1.
A man has 3 different shirts and 4 different pants. How many outfits does the man have?
Solution
We can think of choosing an outfit as a two stage experiment.  Let us proceed in steps:
Step 1) We first select a shirt. There are 3 possible choices.
Step 2) We now select a pants. Regardless of the shirt selected in step 1, there are 4 possible choices.
Hence, by The Basic Principle of Counting, there are \( 3 \times 4 = 12 \) possible outcomes.
Suppose there are four different routes from Ozone Park to Kew Gardens and six different routes from Kew Gardens to Flushing. How many different routes are there from Ozone Park to Flushing which passes through Kew Gardens?
Solution
We must generate a route/path from Ozone Park to Flushing which passes through Kew Gardens. To do this, we can imagine the route/path as being constructed in two steps.
Step 1) We first select a path from Ozone Park to Kew Gardens. There are 4 possible choices.
Step 2) We now select a path from Kew Gardens to Flushing. Regardless of the path selected in step 1, there are 6 possible choices.
Hence, by The Basic Principle of Counting, there are \( 4 \times 6 = 24 \) possible paths.
There is no need to restrict ourselves to studying experiments where the experiment can be thought of as a two step process. Instead, we may wish to study experiments that have three steps or four steps or in general \( k \) steps. For instance, if we now consider an outfit to be a specification of shirt, pants, and shoes worn, then we can ask the following question: A man has 3 different shirts, 4 different pants, and 2 different shoes. How many outfits does the man have?
In situations like these, we must generalize The Basic Principle of Counting. Doing so yields The Generalized Principle of Counting.
Suppose an experiment satisfies the following two criteria:
- The experiment is performed in \(k \) distinct parts or steps.
- The first part of the experiment has \( n_1 \) possible outcomes and regardless of which of these outcomes may occur, the second part of the experiment has \( n_2 \) possible outcomes and regardless of which of these outcomes may occur, the third part of the experiment has \{ n_3 \) possible outcomes, and \( \ldots \), the \( (k-1)^{st} \) part of the experiment has \( n_{k-1} \) possible outcomes and regardless of which of these outcomes may occur, the \( (k)^{th} \) part of the experiment has \( n_{k} \) possible outcomes.
Then the sample space, \( S \), has a total of \( n_1 \times n_2 \times n_3 \times \ldots \times n_k \) possible outcomes.
A man has 3 different shirts, 4 different pants, and 2 different shoes. How many outfits does the man have?
Solution
We can now think of choosing an outfit as a three stage experiment. We first must select a shirt, then a pants, and then select the shoes. Let us proceed in these steps:
Step 1) We first select a shirt. There are 3 possible choices.
Step 2) We now select a pants. Regardless of the shirt selected in step 1, there are 4 possible choices.
Step 3) We now select the shoes. Regardless of the shirt and pants selected, there are 2 possible choices.
Hence, by The Generalized Principle of Counting, there are \( 3 \times 4 \times 2= 24 \) possible outcomes.
Although this completes the above example, it might be useful to note how The Generalized Principle of Counting and be thought of as an iterated version of The Basic Principle of Counting. To show this, allow us to consider the three stage experiment above as two separate two-stage experiments. In the first stage, we select a shirt and pants. Then in the second stage, we select a pair of shoes. Similar to what was done before, allow us to denote the three shirts by \( S_1, S_2 \) and \( S_3 \), the four pants by \(P_1, P_2, P_3 \) and \( P_4 \), and the two shoes by \( J_1 \) and \(J_2 \). Doing so gives us the following chart:
| \( J_1 \) | \( J_2 \) | |
| \( (S_1, P_1)\) | \( \big((S_1, P_1) , J_1 \big) \) | \( \big((S_1, P_1) , J_2 \big) \) |
| \( (S_1, P_2)\) | \( \big((S_1, P_2) , J_1 \big) \) | \( \big((S_1, P_2) , J_2 \big) \) |
| \( (S_1, P_3)\) | \( \big((S_1, P_3) , J_1 \big) \) | \( \big((S_1, P_3) , J_2 \big) \) |
| \( (S_1, P_4)\) | \( \big((S_1, P_4) , J_1 \big) \) | \( \big((S_1, P_4) , J_2 \big) \) |
| \( (S_2, P_1)\) | \( \big( (S_2, P_1) , J_1 \big) \) | \( \big((S_2, P_1) , J_2 \big) \) |
| \( (S_2, P_2)\) | \( \big((S_2, P_2) , J_1 \big) \) | \( \big((S_2, P_2) , J_2 \big) \) |
| \( (S_2, P_3)\) | \( \big((S_2, P_3) , J_1 \big) \) | \( \big((S_2, P_3) , J_2 \big) \) |
| \( (S_2, P_4)\) | \( \big((S_2, P_4) , J_1 \big) \) | \( \big((S_2, P_4) , J_2 \big) \) |
| \( (S_3, P_1)\) | \( \big((S_3, P_1) , J_1 \big) \) | \( \big((S_3, P_1) , J_2 \big) \) |
| \( (S_3, P_2)\) | \( \big((S_3, P_2) , J_1 \big) \) | \( \big((S_3, P_2) , J_2 \big) \) |
| \( (S_3, P_3)\) | \( \big((S_3, P_3) , J_1 \big) \) | \( \big((S_3, P_3) , J_2 \big) \) |
| \( (S_3, P_4)\) | \( \big( (S_3, P_4) , J_1 \big) \) | \( \big((S_3, P_4) , J_2 \big) \) |
How many diferent 7-place license plates are possible if the first three places are to be occupied with letters and the last four places are to be occupied with the digits where each digit ranges from 0-9 inclusively? You may assume that repetition of letters and digits are allowed.
Solution
A license plate may be thought of as a seven stage experiment. We proceed in steps:
- Select a letter for position 1: 26 choices
- Select a letter for position 2: 26 choices
- Select a letter for position 3: 26 choices
- Select a digit for position 4: 10 choices
- Select a digit for position 5: 10 choices
- Select a digit for position 6: 10 choices
- Select a digit for position 7: 10 choices
By The Generalized Principle of Counting, there are \( 26^3 \times 10^4 = 175,760,000 \) different license plates.
Allow us to modify the above example by considering how the solution would change if repetition was not allowed.
How many diferent 7-place license plates are possible if the first three places are to be occupied with letters and the last four places are to be occupied with the digits where each digit ranges from 0-9 inclusively?
Solution
A license plate may be thought of as a seven stage experiment. We proceed in steps:
- Select a letter for position 1: 26 choices
- Select a letter for position 2: 25 choices
- Select a letter for position 3: 24 choices
- Select a digit for position 4: 10 choices
- Select a digit for position 5: 9 choices
- Select a digit for position 6: 8 choices
- Select a digit for position 7: 7 choices
By The Generalized Principle of Counting, there are \( 26^3 \times 10^4 = 78,624,000 \) different license plates.
At this point in time, we may feel that the above framework seems like overkill for such a simple problem and it appears to be unnecessary. However, our counting questions will become more complicated and the process will not be as short as the above problem. For those types of complex problems, we will need the above framework and so we will continue to stick to this framework to gain familiarity with it. Moreover, our focus at this point is not the end answer. Instead, our objective is to write a clear solution.
Let us suppose we have the following simplified model of Chipotle. At Chipotle, we must select:
- One of five possible entrees (a burrito, burrito bowl, Quesadilla, salad, or taco).
- One of seven different proteins (chicken al pastor, chicken, steak, barbacoa, carnitas, sofritas, veggie).
- One of three possible rices (white rice, brown rice, no rice).
- One of three possible beans (black beans, pinto beans, no beans).
- Any amount of ten different toppings (guacamole, fresh tomato salsa, roasted chili-corn salsa, tomatillo green chili salsa, tomatillo-red chili salsa, sour cream, fajita veggies, cheese, romaine lettuce, queso blanco).
How many different dishes are available to a customer?
Critique This Solution:
By The Generalized Principle of Counting, there are \( 5 \times 7 \times 3 \times 3 \times 10 = 3150 \) possible outcomes. Is this solution correct? What do you think?
- Answer
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The above solution is incorrect! Do not fall into the trap that The Basic Principle of Counting states that we simply multiple a bunch of numbers. Allow us to use our framework to fully answer this question:
- Select one entree: 5 choices
- Select one protein: 7 choices.
- Select one option for rice: 3 choices
- Select one option for beans: 3 choices
- Select one option for guacamole: 2 choices
- Select one option for fresh tomato salsa: 2 choices
- Select one option for roasted chili-corn salsa: 2 choices
- Select one option for tomatillo green chili salsa: 2 choices
- Select one option for tomatillo red salsa: 2 choices
- Select one option for sour cream: 2 choices
- Select one option for fajita veggies: 2 choices
- Select one option for cheese: 2 choices
- Select one option for romaine lettuce: 2 choices
- Select one option for queso blanco: 2 choices
By The Generalized Principle of Counting, there are \( 5 \times 7 \times 3 \times 3 \times 2^{10} = 322,560\) possible outcomes.
The above example illustrates that we should be careful while applying The Generalized Principle of Counting. Specifically, proceeding in steps can help us minimize mistakes.


9.1: Reading Strategies
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Introduction
Picture yourself sitting at a desk, a book in your hands.   You blink. You realize your eyes have been scanning the page for 10 minutes, and you can’t remember a single thing you have read.
Or picture this: You’ve had a hard day. You are up at 6:00 a.m. to get the kids ready for school.  You pick up the kids and then shop for dinner. Dinner is late, of course, and the kids are grumpy.
Finally, you get to your books at 8:00 p.m. You begin a reading assignment on something called the equity method of accounting for common stock investments . “I am preparing for the future,” you tell yourself as you plod through two paragraphs and begin the third. Suddenly, everything in the room looks different. Your head is resting on your elbow, which is resting on the equity method of accounting. The clock reads 11:00 p.m. Say goodbye to three hours.
Contrast this scenario with the image of an active reader, who exhibits the following behaviors:
- Stays alert, poses questions about what he reads, and searches for the answers
- Recognizes levels of information within the text, separating the main points and general principles from supporting details
- Quizzes himself about the material, makes written notes, and lists unanswered questions
- Instantly spots key terms and takes the time to find the definitions of unfamiliar words
- Thinks critically about the ideas in the text and looks for ways to apply them
That sounds like a lot to do. Yet skilled readers routinely accomplish all these things and more— while enjoying the process. Successful students engage actively with reading material. They’re willing to grapple with even the most challenging texts. They wrestle meaning from each page. They fill the margins with handwritten questions. They underline, highlight, annotate, and nearly rewrite some books to make them their own.
Successful students also commit to change their lives on the basis of what they read. Of every chapter, they ask, What’s the point? And what’s the payoff? How can I use this to live my purpose and achieve my goals? These students are just as likely to create to-do lists as to take notes on their reading. And when they’re done with a useful book, successful students share its points with others for continuing conversation. Reading becomes a creative act and a tool for building a community.
One way to experience this kind of success is to approach reading with a system in mind. You can use active reading to avoid mental minivacations and reduce the number of unscheduled naps during study time, even after a hard day. Active reading is a way to decrease difficulty and struggle by increasing energy and skill. Once you learn these strategies, you might actually spend less time on your reading but get more out of it.
Active Reading Phase 1—Before You Read
Active reading is a three-phase technique you can use to extract the ideas and information you want. They include strategies to use before, while, and after you read. Using these strategies will help you engage with the material more effectively and therefore remember more of what you read.
Phase 1 happens before you read and includes the following steps:
Step 1: Preview. Before you start reading, preview the entire assignment. You don’t have to memorize what you preview to get value from this step.
- Look over the table of contents and flip through the text page by page, if you are starting a new book. If you’re going to read one chapter, flip through the pages of that chapter.
- Read all chapter headings and subheadings. Like the headlines in a newspaper, these are usually printed in large, bold type. Often, headings are brief summaries themselves.
- Keep an eye out for summary statements. If the assignment is long or complex, read the summary first. Many textbooks have summaries in the introduction or at the end of each chapter.
- Seek out familiar concepts, facts, or ideas when previewing. These items can help increase comprehension by linking new information to previously learned material. Take a few moments to reflect on what you already know about the subject—even if you think you know nothing. This technique prepares your brain to accept new information.
- Look for ideas that spark your imagination or curiosity. Inspect drawings, diagrams, charts, tables, graphs, and photographs.
- Imagine what kinds of questions will show up on a test. Previewing helps to clarify your purpose for reading. Ask yourself what you will do with this material and how it can relate to your long-term goals. Keep your preview short. If the entire reading assignment will take less than an hour, your preview might take five minutes. Previewing is also a way to get yourself started when an assignment looks too big to handle. It is an easy way to step into the material.
Step 2: Outline. With complex material, you should take the time to understand the structure of what you are about to read. Outlining actively organizes your thoughts about the assignment and can help make complex information easier to understand.
- Spend some time studying the chapter outline in your textbook, if an outline is provided.
- Sketch a brief outline in the margin of the book or at the beginning of your notes on a separate sheet of paper, if an outline is not provided. Later, as you read and take notes, you can add to your outline. Headings in the text can serve as major and minor entries in your outline.
The amount of time you spend on this outlining step will vary.  For other assignments (fiction and poetry, for example), you can skip this step altogether.
Step 3: Question. Before you begin a careful reading, determine what you want from the assignment.
- Write down a list of questions, including any questions that resulted from your preview of the materials.
- Turn chapter headings and subheadings into questions. For example, if a heading is “Transference and Suggestion,” you can ask yourself, What are transference and suggestion? How does transference relate to suggestion?
- Make up a quiz as if you were teaching this subject to your classmates.
- Write specific questions about a concept if you do not understand it. The more detailed your questions, the more powerful this technique becomes. You don’t need to answer every question that you ask. The purpose of making up questions is to get your brain involved in the assignment. Take your unanswered questions to class, where they can serve as springboards for class discussion.
Active Reading Phase 2—While You Read, Part 1
Phase 1 of active reading is done before reading, but Phase 2 happens while you read, helping you figure out what you are looking for and setting up some context. This phase includes the following steps:
Step 4: Focus. You have previewed the reading assignment, organized it in your mind or on paper, and formulated questions. Now you are ready to begin reading.
It’s easy to fool yourself about reading. Having an open book in your hand and moving your eyes across a page don’t mean that you are reading effectively. Reading takes mental focus. As you read, be conscious of where you are and what you are doing.
To begin, get in a position to stay focused. If you observe chief executive officers, you’ll find that some of them wear out the front of their chair first. They’re literally on the edge of their seat. Approach your reading assignment in the same way. Sit up. Keep your spine straight. Avoid reading in bed, except for fun.
Avoid marathon reading sessions. Schedule breaks, and set a reasonable goal for the entire session. Then, reward yourself with an enjoyable activity for 10 or 15 minutes every hour or two.
For difficult reading, set more limited goals. Read for a half-hour and then take a break. Most students find that shorter periods of reading distributed throughout the day and week can be more effective than long sessions.
Visualize the material. Form mental pictures of the concepts as they are presented. If you read that a voucher system can help control cash disbursements, picture a voucher handing out dollar bills. Using visual imagery in this way can help deepen your understanding of the text while allowing information to be transferred into your long-term memory.
Read the material out loud, especially if it is complicated. Some of us remember better and understand more quickly when we hear an idea.
Get a feel for the subject. For example, let’s say you are reading about a microorganism—a paramecium—in your biology text. Imagine what it would feel like to run your finger around the long, cigar-shaped body of the organism. Imagine feeling the large fold of its gullet on one side and the tickle of the hairy little cilia as they wiggle in your hand.
In addition, predict how the author will answer your key questions. Then read to find out if your predictions were accurate.
Active Reading Phase 2—While You Read, Part 2
Once you have taken the important step of creating your focus and are aware of what you are looking for while reading, you are ready to use Step 5. During this step in Phase 2 of active reading, you use strategies for marking your text to identify the important elements.
Step 5: Flag answers. As you read, seek out the answers to your questions.  When you do find an answer, flag it so that it stands out on the page.
Deface your books. Flag answers by highlighting, underlining, writing comments, filling in your outline, or marking up pages in any other way that helps you. Indulge yourself as you never could with your grade school books.
Marking up your books offers other benefits. When you read with a highlighter, pen, or pencil in your hand, you involve your kinesthetic senses of touch and motion. Being physical with your books can help build strong neural pathways in your memory. You can mark up a text in many ways. For example:
- Place an asterisk (*) or an exclamation point (!) in the margin next to an especially important sentence or term.
- Circle key terms and words to look up later in a dictionary.
- Write short definitions of key terms in the margin.
- Write a Q in the margin to highlight possible test questions, passages you don’t understand, and questions to ask in class.
- Write personal comments in the margin—points of agreement or disagreement with the author.
- Write mini-indexes in the margin—that is, the numbers of other pages in the book where the same topic is discussed.
- Write summaries in your own words.
- Rewrite chapter titles, headings, and subheadings so that they’re more meaningful to you.
- Draw diagrams, pictures, tables, or maps that translate text into visual terms.
- Number each step in a list or series of related points.
- In the margins, write notes about the relationships between elements in your reading. For example, note connections between an idea and examples of that idea.
- If you infer an answer to a question or come up with another idea of your own, write that down as well.
Avoid marking up a textbook too soon. Wait until you complete a chapter or section to make sure you know the key points and then mark it up. Sometimes, flagging answers after you read each paragraph works best.
Also remember that the purpose of making marks in a text is to call out important concepts or information that you will review later. Flagging key information can save lots of time when you are studying for tests. With this in mind, highlight or underline sparingly—usually less than 10 percent of the text. If you mark up too much on a page, you defeat the purpose: to flag the most important material for review.
Finally, jot down new questions, and note when you don’t find the answers you are looking for. Ask these questions in class, or see your instructor personally. Demand that your textbooks give you what you want—answers.
Active Reading Phase 3—After You Read
At the end of Phase 2, your reading is complete, but to get the most out of what you just read, it’s important to complete the final phase of active reading—Phase 3. This phase happens after you read and includes the following steps:
Step 6: Recite. Talk to yourself about what you’ve read. Or talk to someone else. When you finish a reading assignment, make a speech about it. When you recite, you practice an important aspect of metacognition— synthesis , or combining individual ideas and facts into a meaningful whole.
One way to recite is to look at each underlined point. Note what you marked; then, put the book down and start talking out loud. Explain as much as you can about that particular point. To make this technique more effective, do it in front of a mirror. It might seem silly, but the benefits can be enormous. Reap them at exam time.
A related technique is to stop reading periodically and write a short, free-form summary of what you just read. In one study, this informal “retrieval practice” helped students recall information better than other study techniques did (Karpicke and Blunt 2011).
Classmates are even better than mirrors. Form a group to practice teaching one another what you have read. One of the best ways to learn anything is to teach it to someone else. In addition, talk about your reading whenever you can. Tell friends and family members what you’re learning.
Talking about your reading reinforces a valuable skill—the ability to summarize. To practice this skill, pick one chapter (or one section of one chapter) from any of your textbooks. State the main topic covered in the chapter. Then, state the main points that the author makes about the topic.
Step 7: Review. Plan to do your first complete review within 24 hours of reading the material. Sound the trumpets! This point is critical: A review within 24 hours moves information from your short-term memory to your long-term memory.
Review within one day. If you read it on Wednesday, review it on Thursday. During this review, look over your notes and clear up anything you don’t understand. Recite some of the main points.
This review can be short. You might spend as little as 15 minutes reviewing a difficult two-hour reading assignment. Investing that time now can save you hours later when studying for exams.
Step 8: Review again. This final step can be very short—perhaps only four or five minutes per assignment. Simply go over your notes. Read the highlighted parts of your text. Recite one or two of the more complicated points.
The purpose of these reviews is to keep the neural pathways to the information open and to make them more distinct. That way, the information can be easier to recall. You can accomplish these short reviews anytime, anywhere, if you are prepared.
Sometimes, longer review periods are appropriate. For example, if you found an assignment difficult, consider rereading it. Start over, as if you had never seen the material before. Sometimes, a second reading will provide you with surprising insights.
Decades ago, psychologists identified the primacy-recency effect , which suggests that we most easily remember the first and last items in any presentation (Pineño and Miller 2005). Previewing and reviewing your reading can put this theory to work for you.
Dealing With Challenging Texts
Successful readers monitor their understanding of reading material. They do not see confusion as a mistake or a personal shortcoming. Instead, they take it as a cue to change reading strategies and process ideas at a deeper level.
Read it again. Somehow, students get the idea that reading means opening a book and dutifully slogging through the text—line by line, page by page—moving in a straight line from the first word to the last. Feel free to shake up your routine. Make several passes through tough reading material. During a preview, for example, just scan the text to look for key words and highlighted material. Next, skim the entire chapter or article again, spending a little more time and taking in more than you did during your preview. Finally, read in more depth.
Read it out loud. Make noise. Read a passage out loud several times, each time using a different inflection and emphasizing a different part of the sentence. Be creative. Imagine that you are the author talking.
Use another text.
Find a similar text in the library. Sometimes a concept is easier to understand if it is expressed another way. Children’s books—especially children’s encyclopedias—can provide useful overviews of baffling subjects.
Talk to someone who can help.
Admit when you are stuck. Then, bring questions about reading assignments to classmates and members of your study group. Also, make an appointment with your instructor. Most teachers welcome the opportunity to work individually with students. Be specific about your confusion. Point out the paragraph that you found toughest to understand.
Reading Faster
One way to read faster is to read faster. This idea might sound like double-talk, but it is a serious suggestion. The fact is, you can probably read faster—without any loss in comprehension— simply by making a conscious effort to do so. Your comprehension might even improve. You might try the following suggestions.
Move your eyes faster. When we read, our eyes leap across the page in short bursts called saccades (pronounced “să-käds”).  Our eyes stop like that, too, in pauses called fixations .
Although we experience the illusion of continuously scanning each line, we actually take in groups of words—usually about three at a time. For more than 90 percent of reading time, our eyes are at a dead stop, in those fixations.
One way to decrease saccades is to follow your finger as you read. The faster your finger moves, the faster your eyes move. You can also use a pen, pencil, or 3 × 5 card as a guide.
Your eyes can move faster if they take in more words with each burst—for example, six instead of three. To practice taking in more words between fixations, find a newspaper with narrow columns. Read down one column at a time, and fixate only once per line.
In addition to using the above techniques, simply make a conscious effort to fixate less. You might feel a little uncomfortable at first. That’s normal. Just practice often, for short periods.
Notice and release ineffective habits. Our eyes make regressions; that is, they back up and reread words. You can reduce regressions by paying attention to them. Use the handy 3 × 5 card to cover words and lines that you have just read. You can then note how often you stop and move the card back to reread the text. Don’t be discouraged if you stop often at first. Being aware of it helps you regress less frequently.
Also notice vocalizing. You are more likely to read faster if you don’t read out loud or move your lips. You can also increase your speed if you don’t subvocalize—that is, if you don’t mentally “hear” the words as you read them. To stop doing it, just be aware of it.
When you first attempt to release these habits, choose simpler reading material. That way, you can pay closer attention to your reading technique. Gradually work your way up to more complex material.
Stay flexible. Remember that speed isn’t everything. Skillful readers vary their reading rate according to their purpose and the nature of the material. An advanced text in analytic geometry usually calls for a different reading rate than the Sunday comics.
You also can use different reading rates on the same material. For example, you might first sprint through an assignment for the key words and ideas, and then return to the difficult parts for a slower and more thorough reading.
Another option is to divide a large reading assignment into smaller sections and use different reading strategies for each one. You might choose to read the first and last sections in detail, for example, and skim the middle sections.
As a general guideline, slow down your reading pace for material that’s technical and unfamiliar to you. Speed up for material that’s familiar, staying alert for anything that seems new or significant.
Also remember that reading faster without comprehension can actually increase the amount of time that you study. Balance the desire for speed with the need for understanding what you read.
Finally, remember the first rule of reading fast: Just do it!
Building Your Vocabulary
Having a large vocabulary makes reading more enjoyable and increases the range of materials you can explore. In addition, building your vocabulary gives you more options for self-expression when speaking or writing. With a larger vocabulary, you can think more precisely by making finer distinctions between ideas. And you won’t have to stop to search for words at crucial times, such as a job interview.
Strengthen your vocabulary by looking up unfamiliar terms. A desk dictionary is an easy-to- handle abridged dictionary that you can use many times in the course of a day. In contrast, an unabridged dictionary is large and not made for you to carry around. It provides more complete information about words and definitions not included in your desk dictionary as well as synonyms, usage notes, and word histories. Look for unabridged dictionaries in libraries and bookstores.
You might prefer using one of several online dictionaries, such as Dictionary.com. Another common option is to use a search engine such as Google.com. If you do this, inspect the results carefully. They can vary in quality and be less useful than the definitions you’d find in a good dictionary or thesaurus.
Construct a word stack. When you come across an unfamiliar word, write it down on a 3 × 5 card. Below the word, copy the sentence in which it was used, along with the page number. You can look up each word immediately, or you can accumulate a stack of these cards and look up the words later. Write the definition of each word on the back of the 3 × 5 card, adding the diacritics—marks that tell you how to pronounce it.
To expand your vocabulary and learn the history behind the words, take your stack of cards to an unabridged dictionary. As you find related words in the dictionary, add them to your stack. These cards become a portable study aid that you can review in your spare moments.
Learn—even when your dictionary is across town. When you are listening to a lecture and hear an unusual word or when you are reading on the bus and encounter a word you don’t know, you can still build your word stack. Pull out a 3 × 5 card and write down the word and its sentence. Or make a note of the word on your cell phone. Later, you can look up the definition and write it on the back of the card.
Divide words into parts. Another suggestion for building your vocabulary is to divide an unfamiliar word into syllables and look for familiar parts. This strategy works well if you make it a point to learn common prefixes (beginning syllables) and suffixes (ending syllables).
For example, the suffix - tude usually refers to a condition or state of being. Knowing this makes it easier to conclude that habitude refers to a usual way of doing something and that similitude means being similar or having a quality of resemblance.
Infer the meaning of words from their context. You can often deduce the meaning of an unfamiliar word simply by paying attention to its context—the surrounding words, phrases, sentences, paragraphs, or images. Practice looking for context clues such as these:
- Definitions. A key word might be defined right in the text. Look for phrases such as defined as or in other words .
- Examples. Authors often provide examples to clarify a word meaning. If the word is not explicitly defined, then study the examples. They’re often preceded by the phrases for example, for instance, or such as .
- Lists. When a word is listed in a series, pay attention to the other items in the series. They might define the unfamiliar word through association.
- Comparisons. You might find a new word surrounded by synonyms—words with a similar meaning. Look for synonyms after words such as like and as .
- Contrasts. A writer might juxtapose a word with its antonym. Look for phrases such as on the contrary and on the other hand .
References
Karpicke, Jeffrey D., and Janell R. Blunt. “Retrieval practice produces more learning than elaborative studying with concept
mapping.”
Science.
www.sciencemag.org/content/early/2011/01/19/science.1199327.abstract (accessed January 21, 2011).
Pineño, Oskar and Ralph R. Miller. “Primacy and recency effects in extinction and latent inhibition: A selective review with implications for models of learning.” Behavioural Processes 69, 2 (2005): 223–235.


7.1B: Cranial Bones
- Differentiate the bones of the neurocranium
The neurocranium forms the cranial cavity that surrounds and protects the brain and brainstem. The neurocranium consists of the occipital bone, two temporal bones, two parietal bones, the sphenoid, ethmoid, and frontal bones—all are joined together with sutures.
Evolutionary,it is the expansion of the neurocranium that has facilitated the expansion of the brain and its associated developments.
Neurocranium : A lateral view showing the bones that structure the neurocranium.
Occipital Bone
The occipital bone forms the base of the skull at the rear of the cranium. It articulates with the first vertebra of the spinal cord and also contains the foramen magnum, the large opening of the skill through which the spinal cord passes as it enters the vertebral column. The occipital bone borders the parietal bones through the heavily serrated lambdoidal suture, and also the temporal bones through occipitomastoid suture.
Temporal Bones
The temporal bones are situated at the base and sides of the skull, lateral to the temporal lobes of the brain. The temporal bones consist of four regions the squamous, mastoid, petrous and tympanic regions.
The squamous region is the largest and most superior region. Inferior to the squamous is the mastoid region, and fused between the squamous and mastoid regions is the petrous region. Finally, the small and inferior tympanic region lies anteriorly to the mastoid.
There are two processes that originate from the temporal bone:
- The zygomatic process that projects from the lower squamous region and articulates with the zygomatic bone of the cheek.
- The styloid process projects downwards from the interior of the temporal bone and provides attachment for several muscles associated with the tongue.
The temporal bones have four borders:
- The occipitomastoid suture separates the occipital bone and mastoid portion of temporal bone.
- The squamosal suture separates the parietal bone and squama portion of temporal bone.
- The sphenosquamosal suture separates the sphenoid bone and squama portion of temporal bone.
- The zygomaticotemporal suture separates the zygomatic bone and zygomatic process of temporal bone.
Parietal Bones
The two large parietal bones are connected and make up part of the roof and sides of the human skull. The two bones articulate to form the sagittal suture. In the front, the parietal bones form the coronal suture with the frontal bone, and in the rear, the lambdoid suture is formed by the occipital bone. Finally, the squamosal suture separates the parietal and temporal bones.
Sphenoid Bone
The sphenoid bone is situated in the middle of the skull towards the front and forms the rear of the orbit. It has been described as resembling a butterfly due to its wing-like processes. The sphenoid bone is divided into several parts: the body of the bone, two greater wings, two lesser wings, and the pterygoid processes.
The sphenoid bone is one of the most complex in the body due to its interactions with numerous facial bones, ligaments, and muscles. The body that forms the middle of the sphenoid bone articulates with the ethmoid and occipital bone and forms a key part of the nasal cavity; it also contains the sphenoidal sinuses.
The greater wings form the floor of the middle cranial fossa that houses the frontal lobes and pituitary gland, and also the posterior wall of the orbit. The lesser wings project laterally and form the floor of the anterior cranial fossa and the superior orbital fissure through which several key optical nerves pass.
Ethmoid Bone
The ethmoid bone is a small bone in the skull that separates the nasal cavity from the brain. It is lightweight due to its spongy, air-filled construction and is located at the roof of the nose and between the two orbits.
The ethmoid bone forms the medial wall of the orbit, the roof of the nasal cavity, and due to its central location it articulates with numerous bones of the viscerocranium. Inside the neurocranium it articulates with the frontal and sphenoid bones.
Frontal Bone
The frontal bone forms the front of the skull and is divided into three parts:
- Squamous: This part is large and flat and forms the main region of the forehead.
- Orbital: This part lies inferiorly and forms the superior border of the orbit.
- Nasal: this part is smaller and articulates with the nasal bones and maxilla to contribute to the roof of the nose.
The frontal bone borders two other neurocranial bones—the parietal bones through the coronal sutures and the sphenoid bone through the sphenofrontal suture. It also articulates with the zygomatic and nasal bones and the maxilla.
Key Points
- The eight bones of the neurocranium form major portions of the skull and protect the brain.
- The neurocranium consists of two temporal bones situated to the base and side of the skull, and two parietal bones that make up the roof of the skull.
- A single occipital bone forms the base of the skull, and the frontal bone forms the forehead.
- The sphenoid and ethmoid bones located to the front of the skull form parts of the orbital sockets and nasal cavity; they also support and protect key organs found in the skull.
Key Terms
- neurocranium : The part of the skull that encloses and protects the brain and brain stem.


8.3: A Single Population Mean using the Student t-Distribution
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In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation \(s\) as an estimate for \(\sigma\) and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Goset (1876–1937) of the Guinness brewery in Dublin, Ireland ran into this problem. His experiments with hops and barley produced very few samples. Just replacing \(\sigma\) with \(s\) did not produce accurate results when he tried to calculate a confidence interval. He realized that he could not use a normal distribution for the calculation; he found that the actual distribution depends on the sample size. This problem led him to "discover" what is called the Student's t-distribution. The name comes from the fact that Gosset wrote under the pen name "Student."
Up until the mid-1970s, some statisticians used the normal distribution approximation for large sample sizes and only used the Student's \(t\)-distribution only for sample sizes of at most 30. With graphing calculators and computers, the practice now is to use the Student's t-distribution whenever \(s\) is used as an estimate for \(\sigma\). If you draw a simple random sample of size \(n\) from a population that has an approximately a normal distribution with mean \(\mu\) and unknown population standard deviation \(\sigma\) and calculate the \(t\)-score
\[t = \dfrac{\bar{x} - \mu}{\left(\dfrac{s}{\sqrt{n}}\right)},\]
then the \(t\)-scores follow a Student's t-distribution with \(n – 1\) degrees of freedom. The \(t\)-score has the same interpretation as the z -score. It measures how far \(\bar{x}\) is from its mean \(\mu\). For each sample size \(n\), there is a different Student's t-distribution.
The degrees of freedom, \(n – 1\), come from the calculation of the sample standard deviation \(s\). Previously, we used \(n\) deviations (\(x - \bar{x}\) values) to calculate \(s\). Because the sum of the deviations is zero, we can find the last deviation once we know the other \(n – 1\) deviations. The other \(n – 1\) deviations can change or vary freely. We call the number \(n – 1\) the degrees of freedom (df).
For each sample size \(n\), there is a different Student's t-distribution.
Properties of the Student's \(t\)-Distribution
- The graph for the Student's \(t\)-distribution is similar to the standard normal curve.
- The mean for the Student's \(t\)-distribution is zero and the distribution is symmetric about zero.
- The Student's \(t\)-distribution has more probability in its tails than the standard normal distribution because the spread of the \(t\)-distribution is greater than the spread of the standard normal. So the graph of the Student's \(t\)-distribution will be thicker in the tails and shorter in the center than the graph of the standard normal distribution.
- The exact shape of the Student's \(t\)-distribution depends on the degrees of freedom. As the degrees of freedom increases, the graph of Student's \(t\)-distribution becomes more like the graph of the standard normal distribution.
- The underlying population of individual observations is assumed to be normally distributed with unknown population mean \(\mu\) and unknown population standard deviation \(\sigma\). The size of the underlying population is generally not relevant unless it is very small. If it is bell shaped (normal) then the assumption is met and doesn't need discussion. Random sampling is assumed, but that is a completely separate assumption from normality.
Calculators and computers can easily calculate any Student's \(t\)-probabilities. The TI-83,83+, and 84+ have a tcdf function to find the probability for given values of \(t\). The grammar for the tcdf command is tcdf(lower bound, upper bound, degrees of freedom). However for confidence intervals, we need to use inverse probability to find the value of t when we know the probability.
For the TI-84+ you can use the invT command on the DISTRibution menu. The invT command works similarly to the invnorm. The invT command requires two inputs: invT(area to the left, degrees of freedom) The output is the t-score that corresponds to the area we specified.
The TI-83 and 83+ do not have the invT command. (The TI-89 has an inverse T command.)
A probability table for the Student's \(t\)-distribution can also be used. The table gives \(t\)-scores that correspond to the confidence level (column) and degrees of freedom (row). (The TI-86 does not have an invT program or command, so if you are using that calculator, you need to use a probability table for the Student's \(t\)-Distribution.) When using a \(t\)-table, note that some tables are formatted to show the confidence level in the column headings, while the column headings in some tables may show only corresponding area in one or both tails.
A Student's \(t\)-table gives \(t\)-scores given the degrees of freedom and the right-tailed probability. The table is very limited. Calculators and computers can easily calculate any Student's \(t\) -probabilities.
The notation for the Student's t-distribution (using T as the random variable) is:
- \(T \sim t_{df}\) where \(df = n – 1\).
- For example, if we have a sample of size \(n = 20\) items, then we calculate the degrees of freedom as \(df = n - 1 = 20 - 1 = 19\) and we write the distribution as \(T \sim t_{19}\).
If the population standard deviation is not known , the error bound for a population mean is:
- \(EBM = \left(t_{\frac{\alpha}{2}}\right)\left(\frac{s}{\sqrt{n}}\right)\),
- \(t_{\frac{\alpha}{2}}\) is the \(t\)-score with area to the right equal to \(\frac{\alpha}{2}\),
- use \(df = n – 1\) degrees of freedom, and
- \(s =\) sample standard deviation.
The format for the confidence interval is:
\[(\bar{x} - EBM, \bar{x} + EBM). \label{confint}\]
To calculate the confidence interval directly:
Press STAT.
Arrow over to TESTS.
Arrow down to 8:TInterval and press ENTER (or just press 8).
Example \(\PageIndex{1}\): Acupuncture
Suppose you do a study of acupuncture to determine how effective it is in relieving pain. You measure sensory rates for 15 subjects with the results given. Use the sample data to construct a 95% confidence interval for the mean sensory rate for the population (assumed normal) from which you took the data.
The solution is shown step-by-step and by using the TI-83, 83+, or 84+ calculators.
8.6; 9.4; 7.9; 6.8; 8.3; 7.3; 9.2; 9.6; 8.7; 11.4; 10.3; 5.4; 8.1; 5.5; 6.9
Answer
- The first solution is step-by-step (Solution A).
- The second solution uses the TI-83+ and TI-84 calculators (Solution B).
Solution A
To find the confidence interval, you need the sample mean, \(\bar{x}\), and the \(EBM\).
\(\bar{x} = 8.2267 \)
\(s = 1.6722\) \(n = 15\)
\(df = 15 – 1 = 14 CL so \alpha = 1 – CL = 1 – 0.95 = 0.05\)
\(\frac{\alpha}{2} = 0.025 t_{\frac{\alpha}{2}} = t_{0.025}\)
The area to the right of \(t_{0.025}\) is 0.025, and the area to the left of \(t_{0.025}\) is 1 – 0.025 = 0.975
\(t_{\frac{\alpha}{2}} = t_{0.025} = 2.14\) using invT(.975,14) on the TI-84+ calculator.
\[ \begin{align*} EBM &= \left(t_{\frac{\alpha}{2}}\right)\left(\frac{s}{\sqrt{n}}\right) \\[4pt] &= (2.14)\left(\frac{1.6722}{\sqrt{15}}\right) = 0.924 \end{align*}\]
Now it is just a direct application of Equation \ref{confint}:
\[ \begin{align*} \bar{x} – EBM &= 8.2267 – 0.9240 = 7.3 \\[4pt] \bar{x} + EBM &= 8.2267 + 0.9240 = 9.15 \end{align*}\]
The 95% confidence interval is (7.30, 9.15).
We estimate with 95% confidence that the true population mean sensory rate is between 7.30 and 9.15.
Solution B
Press
STAT
and arrow over to
TESTS
.
Arrow down to
8:TInterval
and press
ENTER
(or you can just press
8
).
Arrow to
Data
and press
ENTER
.
Arrow down to
List
and enter the list name where you put the data.
There should be a 1 after
Freq
.
Arrow down to
C-level
and enter 0.95
Arrow down to
Calculate
and press
ENTER
.
The 95% confidence interval is (7.3006, 9.1527)
When calculating the error bound, a probability table for the Student's t-distribution can also be used to find the value of \(t\). The table gives \(t\)-scores that correspond to the confidence level (column) and degrees of freedom (row); the \(t\)-score is found where the row and column intersect in the table.
Exercise \(\PageIndex{1}\)
You do a study of hypnotherapy to determine how effective it is in increasing the number of hourse of sleep subjects get each night. You measure hours of sleep for 12 subjects with the following results. Construct a 95% confidence interval for the mean number of hours slept for the population (assumed normal) from which you took the data.
8.2; 9.1; 7.7; 8.6; 6.9; 11.2; 10.1; 9.9; 8.9; 9.2; 7.5; 10.5
- Answer
-
(8.1634, 9.8032)
Example \(\PageIndex{2}\): The Human Toxome Project
The Human Toxome Project (HTP) is working to understand the scope of industrial pollution in the human body. Industrial chemicals may enter the body through pollution or as ingredients in consumer products. In October 2008, the scientists at HTP tested cord blood samples for 20 newborn infants in the United States. The cord blood of the "In utero/newborn" group was tested for 430 industrial compounds, pollutants, and other chemicals, including chemicals linked to brain and nervous system toxicity, immune system toxicity, and reproductive toxicity, and fertility problems. There are health concerns about the effects of some chemicals on the brain and nervous system. Table \(\PageIndex{1}\) shows how many of the targeted chemicals were found in each infant’s cord blood.
| 79 | 145 | 147 | 160 | 116 | 100 | 159 | 151 | 156 | 126 |
| 137 | 83 | 156 | 94 | 121 | 144 | 123 | 114 | 139 | 99 |
Use this sample data to construct a 90% confidence interval for the mean number of targeted industrial chemicals to be found in an in infant’s blood.
Solution A
From the sample, you can calculate \(\bar{x} = 127.45\) and \(s = 25.965\). There are 20 infants in the sample, so \(n = 20\), and \(df = 20 – 1 = 19\).
You are asked to calculate a 90% confidence interval: \(CL = 0.90\), so
\[\alpha = 1 – CL = 1 – 0.90 = 0.10 \frac{\alpha}{2} = 0.05, t_{\frac{\alpha}{2}} = t_{0.05}\]
By definition, the area to the right of \(t_{0.05}\) is 0.05 and so the area to the left of \(t_{0.05}\) is \(1 – 0.05 = 0.95\).
Use a table, calculator, or computer to find that \(t_{0.05} = 1.729\).
\(EBM = t_{\frac{\alpha}{2}}\left(\frac{s}{\sqrt{n}}\right) = 1.729\left(\frac{25.965}{\sqrt{20}}\right) \approx 10.038\)
\(\bar{x} – EBM = 127.45 – 10.038 = 117.412\)
\(\bar{x} + EBM = 127.45 + 10.038 = 137.488\)
We estimate with 90% confidence that the mean number of all targeted industrial chemicals found in cord blood in the United States is between 117.412 and 137.488.
Solution B
Enter the data as a list.
Press
STAT
and arrow over to
TESTS
.
Arrow down to
8:TInterval
and press
ENTER
(or you can just press
8
). Arrow to Data and press
ENTER
.
Arrow down to
List
and enter the list name where you put the data.
Arrow down to
Freq
and enter 1.
Arrow down to
C-level
and enter 0.90
Arrow down to
Calculate
and press
ENTER
.
The 90% confidence interval is (117.41, 137.49).
Example \(\PageIndex{3}\)
A random sample of statistics students were asked to estimate the total number of hours they spend watching television in an average week. The responses are recorded in Table \(\PageIndex{2}\). Use this sample data to construct a 98% confidence interval for the mean number of hours statistics students will spend watching television in one week.
| 0 | 3 | 1 | 20 | 9 |
| 5 | 10 | 1 | 10 | 4 |
| 14 | 2 | 4 | 4 | 5 |
Solution A
- \(\bar{x} = 6.133\),
- \(s = 5.514\),
- \(n = 15\), and
- \(df = 15 – 1 = 14\).
\(CL = 0.98\), so \(\alpha = 1 - CL = 1 - 0.98 = 0.02\)
\(\frac{\alpha}{2} = 0.01 t_{\frac{\alpha}{2}} = t_{0.01} 2.624\)
\(EBM = t_{\frac{\alpha}{2}}\left(\frac{s}{\sqrt{n}}\right) = 2.624\left(\frac{5.514}{\sqrt{15}}\right) - 3.736\)
\(\bar{x} – EBM = 6.133 – 3.736 = 2.397\)
\(\bar{x} + EBM = 6.133 + 3.736 = 9.869\)
We estimate with 98% confidence that the mean number of all hours that statistics students spend watching television in one week is between 2.397 and 9.869.
Solution B
Enter the data as a list.
Press
STAT
and arrow over to
TESTS
.
Arrow down to
8:TInterval
.
Press
ENTER
.
Arrow to
Data
and press
ENTER
.
Arrow down and enter the name of the list where the data is stored.
Enter
Freq
: 1
Enter
C-Level
: 0.98
Arrow down to
Calculate
and press
Enter
.
The 98% confidence interval is (2.3965, 9,8702).
Reference
- “America’s Best Small Companies.” Forbes, 2013. Available online at http://www.forbes.com/best-small-companies/list/ (accessed July 2, 2013).
- Data from Microsoft Bookshelf .
- Data from http://www.businessweek.com/ .
- Data from http://www.forbes.com/ .
- “Disclosure Data Catalog: Leadership PAC and Sponsors Report, 2012.” Federal Election Commission. Available online at www.fec.gov/data/index.jsp (accessed July 2,2013).
- “Human Toxome Project: Mapping the Pollution in People.” Environmental Working Group. Available online at www.ewg.org/sites/humantoxome...tero%2Fnewborn (accessed July 2, 2013).
- “Metadata Description of Leadership PAC List.” Federal Election Commission. Available online at www.fec.gov/finance/disclosur...pPacList.shtml (accessed July 2, 2013).
Glossary
- Degrees of Freedom (\(df\))
- the number of objects in a sample that are free to vary
- Normal Distribution
- a continuous random variable (RV) with pdf \(f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-(x-\mu)^2/2\sigma^{2}}\), where \(\mu\) is the mean of the distribution and \(\sigma\) is the standard deviation, notation: \(X \sim N(\mu,\sigma)\). If \(\mu = 0\) and \(\sigma = 1\), the RV is called the standard normal distribution .
- Standard Deviation
- a number that is equal to the square root of the variance and measures how far data values are from their mean; notation: \(s\) for sample standard deviation and \(\sigma\) for population standard deviation
- Student's t -Distribution
-
investigated and reported by William S. Gossett in 1908 and published under the pseudonym Student; the major characteristics of the random variable (RV) are:
- It is continuous and assumes any real values.
- The pdf is symmetrical about its mean of zero. However, it is more spread out and flatter at the apex than the normal distribution.
- It approaches the standard normal distribution as n get larger.
- There is a "family" of t–distributions: each representative of the family is completely defined by the number of degrees of freedom, which is one less than the number of data.


4.1.1: The Nature of Electromagnetic Radiation
Unlike convection or conduction, heat transfer by electromagnetic radiation requires no intervening medium to transmit it. Electromagnetic radiation travels through space in the form of waves. It's hard to imagine radiation moving as waves through empty space without a medium to transfer the wave form. The waves created when you drop a rock into a pool require molecules of water to propagate them, but not so for radiation.
Energy as electromagnetic waves
The quantity of energy carried in a wave is associated with the height or amplitude of the wave. Everything else being equal, the amount of energy carried in a wave is directly proportional to the amplitude of the wave. The type or "quality" of radiation depends on the wavelength , the distance between successive crests. The greater the distance between wave crests, the longer the wavelength.
Any body that has a temperature is emitting electromagnetic radiation. There are an infinite number of wavelengths that make up the electromagnetic spectrum though we group them into a number of bands (Figure \(\PageIndex{2}\)). The shortest wavelengths fall into the gamma rays, the electromagnetic radiation we can see with our eyes and processed by our brains falls into the visible band, and radio waves are comprised of the longest wavelengths.
The maximum wavelength at which a body emits radiation depends on its temperature. Wein's (pronounced "weens") Law states that the peak wavelength of radiation emission is inversely related to the temperature of the emitting body. That is, the hotter the body, the shorter the wavelength of peak emission. Figure \(\PageIndex{3}\) shows the wavelengths over which the Sun and Earth emit most of their radiation. The Sun being a much hotter body emits most of its radiation in the shortwave end and the Earth in the longwave end of the spectrum. The division between shortwave and longwave radiation occurs at about 3 micrometers.


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4.1: Overview, Objectives, and Key Terms
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4.2: Critical Thinking
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4.3: Introduction to Criminal Investigation
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4.4: Collecting Evidence
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In this module, we will look at the role the police/law enforcement plays in collecting evidence for presentation in the trial. As we progress in this course, you will see how vital it is that officers understand the rights of suspects and the importance of following criminal court processes and procedures. If an officer fails to properly collect evidence, that evidence could be excluded from the court and a guilty person could go free.
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4.5: Criminal Profiling
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Criminal profiling is synonymous with offender profiling both of which are useful investigative tools used in policing (Woodhams and Toye, 2007). Profiling’s purpose is to identify likely suspects that are correlated with criminal characteristics and patterns. Berg (200) maintains that a Psychological profiler relies heavily upon perpetrators' methods of operation or evidence left at a crime scene to identify a person’s personality or state of mind.
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4.6: Racial Profiling
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4.7: Terry Stops
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4.8: A Critical Look at Terry Stops
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17.2: Appendix B to Applied Probability- some mathematical aids
Series
1. Geometric series From the expression \((1 - r) (1 + r + r^2 + \cdot\cdot\cdot + r^n) = 1 - r^{n + 1}\), we obtain
\(\sum_{k = 0}^{n} r^k = \dfrac{1 - r^{n + 1}}{1 - r}\) for \(r \ne 1\)
For \(|r| < 1\), these sums converge to the geometric series \(\sum_{k = 0}^{\infty} r^k = \dfrac{1}{1 - r}\)
Differentiation yields the following two useful series:
\(\sum_{k = 1}^{\infty} kr^{k - 1} = \dfrac{1}{(1 - r)^2}\) for \(|r| < 1\) and \(\sum_{k = 2}^{\infty} k(k - 1)r^{k - 2} = \dfrac{2}{(1 - r)^3}\) for \(|r| < 1\)
For the finite sum, differentiation and algebraic manipulation yields
\(\sum_{k = 0}^{n} k r^{k - 1} = \dfrac{1 - r^n [1 + n(1 - r)]}{(1 - r)^2}\) which converges to \(\dfrac{1}{(1 - r)^2}\) for \(|r| < 1\)
2. Exponential series . \(e^k = \sum_{k = 0}^{\infty} \dfrac{x^k}{k!}\) and \(e^{-s} = \sum_{k = 0}^{\infty} (-1)^k \dfrac{x^k}{k!}\) for any \(x\)
Simple algebraic manipulation yields the following equalities useful for the Poisson distribution:
\(\sum_{k = n}^{\infty} k \dfrac{x^k}{k!} = x \sum_{k = n - 1}^{\infty} \dfrac{x^k}{k!}\) and \(\sum_{k = n}^{\infty} k (k - 1) \dfrac{x^k}{k!} = x^2 \sum_{k = n - 2}^{\infty} {x^k}{k!}\)
3. Sums of powers of integers \(\sum_{i = 1}^{n} i = \dfrac{n(n + 1)}{2}\) \(\sum_{i = 1}^{n} i^2 = \dfrac{n(n + 1)(2n + 1)}{6}\)
Some useful integrals
1. The gamma function \(\Gamma(r) = \int_{0}^{\infty} t^{r - 1} e^{-t}\ dt\) for \(r > 0\)
Integration by parts shows \(Gamma (r) = (r - 1) \Gamma (r - 1)\) for \(r > 1\)
By induction \(\Gamma (r) = (r - 1)(r - 2) \cdot\cdot\cdot (r - k) \Gamma (r - k)\) for \(r > k\)
For a positive integer \(n\), \(\Gamma (n) = (n - 1)!\) with \(\Gamma (1) = 0! = 1\)
2. By a change of variable in the gamma integral, we obtain
\(\int_{0}^{\infty} t^r e^{-\lambda t}\ dt = \dfrac{\Gamma (r+1)}{\lambda^{r + 1}}\) \(r > -1\), \(\lambda > 0\)
3. A well known indefinite integral gives
\(\int_{a}^{\infty} te^{-\lambda t}\ dt = \dfrac{m!}{\lambda^{m + 1}} e^{-\lambda a} [1 + \lambda a + \dfrac{(\lambda a)^2}{2!} + \cdot\cdot\cdot + \dfrac{(\lambda a)^m}{m!}]\)
4. The following integrals are important for the Beta distribution.
\(\int_{0}^{1} u^r (1 - u)^s\ du = \dfrac{\Gamma (r + 1) \Gamma (s + 1)}{\Gamma (r + s + 2)}\) \(r > -1\), \(s > -1\)
For nonnegative integers \(m, n\) \(\int_{0}^{1} u^m (1 - u)^n\ du = \dfrac{m! n!}{(m + n + 1)!}\)
Some basic counting problems
We consider three basic counting problems, which are used repeatedly as components of more complex problems. The first two, arrangements and occupancy are equivalent. The third is a basic matching problem.
Arrangements
of \(r\) objects selected from among \(n\) distinguishable objects.
a. The order is significant.
b. The order is irrelevant.
For each of these, we consider two additional alternative conditions.
1. No element may be selected more than once.
2. Repitition is allowed.
Occupancy
of \(n\) distinct cells by \(r\) objects. These objects are
a. Distinguishable.
b. Indistinguishable.
The occupancy may be
1. Exclusive.
2. Nonexclusive (i.e., more than one object per cell)
The results in the four cases may be summarized as follows:
a. 1. Ordered arrangements, without repetition (
permutations
). Distinguishable objects, exclusive occupancy.
\(P(n, r) = \dfrac{n!}{(n - r)!}\)
2. Ordered arrangements, with repitition allowed. Distinguishable objects, nonexclusive occupancy.
\(U(n,r) = n^r\)
b. 1. Arrangements without repetition, order irrelevant ( combinations ). Indistinguishable objects, exclusive occupancy.
\(C(n, r) = \dfrac{n!}{r!(n - r)!} = \dfrac{P(n, r)}{r!}\)
2. Unordered arrangements, with repetition. Indistinguishable objects, nonexclusive occupancy.
\(S(n, r) = C(n + r - 1, r)\)
Matching \(n\) distinguishable elements to a fixed order. Let \(M(n, k)\) be the number of permutations which give \(k\) matches.
\(n = 5\)
Natural order 1 2 3 4 5
Permutation 3 2 5 4 1 (Two matches– positions 2, 4)
We reduce the problem to determining \(m(n, 0)\), as follows:
Select \(k\) places for matches in \(C(n, k)\) ways.
Order the \(n - k\) remaining elements so that no matches in the other \(n - k\) places.
\(M(n, k) = C(n, k) M(n - k, 0)\)
Some algebraic trickery shows that \(M(n, 0)\) is the integer nearest \(n!/e\). These are easily calculated by the MATLAB command
M = round(gamma(n+1)/exp(1))
For example
>> M = round(gamma([3:10]+1)/exp(1)); >> disp([3:6;M(1:4);7:10;M(5:8)]') 3 2 7 1854 4 9 8 14833 5 44 9 133496 6 265 10 1334961
Extended binomial coefficients and the binomial series
The ordinary binomial coefficient is \(C(n, k) = \dfrac{n!}{k!(n - k)!}\) for integers \(n > 0\), \(0 \le k \le n\)
For any real \(x\), any integer \(k\), we extend the definition by
\(C(x, 0) = 1\), \(C(x, k) = 0\) for \(k < 0\), and \(C(n, k) = 0\) for a positive integer \(k > n\)
and
\(C(x, k) = \dfrac{x(x - 1) (x - 2) \cdot\cdot\cdot (x - k + 1)}{k!}\) otherwise
The
Pascal's relation
holds: \(C(x, k) = C(x - 1, k - 1) + C(x - 1, k)\)
The power series expansion about \(t = 0\) shows
\((1 + t)^x = 1 + C(x, 1)t + C(x, 2)t^2 + \cdot\cdot\cdot\) \(\forall x\), \(-1 < t < 1\)
For \(x = n\), a positive integer, the series becomes a polynomial of degree \(n\)
Cauchy's equation
Let \(f\) be a real-valued function defined on \((0, \infty)\), such that
a. \(f(t + u) = f(t) + f(u)\) for \(t, u > 0\), and
b. There is an open interval \(I\) on which \(f\) is bounded above (or is bounded below).
Then \(f(t) = f(1) t\) \(\forall t > 0\)
Let \(f\) be a real-valued function defined on (\(0, \infty\)) such that
a. \(f(t + u) = f(t)f(u)\) \(\forall t, u > 0\), and
b. There is an interval on which \(f\) is bounded above.
Then, either \(f(t) = 0\) for \(t > 0\), or there is a constant \(a\) such that \f(t) = e^{at}\) for \(t >0\)
[For a proof, see Billingsley, Probability and Measure , second edition, appendix A20]
Countable and uncountable sets
A set (or class) is countable iff either it is finite or its members can be put into a one-to-one correspondence with the natural numbers.
Examples
- The set of odd integers is countable.
- Thee finite set \(\{n: 1 \le n \le 1000\}\) is countable.
- The set of all rational numbers is countable. (This is established by an argument known as diagonalization).
- The set of pairs of elements from two countable sets is countable.
- The union of a countable class of countable sets is countable.
A set is uncountable iff it is neither finite nor can be put into a one-to-one correspondence with the natural numbers.
Examples
- The class of positive real numbers is uncountable. A well known operation shows that the assumption of countability leads to a contradiction.
- The set of real numbers in any finite interval is uncountable, since these can be put into a one-to-one correspondence of the class of all positive reals.


1.5: Social Theories
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Making Sense of Abstract Theories
Sociological theories are the core and underlying strength of the discipline. They guide researchers in their studies. They also guide practitioners in their intervention strategies. And they will provide you with a basic understanding of how to see the larger social picture in your own personal life. A Theory is a set of interrelated concepts used to describe, explain, and predict how society and its parts are related to each other . The metaphor I've used for many years to illustrate the usefulness of a theory is what I call the "goggles metaphor." Goggles are a set of inter-related parts that help us see things more clearly. Goggles work because the best scientific components work together to magnify, enlarge, clarify, and expand to our view of the thing we are studying.
Theories are sets of inter-related concepts and ideas that have been scientifically tested and combined to magnify, enlarge, clarify, and expand our understanding of people, their behaviors, and their societies. Without theories, science would be a futile exercise in statistics. In the diagram below you can see the process by which a theory leads sociologist to perform a certain type of study with certain types of questions that can test the assumptions of the theory. Once the study is administered the findings and generalizations can be considered to see if they support the theory. If they do, similar studies will be performed to repeat and fine-tune the process. If the findings and generalizations do not support the theory, the sociologist rethinks and revisits the assumptions they made.
Here's a real-life scientific example. In the 1960's two researchers named Cumming and Henry studied the processes of aging. They devised a theory on aging that had assumptions built into it. These were simply put, that all elderly people realize the inevitability of death and begin to systematically disengage from their previous youthful roles while at the same time society prepares to disengage from them (see Maddox et al. 1987 The Encyclopedia of Aging, Springer Pub. NY for much more detail. Cumming and Henry tested their theory on a large number of elderly persons. Findings and generalization consistently yielded a "no" in terms of support for this theory. For all intents and purposes this theory was abandoned and is only used in references such as these (for a more scientifically supported theory on aging Google "Activity Theory and/or Continuity Theory"). Theories have to be supported by research and they also provide a framework for how specific research should be conducted.
By the way, theories can be used to study society-millions of people in a state, country, or even at the world level. When theories are used at this level they are referred to as Macro Theories, theories which best fit the study of massive numbers of people (typically Conflict and Functional theories). When theories are used to study small groups or individuals, say a couple, family, or team, they are referred to as being Micro Theories, theories which best fit the study of small groups and their members (typically Symbolic Interactionism or Social Exchange theories). In many cases, any of the four main theories can be applied at either the macro or micro levels.
There are really two distinct types of theories: first, Grand Theory, which is a theory which deals with the universal aspects of social processes or problems and is based on abstract ideas and concepts rather than on case specific evidence . These include Conflict, Functionalism, Symbolic Interactionism, and Social Exchange Theories; second, Middle-Range Theory, which is a theory derived from specific scientific findings and focuses on the interrelation of two or more concepts applied to a very specific social process or problem. Robert K. Merton (1910-2003) was a functional theory-based sociologist who taught the value of using smaller more specifically precise theories in trying to explain smaller and more specific social phenomena. These theories include: Continuity, Activity, Differential Association, and Labeling theories. (see American Sociology Association, Theory http://www.asatheory.org/ ).
Let's consider the four grand theories one at a time. The Conflict Theory is a macro theory. A Macro Theory is a sociological theory designed to study the larger social, global, and societal level of sociological phenomena . This theory was founded by a German philosopher, economist, sociologist, and revolutionary (1818-1883). Marx was a witness to oppression perpetrated by society's elite members against the masses of poor. He had very little patience for the capitalistic ideals that undergirded these powerful acts of inhumane exploitation of the average person. To him struggle was innate to all human societies. Later another German named Max Weber (1864-1920; pronounced "Veybur") further developed this sociological theory and refined it to a more moderate position. Weber studied capitalism further but argued against Marx's outright rejection of it.
Conflict Theory
Conflict theory is especially useful in understanding: war, wealth and poverty, the haves and the have nots, revolutions, political strife, exploitation, divorce, ghettos, discrimination and prejudice, domestic violence, rape, child abuse, slavery, and more conflict-related social phenomena. Conflict Theory claims that society is in a state of perpetual conflict and competition for limited resources . Marx and Weber, were they alive today, would likely use Conflict Theory to study the unprecedented bail outs by the US government which have proven to be a rich-to-rich wealth transfer.
 It is a power struggle which is most often won by wealthy elite and lost by the common person of common means. Power is the ability to get what one wants even in the presence of opposition. Authority is the institutionalized legitimate power . By far the Bourgeoisie, or wealthy elite (royalty, political, and corporate leaders), have the most power. Bourgeoisie are the "Goliaths" in society who often bully their wishes into outcomes. The Proletariat are the common working class, lower class, and poor members of society. According to Marx (see diagram below) the Bourgeoisie and Proletariat cannot both have it their way and in order to offset the wealth and power of the Bourgeoisie the proletariat often rise up and revolt against their oppressors (The French, Bolshevik, United States, Mexican, and other revolutions are examples).
In fact Marx and Weber realized long ago that society does have different classes and a similar pattern of relatively few rich persons in comparison to the majority who are poor. The rich call the shots. Look below at the photographic montage of homes in one US neighborhood which were run down, poor, trashy, and worth very little. They were on the West side of this gully and frustrated many who lived on the East side who were forced to drive through these "slums" to reach their own mansions.
Figure 3. Photo Montage of Haves and Have Nots in a US Neighborhood
© 2009 Ron J. Hammond, Ph.D.
The Conflict Theory has been repeatedly tested against scientifically derived data and it repeatedly proves to have a wide application among many different levels of sociological study. That is not to say that all sociological phenomena are conflict-based. But, most Conflict theorists would argue that more often than not Conflict assumptions do apply. Feminist theory is a theoretical perspective that is couched primarily in Conflict Theory assumptions.
Functionalism or Structural Functionalism Theory
The next grand theory is called Functionalism or Structural Functionalism. The Functionalist Theory claims that society is in a state of balance and kept that way through the function of society's component parts. This theory has underpinnings in biological and ecological concepts (see diagram below). Society can be studied the same way the human body can be studied - by analyzing what specific systems are working or not working, diagnosing problems, and devising solutions to restore balance. Socialization, religious involvement, friendship, health care, economic recovery, peace, justice and injustice, population growth or decline, community, romantic relationships, marriage and divorce, and normal and abnormal family experiences are just a few of the evidences of functional processes in our society.
Sure, Functionalists would agree with Conflict Theorists that things break down in society and that unfair treatment of others is common. These break downs are called Dysfunctions, which are breakdowns or disruptions in society and its parts that threaten social stability. Enron's collapse, the ruination of 14,000 employees' retirement funds, the loss of millions in shareholder investments, and the serious doubt it left in the mind of US investors about the Stock Market's credibility and reliability which lasted for nearly a decade are examples of dysfunctions in the economic sector of the economy. But, Functionalists also look at two types of functions: manifest and latent functions. Manifest Functions are the apparent and intended functions of institutions in society. Latent Functions are the less apparent, unintended, and often unrecognized functions in social institutions and processes .
Back to Enron, the government's manifest function includes regulation of investment rules and laws in the Stock market to ensure credibility and reliability. After the Enron collapse, every company offering stocks for trade underwent a government supervised audit of its accounting processes in order to restore the public trust. For the most part balance was restored in the Stock Market (to a certain degree at least). There are still many imbalances in the investment, mortgage, and banking sectors which have to be readjusted; but, that's the point - society does readjust and eventually recover some degree of function.
Does the government also provide latent or accidental functions to society? Yes. Take for example the US military bases. Of all the currently open US military bases, all are economic boons for the local communities surrounding them. All provide jobs, taxes, tourism, retail, and government contract monies that would otherwise go somewhere else. When the discussion about closing military bases comes up in Washington DC, Senators and members of Congress go to work trying to keep their community's bases open.
As you can already tell, Functionalism is more positive and optimistic that Conflict Theory (the basis for much criticism by many Conflict Theorists). Functionalists realize that just like the body, societies get "sick" or dysfunction. By studying society's parts and processes, Functionalists can better understand how society remains stable or adjust to destabilizing forces when unwanted change is threatened. According to this theory most societies find that healthy balance and maintain it (unless they don't and collapse as many have in the history of the world. Equilibrium is the state of balance maintained by social processes that help society adjust and compensate for forces that might tilt it onto a path of destruction.
Getting back to the Conflict example of the gully separating extremely wealthy and poor neighborhoods, look at this Habitat for Humanity picture below. I took this close to my own home, because it represents what Functional Theorists claim happens - component parts of society respond to dysfunctions in ways that help to resolve problems. In this house the foundation was dug, poured, and dried within a week. From the foundation to this point was three working days. This house is now finished and lived in, thanks mostly to the Habitat non-profit process and the work of many volunteers. From the Functionalism perspective, optimism is appropriate and fits the empirical data gathered in society.
Figure 2. Photo of a Recently Finished Habitat for Humanity Home
© 2009 Ron J. Hammond, Ph.D.
Symbolic Interactionism Theory
Interactionism comes in two theoretical forms: Symbolic Interaction and Social Exchange. By far, my favorite sociological theory is Symbolic Interactionism. Symbolic Interaction claims that society is composed of ever-present interactions among individuals who share symbols and their meanings . This is a very useful theory for: understanding other people; improving communications; learning and teaching skills in cross-cultural relations; and generally speaking, "not doing harm to your roommates" as many of my students often say after understanding this theory. Values, communication, witch hunting, crisis management, fear from crime, fads, love and all that comes with it, "evil and sin," what's hot and what's not, alien abduction beliefs, "who I am," litigation, mate selection, arbitration, dating joys and woes, and both personal and national meanings and definitions (September 1, 2001-WTC) can all be better understood using Symbolic Interactionism.
Once you realize that individuals are, by their social natures, very symbolic with one another, then you begin to understand how to persuade your friends and family, how to understand others' points of view, and how to resolve misunderstandings. This theory magnifies the concepts of meanings. Think about these three words, LOVE, LUST, and LARD. Each letter is a symbol. When combined in specific order, each word can be defined. Because we memorize words and their meanings we know that there is a striking difference between LOVE and LUST. We also know that LARD has nothing to do with either of these two terms. Contrast these word pairs: hate versus hope; help versus hurt; advise versus abuse; and connect versus corrupt. These words, like many others carry immense meaning and when juxtaposed sound like the beginning of philosophical ideas.
Symbolic Interactionism makes it possible for you to be a college student. It makes it so you understand your professors' expectations and know how to step up to them. Our daily interactions are filled with symbols and an ongoing process of interactions with other people based on the meanings of these symbols. "How's it going?" Ever had anyone you've greeted actually answer that question? Most of us never have. It's a greeting, not a question in the US culture (see culture chapter).
If you want to surprise someone, answer them next time they say "How's it going?" If they have a sense of humor, they might get a kick out of it. If not, you may have to explain yourself. Symbolic Interactionism Theory explores the way we communicate and helps us to understand how we grow up with our self-concept (see socialization chapter). It helps you to know what the expectations of your roles are and if you perceive yourself as doing a good job or not in meeting those expectations.
There are many other Symbolic Interactionism concepts out there to study, let's just talk about one more-The Thomas Theorem or Definition of the Situation. The Thomas Theorem is often called the "Definition of the situation" which is basically if people perceive or define something as being real then it is real in its consequences. I give a few examples from the media: a woman was diagnosed as HIV positive. She made her funeral plans, made sure her children would be cared for then prepared to die. Two-years later she was retested. It turned out her first test results were a false positive, yet she acted as though she had AIDS and was certainly going to die soon from it.
In a hypothetical case, a famous athlete (you pick the sport) defines himself as invincible and too famous to be held legally accountable for his criminal behavior. He is subsequently found guilty. A politician (you pick the party and level of governance) believes that his/her constituents will tolerate anything. When he/she doesn't get reelected no one is surprised. The point is that when we define our situation as being real, we act as though it is real (regardless of the objective facts in the matter).
Symbolic Interactionism is very powerful in helping people to understand each other. Newlyweds, roommates, life-long friends, young adult children and their parents, and teammates can all utilize the principles to "walk a mile in the other's shoes;" "see the world through their glasses;" and/or simply "get it." One of the major realization that comes with Symbolic Interactionism is that you begin to understand the other people in your life and come to know that they are neither right nor wrong, just of a different point of view. They just define social symbols with varying meanings.
To understand the other person's symbols and meanings, is to approach common ground. Listen to this statement by Rosa Parks (1913-2005), "All I was doing was trying to get home from work." In 1955 when she refused to give up her seat on the bus to a White person, it proved to be a spark for the Civil Rights Movement that involved the leadership of Martin Luther King Jr. and many other notable leaders. It was Rosa Parks' simple and honest statement that made her act of defiance so meaningful. The lion share of the nation was collectively tired and sick of the mistreatment of Blacks. Many Whites joined the protests while others quietly sympathized. After all that was written in the history books about it, a simple yet symbolic gesture by Rosa Parks symbolically started the healing process for the United States.
Social Exchange Theory
The remaining theory and second interactionist theory is Social Exchange. Social Exchange claims that society is composed of ever present interactions among individuals who attempt to maximize rewards while minimizing costs. Assumptions in this theory are similar to Conflict theory assumptions yet have their interactistic underpinnings. Basically, human beings are rational creatures, capable of making sound choices once the pros and cons of the choice are understood. This theory uses a formula to measure the choice making processes.
(REWARDS-COSTS)=OUTCOMES
or
("What I get out of it"-"What I lose by doing it")="My decision "
We look at the options available to us and weigh as best we can how to maximize our rewards and minimize our losses. Sometimes we get it right and other times we make a bad choice. One of the powerful aspects of this theory is the concept of Equity. Equity is a sense that the interactions are fair to us and fair to others involved by the consequences of our choices . For example, why is it that women who work 40 hours a week and have husbands who work 40 hours per week do not perform the same number of weekly hours of housework and childcare? Scientists have surveyed many couples to find the answer. Most often, it boils down to a sense of fairness or equity. Because she defines it as her role to do housework and childcare, while he doesn't; because they tend to fight when she does try to get him to perform housework, and because she may think he's incompetent, they live with an inequitable arrangement as though it were equitable (don't get me started on the evidence that supports men sharing the actual roles of housekeepers and childcare providers-see Joseph Pleck, "Working Wives/ Working Husbands" Sage Pub, CA).
Each of us tries constantly to weigh pros and cons and to maximize the outcomes of our choices. I often provide a rhetorical challenge to my students when I ask them to go down to the cafeteria, pick the least attractive person they can find, take them on a date where they drive and they pay for everything, then give the person a 7 second kiss at the end of the date. "Why would we do that?" they typically ask. "That's my point," I typically reply, having increased a bit of their understanding of the Social Exchange Theory.
Any of the four theories can be used to study any individual and collective behaviors. But, some do work better than others because their assumptions more precisely match the issue of interest. Divorce might be studied from the Conflict Theory to understand how things become adversarial and how and why contested divorces sometimes become violent. Divorce might be studied from the Functionalism Theory to understand how divorce is a means to resolving untenable social circumstance-it is a gesture designed to restore balance and equilibrium. Divorce might be studied using the Symbolic Interactionism Theory to identify how people define their roles before, during, and after the divorce and how they reestablish new roles as unmarried adults. Divorce might also be studied using the Social Exchange Theory to understand the processes and choices that lead to the final divorce decision, distribution of assets, child custody decrees and the final legal change of status (see Levinger and Moles, "Divorce and Separation: Context, Causes, and Consequences" 1979, Basic Books).
I've enclosed a simple summary sheet of the four basic theories used most by sociologists. It serves well as a reference guide, but can't really replace your efforts to study sociological theories in more detail. On the next page I've enclosed a self-assessment that may help you to assess your leanings towards these four main theories and two others that are often used by sociologists. On the self-assessment don't be surprised if you find that all four theories fit your world-view. Keep in mind they have been extensively studied for a very long time.
Comparing the Four Sociological Theories
| Introduction to Sociology: 4 Basic Theories | |||
|---|---|---|---|
| Conflict | Functionalism | Symbolic Interactionism | Social Exchange |
| Macro | Macro | Micro | Micro |
|
-Inequality lies at the core of society which leads to conflict -Resources are limited -Power is not evenly distributed -Competition is inevitable (winners & losers) -Negotiations based on influence, threats, promises, and consensus -Threats and coercion -Any resource can be used as tool of power or exploitation -War is natural -Haves and have nots -Privileges are protected by haves -Order is challenged by have nots -Examples of: Gender & Feminist |
-Uses biological model (society is like a living organism) -Society has interrelated parts -What are functions or dysfunctions of parts -Society finds balance and is stable -Equilibrium -Society adjusts to maintain balance -How are parts integrated -Manifest functions -Latent functions and dysfunctions -Example of: Systems Theory |
-Society is an ongoing process of many social interactions -Interactions based on symbolic context in which they occur -Subjective perceptions are critical to how symbols are interpreted -Communications -Meanings -Significant others -Roles -Relative deprivation -Self -Reality shaping in self and with others -Key Ideas: Social construction of reality Thomas Theorem Definition of situation -Example of: theories of self |
-Society is an ongoing series of exchanges which occur during interactions -Interactions based on formula: (Rewards- Costs)=Outcomes Rewards -Costs -Profit/Loss -Comparisons -Limited resources -Power -Legitimacy -Equity -Negotiations -Tradeoffs -Example of: Levinger=s model on divorce: (Attractions +/- Barriers)=/-(Alternative Attractions) |
Prepared by Ron J. Hammond, Ph.D. 20008


15: Integrating Applications Last updated Save as PDF Page ID 32672 OpenStax OpenStax 15.0: Chapter Scenario 15.1: Microsoft 365- Collaboration and Integration 15.2: Microsoft Word- Integration with Microsoft Excel and Microsoft Access 15.3: Microsoft Word and Microsoft PowerPoint Integration 15.4: Microsoft Excel and Microsoft PowerPoint Integration 15.5: Microsoft Excel and Microsoft Access Integration 15.6: Integrating Data from Other Programs into Google Workspace 15.7: New Developments- The Role of Artificial Intelligence 15.8: Mastering Workplace Software Skills- A Project 15.9: Chapter Review - Key Terms 15.10: Chapter Review - Summary 15.11: Chapter Review - Review Questions 15.12: Chapter Review - Practice Exercises 15.13: Chapter Review - Written Questions


1.7: The Green's Function for a Free Particle
We have defined the free-particle Green’s function as the operator \(\hat{G}_0=\big(E-\hat{H}_0\big)^{-1}\) . Its representation in the position basis, \(\langle\mathbf{r}|\hat{G}_0|\mathbf{r}'\rangle\) , is called the propagator . As we have just seen, when the Born series is written in the position basis, the propagator appears in the integrand and describes how the particle “propagates” between discrete scattering events.
The propagator is a solution to a partial differential equation:
\[\begin{align} \begin{aligned}\langle\mathbf{r} |\big(E-\hat{H}_0\big) \hat{G}_0 |\mathbf{r}'\rangle &= \langle\mathbf{r}|\hat{I}|\mathbf{r}'\rangle \\ = \left(E + \frac{\hbar^2}{2m}\nabla^2 \right) \langle\mathbf{r} |\hat{G}_0 |\mathbf{r}'\rangle &= \delta^d(\mathbf{r}-\mathbf{r}') \\ \Rightarrow \qquad\quad \left(\nabla^2 + k^2\right) \langle\mathbf{r} |\hat{G}_0 |\mathbf{r}'\rangle &= \frac{2m}{\hbar^2} \delta^d(\mathbf{r}-\mathbf{r}').\end{aligned}\end{align}\]
As before, \(k = \sqrt{2mE/\hbar^2}\) where \(E\) is the energy of the incident particle. Therefore, up to a factor of \(2m/\hbar^2\) , the propagator is the Green’s function for the \(d\) -dimensional Helmholtz equation (see Section 1.4). Note that the \(\nabla^2\) acts upon the \(\mathbf{r}\) coordinates, not \(\mathbf{r}'\) .
To solve for \(\langle\mathbf{r}|\hat{G}_0|\mathbf{r}'\rangle\) , we can use the momentum eigenstates:
\[\begin{align} \begin{aligned}\langle\mathbf{r}|\hat{G}_0|\mathbf{r}'\rangle &= \langle\mathbf{r}|\hat{G}_0 \Big(\int d^dk' |\mathbf{k}'\rangle\langle\mathbf{k}'| \Big) |\mathbf{r}'\rangle \\ &= \int d^dk' \; \langle\mathbf{r}|\mathbf{k}'\rangle \; \frac{1}{E-\frac{\hbar^2|\mathbf{k}'|^2}{2m}} \; \langle\mathbf{k}'|\mathbf{r}'\rangle \\ &= \frac{2m}{\hbar^2} \frac{1}{(2\pi)^d} \int d^dk' \; \frac{\exp\left[i\mathbf{k}' \cdot (\mathbf{r}-\mathbf{r}')\right]}{k^2-|\mathbf{k}'|^2}.\end{aligned} \label{rGr}\end{align}\]
To proceed, we must specify the spatial dimension \(d\) . Let us set \(d = 3\) ; the calculations for other \(d\) are fairly similar. To calculate the integral over the 3D wave-vector space, we adopt spherical coordinates \((k',\theta,\phi)\) , with the coordinate axes aligned so that \(\mathbf{r}-\mathbf{r}'\) points along the \(\theta=0\) direction. We can now do the integral:
\[\begin{align} \begin{aligned}\langle\mathbf{r}|\hat{G}_0|\mathbf{r}'\rangle &= \frac{2m}{\hbar^2} \frac{1}{(2\pi)^3} \int d^3k' \; \frac{\exp\left[i\mathbf{k}'\cdot (\mathbf{r}-\mathbf{r}')\right]}{k^2-|\mathbf{k}'|^2} \\ &= \frac{2m}{\hbar^2} \frac{1}{(2\pi)^3} \int_0^\infty dk' \int_0^\pi d\theta \int_{0}^{2\pi} d\phi \;{k'}^{2}\sin\theta\; \frac{\displaystyle \exp\left(ik'|\mathbf{r}-\mathbf{r}'|\cos\theta\right)}{k^2-{k'}^2} \\ &= \frac{2m}{\hbar^2} \frac{1}{(2\pi)^2} \int_0^\infty dk' \int_{-1}^1 d\mu \;{k'}^2\; \frac{\displaystyle \exp\left(ik'|\mathbf{r}-\mathbf{r}'|\mu\right)}{k^2-{k'}^2} \qquad(\text{letting}\;\mu = \cos\theta) \\ &= \frac{2m}{\hbar^2} \frac{1}{(2\pi)^2} \int_0^\infty dk' \; \frac{ {k'}^2}{k^2-{k'}^2}\, \frac{\displaystyle \exp\left(ik'|\mathbf{r}-\mathbf{r}'|\right) - \exp\left(-ik'|\mathbf{r}-\mathbf{r}'|\right)}{ik'|\mathbf{r}-\mathbf{r}'|} \\ &= \frac{2m}{\hbar^2} \frac{1}{(2\pi)^2} \frac{i}{|\mathbf{r}-\mathbf{r}'|} \int_{-\infty}^\infty dk' \; \frac{\displaystyle k'\, \exp\left(ik'|\mathbf{r}-\mathbf{r}'|\right)}{(k' - k)(k'+k)}\end{aligned}\end{align}\]
This looks like something we can handle with contour integration techniques. But there’s a snag: the integration contour runs over the real- \(k'\) line, and since \(k \in \mathbb{R}^+\) , there are two poles on the contour (at \(\pm k\) ). Hence, the value of the integral, as written, is singular.
To make the integral non-singular, we must “regularize” it by tweaking its definition. One way is to displace the poles infinitesimally in the complex \(k'\) plane, shifting them off the contour. We have a choice of whether to move each pole upwards or downwards; this choice turns out to be linked to whether the waves described by \(\hat{G}_0\) are incoming, outgoing, or behave some other way at infinity. It turns out that the right choice for us is to move the pole at \(-k\) infinitesimally downwards, and the pole at \(+k\) infinitesimally upwards:
This means replacing the denominator of the integrand as follows:
\[(k' - k)(k'+k) \;\rightarrow\; (k' - k - i\varepsilon)(k'+k+i\varepsilon) = {k'}^2 - (k+i\varepsilon)^2,\]
where \(\varepsilon\) is a positive infinitesimal. This is equivalent to replacing \(E \rightarrow E + i\varepsilon\) in the definition of the Green’s function. The integral can now be computed as follows:
\[\begin{align} \begin{aligned}\int_{-\infty}^\infty dk' \; \frac{\displaystyle k' \exp\left(ik'|\mathbf{r}-\mathbf{r}'|\right)}{(k' - k)(k'+k)} &\rightarrow \lim_{\varepsilon \rightarrow 0^+} \int_{-\infty}^\infty dk' \; \frac{\displaystyle k' \exp\left(ik'|\mathbf{r}-\mathbf{r}'|\right)}{(k' - k - i\varepsilon)(k'+k+i\varepsilon)}\;\;\; (\text{regularize}) \\ &= \lim_{\varepsilon \rightarrow 0^+} \int_C dk' \; \frac{\displaystyle k' \exp\left(ik'|\mathbf{r}-\mathbf{r}'|\right)}{(k' - k - i\varepsilon)(k'+k+i\varepsilon)} \quad\;\;\; (\text{close contour above}) \\ &= 2\pi i \lim_{\varepsilon \rightarrow 0^+} \mathrm{Res}\left[\frac{\displaystyle k' \exp\left(ik'|\mathbf{r}-\mathbf{r}'|\right)}{(k' - k - i\varepsilon)(k'+k+i\varepsilon)}\right]_{k'=k+i\varepsilon^+} \\ &= \pi i \exp\left(ik|\mathbf{r}-\mathbf{r}'|\right).\end{aligned}\end{align}\]
Plugging this into Equation \(\eqref{rGr}\) yields the propagator \(\langle\mathbf{r}|\hat{G}_0|\mathbf{r}'\rangle\) . The final result is given below, along with the results for \(d=1\) and \(d=2\) (which are obtained in a similar fashion):
Definition: Propagator
\[\langle\mathbf{r}|\hat{G}_0|\mathbf{r}'\rangle = \frac{2m}{\hbar^2} \times \begin{cases} \Bigg.\displaystyle\frac{1}{2ik} \exp\left(ik|x-x'|\right),& d=1\\ \Bigg. \displaystyle\frac{1}{4i} H^+_0(k|\mathbf{r}-\mathbf{r'}|), & d=2 \\ \displaystyle \Bigg. - \frac{\exp\left(ik|\mathbf{r}-\mathbf{r}'|\right)}{4\pi|\mathbf{r}-\mathbf{r}'|}, & d = 3. \end{cases}\]
The propagator can be regarded as a function of the position \(\mathbf{r}\) , describing a wave propagating outwards from a source point \(\mathbf{r}'\) . This outgoing behavior comes from our above choice of regularization, which tweaked the definition of the Green’s function to be
Definition: Green's function
\[\hat{G_0} = \lim_{\varepsilon\rightarrow 0^+} \big(E - \hat{H}_0 + i \varepsilon\big)^{-1}.\]
This is called an outgoing or causal Green’s function . The word “causal” refers to the concept of “cause-and-effect”: i.e., a source at one point of space (the “cause”) leads to the emission of waves that move outwards (the “effect”).
Different regularizations produce Green’s functions with alternative features. For instance, we could flip the sign of \(i\varepsilon\) in the Green’s function redefinition, which displaces the \(k\) -space poles in the opposite direction. The resulting propagator \(\langle\mathbf{r}|\hat{G}_0|\mathbf{r}'\rangle\) is complex-conjugated, and describes a wave moving inwards from infinity, “sinking” into the point \(\mathbf{r}'\) . Such a choice of regularization thus corresponds to an incoming Green’s function . In the scattering problem, we will always deal with the outgoing/causal Green’s function.