4.2: Data Collection and Documentation
By the end of this section, you will be able to:
- Identify the major components of a complete health history
- Explain the difference between subjective and objective data
- Describe effective verbal and nonverbal communication techniques to collect patient data
Before meeting the patient and beginning the health assessment, the nurse should review the patient’s health record , if it is available (Figure 4.4). Knowing the patient’s demographic data is useful in deciding how to proceed with the assessment process. Information pertaining to past medical history, medications, and chronic diseases can be used to guide clinical decisions regarding current health status. Nurses should educate themselves regarding the patient’s current medications or previous test performed to determine interview questions and apply critical reasoning. The nurse should validate all data with the patient and add to the information if needed.
Once you have reviewed the patient’s basic health data, it is time to take a moment and reflect on your own personal feelings. Sometimes patient situations can illicit personal feelings for the nurse. Examples of situations that may require the nurse to reflect on personal biases include:
- drug addiction
- mental illness
- teenage pregnancy
- active sexually transmitted disease(s)
- LGBTQIA+
- special needs or cognitive challenges
- obesity or lack of health consciousness
It is imperative that the nurse remove all personal biases before interacting with the patient. According to the American Nurses Association’s Nursing Code of Ethics , personal biases can cause judgment, and the potential to project those judgments on the patient, and need to be avoided (ANA, 2022). Nurses should be as objective and open as possible when interacting with patients. Patients need to feel safe and free from negativity while receiving care.
Components of the Health History
During the assessment phase of the nursing process (see Table 4.1), the nurse completes four steps. The four steps include collecting subjective data, collecting objective data, validating the data, and documenting the data. Although there are four separate steps, they are fluid in nature and often overlap, and you may find yourself performing multiple steps concurrently. For example, you may ask your patient about skin concerns while you are inspecting their skin. If they answer that they do not have any concerns but you notice that the skin on the lower extremities is weeping (leaking fluid) and red, validation is initiated.
Documentation of data in the health record is broken down into different components.
The structure of health records can vary between practice settings, but they are all composed of these common components: demographic data , chief complaint or history of present illness, past medical history, past family and/or social history, and review of systems (ROS). Health records also contain areas to document current findings such as assessment data, diagnosis , treatment plans, and a follow-up plan for evaluation (Figure 4.5 and Figure 4.6).
Subjective Data
The information a patient provides the nurse with can indicate diseases or conditions in the body. The subjective data is information obtained from the patient and/or family members and can provide important cues about functioning and unmet needs requiring assistance. Subjective data is considered a symptom because it is something the patient reports. Examples include pain, dizziness, sadness, hunger, or restlessness. The major areas of subjective data include demographic data, chief complaint, ROS, and the patient’s past personal, family, and social history. Subjective data can be collected from a primary source , the patient, or from a secondary source. Examples of a secondary source include information obtained from the history section of the health record, or the patient’s family. If data are gathered from someone other than the patient, the nurse should document where the information is obtained.
Demographic Data
The demographic data consist of identifying basic patient information. Data such as name, date of birth, contact information, race, gender, preferred pronouns, marital status, occupation, religion, allergies, and resuscitation status are included in this component. Other data in this section may include consent forms and the patient’s health insurance or entity responsible for payment of services rendered.
Demographic data are used for both administrative and clinical purposes. Administrative examples include payment collection and confirming upcoming appointments. Clinical examples include health promotion, like cancer screening for certain populations. Demographic data should be reviewed with each patient encounter to ensure changes to the information are documented.
Patient’s Chief Complaint
After recording the patient’s basic information and reviewing it for accuracy, it is time to find out why the patient is seeking medical care. The chief complaint is a statement of the signs and symptoms that have caused the patient to seek medical attention. The chief complaint can be revealed through open-ended questioning . The interview technique of open-ended questioning involves more than a yes or no answer; it requires the patient to provide details in their response. Sample questions the nurse can ask are outlined in Table 4.2. Examples of a patient’s response to an open-ended question about onset may include, “I have been unable to sleep for the last five nights,” “I was planting rose bushes and a thorn stabbed my finger, and it has been throbbing for three days,” or “Yesterday, I slipped coming down my stairs and hurt my back.” Once the nurse has determined the patient’s chief complaint, it is important to continue to probe for more detailed information surrounding the event. Avoid using questions that suggest a response from the patient, such as “Your pain is sharp, right?”; instead, the nurse should ask the patient to describe the pain in their own words.
| Information Domain | Examples of Open-Ended Questions |
|---|---|
| Onset | “How long ago did this first begin” |
| Severity | “Please describe the pain” |
| Location | “Please explain exactly where the pain is located” |
| Self-care | “What medications have you taken for this” or “Please describe anything that has helped in the past” |
| Quality | “Please describe anything you do that makes it feel better or worse” |
| Duration | “When was the last time you experienced pain like this?” |
The nurse should ask if the patient is taking an over-the-counter medication , or a medication that does not require a prescription, as well as an alternative substance such as a nonregulated herbal substance or homeopathic medication. It is valuable for the nurse to recognize that what Western medicinal practice considers “alternative” may be the mainstay of home remedies used for specific cultures and not alternative at all. A good question for the nurse to ask would be, “What nonprescription medications, herbal supplements, or vitamins do you currently take?” It is also important to remember that not all patients visit the physician’s office because of an illness or acute event. Some patients are seeking health promotion such as routine examinations or yearly bloodwork. These patients would have a chief complaint documented as annual examination or routine physical appointment.
Culture is complex and changes with external and internal influences. Meeting patients’ cultural needs is just as important as meeting their physical needs. Mastering cultural competency is essential in providing high-quality care and improving patient outcomes. Nurses should ask questions related to the patient’s cultural preference and traditions to improve patient care outcomes. Examples might include dietary preferences, or end-of-life issues . Nurses should attempt to extract information the patient might not be so forthcoming to share. Examples include how modesty or gender roles are practiced in a specific culture. Using therapeutic communication techniques, the nurse can build a trusting relationship that will enhance the patient’s willingness to share information. Promoting cultural communication contributes to reducing racial and ethnic disparities (Weatherspoon et al., 2016).
Past Health
The past health component of a medical record addresses all of a patient’s previous medical history including personal, family, and social history. The personal history component includes any diagnosis or previously experienced condition, treatment, or surgery. Some examples include diagnoses of diabetes, hypertension, myocardial infarction, pregnancy, or birth. The nurse should ask questions pertaining to childhood illnesses and immunizations to date, and any food, medication, or environmental allergies. Other examples of information include past surgeries or hospitalizations. Information regarding past health status allows the healthcare team to plan appropriate care for any present illness.
Family Health
The family history component contains information related to familial diseases or hereditary conditions. A familial disease tends to occur more often in a particular family. Familial diseases often have a genetic component affecting more members of the family than perhaps they would be affected by chance alone, for instance, high blood pressure or cholesterol. Consider the scenario that follows:
- A 42-year-old male patient presents to your clinic for treatment of consistently high blood pressure readings at home. The patient describes an active lifestyle and dietary choices that are nutritious in nature. He goes on to explain his father had high blood pressure and died of heart disease at age seventy-four years, and his mother, while still living, has had two strokes related to high blood pressure. Based on the patient’s family health history and no other risk factors, it is considered his high blood pressure is familial in nature.
Hereditary Diseases
A hereditary disease is passed down from generation to generation. Some examples of hereditary diseases include Down syndrome , cystic fibrosis , Tay-Sachs disease , Marfan syndrome, and sickle cell anemia. It is important to ask the patient to recall any information regarding the health status of maternal and paternal grandparents, parents, and siblings. Often genograms are used to help visualize a patient’s family history. Figure 4.7 shows a sample genogram.
Social Health
The social history component includes the patient’s lifestyle choices and daily behaviors. Questions related to diet, hobbies, alcohol and tobacco use, sexual history, or other behaviors that place a patient at risk can also be found in this component. Health promotion education provided to a patient can often be derived from information discovered in the social history section of a health record . For example, if a nurse discovered a patient uses tobacco products, then they could provide smoking cessation education.
The Agency for Healthcare Research and Quality has provided the “five A’s” to intervention as an example of how to coach patients for health promotion and disease prevention.
Review of Systems
A comprehensive review of each body system (ROS), such as cardiovascular, musculoskeletal, or urinary, can reveal information the patient may have forgotten to mention previously. Sometimes patients think information may be irrelevant and do not want to take up valuable time discussing what they think is not pertinent information. For example, brittle, peeling, or indented nails can seem like a harmless condition for most people, but it can also be a symptom of cardiac issues. Symptoms or questions related to certain diseases are included in this component. Asking the patient questions in terminology they understand will allow them to describe the status of each body system. A sample review of systems is shown in Figure 4.8 and Figure 4.9.
The ability to smell and taste decreases with age. Medications and some degenerative conditions can also decrease sense of smell and taste in older people. Nurses should ask questions regarding current nutritional status and analyze correlation to age-related decreased sense of smell and taste (National Institute on Aging, 2020).
Suggested questions:
- How has your appetite been lately?
- Do you eat alone or with others?
- How many times a day do you eat?
- Do you use a lot of salt or sugar on your food?

- Have you noticed a change in the way food smells?
Objective Data
Information observed through your senses of hearing, sight, smell, and touch while assessing the patient is objective data , also called signs. Examples include vital sign s, laboratory results, and physical examination findings. Subjective and objective data are often collected and documented together. For example, if a patient says “I am feeling dizzy” and has a recorded blood pressure of 88/52, the feeling of dizziness is a subjective symptom, and the recorded blood pressure is an objective sign. It is important for the nurse to recognize the difference between subjective versus objective information. Table 4.3 provides a comparison.
| Subjective Data (Symptoms) | Conceptual Problem | Objective Data (Signs) |
|---|---|---|
|
Patient states,
“I am worried about my breast cancer diagnosis. How do I tell my children and husband?” |
Anxiety and coping |
Patient presents with:
|
|
Patient states,
“I think may be sick. My skin feels warm, but I am so cold my body is shivering. My head and chest hurt.” |
Thermoregulation, oxygenation, and immunity |
Patient presents with:
|
|
Patient states,
“I have been nauseous and had diarrhea for two weeks. The smell of food makes me want to vomit.” |
Fluid volume deficient, nutrition, elimination, and immunity |
Patient presents with:
|
Vital Signs and Anthropometric Measurements
The first set of objective data collected is called vital sign s, also known as cardinal signs, which are measurements of the body functions that are essential to maintain life. Vital signs frequently assessed and recorded by the nurse include temperature, heart rate, respiration rate, blood pressure, and pain level. Also recorded in this component of the health record are anthropometric measurements, a patient’s height and weight. Each specific vital sign is discussed in detail in Chapter 15 General Survey, Anthropometric Measurement, and Vital Signs. Vital signs are the most important objective data to be collected (Sapra et al., 2022). Vital signs give the healthcare team an overall glimpse of what is going on inside the body at any given moment.
Scenario: Nurse walks into the patient’s room to complete an assessment and the patient starts exhibiting signs of fear and anxiety. The patient pulls the covers up to their chest and is clenching them tightly. The patient will not take their eyes off the portable vital sign machine.
Nurse: Hi, my name is Sara, and I am going to be your nurse today. Do you mind verifying your name and date of birth for me?
Patient: uhm . . ., sure its Jon Blankenship and 01/12/1952. What are you going to do to me?
Nurse: Hi, Mr. Blankenship, I would like to assess your vital sign s, if that is okay?
Patient: What does that mean? Does it hurt? Just a minute ago, they brought a machine in here and took blood from my finger and it still hurts. They said they were assessing my blood sugar levels, but I don’t have blood sugar problems or diabetes. I am afraid you all do not know what you are doing, and I want to go home.
Nurse: Mr. Blankenship, I can understand your fear. Let me try and explain; we recognize you do not have a diagnosis of diabetes. Your physician wants us to monitor your blood sugar levels because the new medication he started you on for your chronic obstructive pulmonary disease (COPD) can cause blood sugar levels to get extremely high in some patients, and we just want to keep an eye on things for your safety.
Patient: Oh, well that makes sense. Then what is that machine for again?
Nurse: This is a portable vital sign machine that lets me check your heart rate, blood pressure, and temperature. If all is well, we only need to check your vital signs every six hours while you are here. If any of your vital signs changes drastically, then we may have to monitor more frequently. While obtaining vital signs does not hurt, the pressure on the blood pressure cuff can feel like a tight squeeze but won’t last long. May I go ahead and complete them now?
Patient: It is okay. You can go ahead but go slow.
Scenario follow-up: The nurse collects the patient’s pain level, heart rate, respiration rate, blood pressure, and temperature. She uses the bed scale to collect his current weight.
Nurse: Okay, I am done; your vital signs are good and are within the range of what we like to see. I know you said you want to go home. I understand being in the hospital is scary and not what you are accustomed to, but it truly is the best place for us to monitor your health right now. Do you think you would be willing to stay a bit longer with us?
Patient: Yes, I will stay for a little longer. It really is not bad here. I think, maybe, I just really miss my wife. We have been married 53 years and have never stayed a day apart.
Nurse: Would you like me to call her so you can talk to her on the phone?  We always encourage a loved one to visit. It helps with your healing process.
Patient: Oh, that would be wonderful! Yes, please call her, here is my phone number.
Physical Findings, Laboratory and Radiology Reports
A physical finding refers to the assessment of each body system. The nurse completes a head-to-toe organized approach to discover physical findings within each body system. Unit 4: Health Assessment and Promotion discusses each body system and accompanying assessment in detail. Other diagnostic information found in this component of the health record includes laboratory and radiology reports. Laboratory reports include data related to the body’s blood and chemistry. Radiology reports consist of information related to x-rays, MRI, computed tomography, or any other form of imaging. Physical findings and laboratory and radiology reports can assist the nurse in determining correct interventions for the care plan as well as evaluating effectiveness of the selected intervention.
Diagnosis
The diagnosis component of the chart is where the provider will document the identification of any diseases or conditions. The physician will use information from all the components of the health record to determine a plan of action and diagnosis for the patient. It is important to remember the professional nurse does not provide the patient with a medical diagnosis. Only the physician and other advance practice certified providers, such as a nurse practitioner or physician’s assistant, can provide a medical diagnosis. The medical diagnosis directs medical treatment and guides the whole healthcare team on how to treat the patient.
The professional nurse develops nursing diagnoses as part of the patient’s plan of care. A nursing diagnosis describes a problem that a nurse can treat within their scope of knowledge, skills, and license. It provides guidance for the nurse on which interventions should be utilized to alleviate signs and symptoms associated with the medical diagnosis.
A collaborative problem is a certain physiological response the nurse monitors that can be addressed through nursing intervention and physician-prescribed treatments. This type of problem can be described as a potential risk for status, such as increased respirations or increased heart rate. Collaborative problems involve all healthcare team members to achieve the goal. Table 4.4 shows examples of all three types of diagnoses. This component of the health record is reserved for the medical diagnosis.
| Patient has come into the emergency room and is experiencing a heart attack | ||
|---|---|---|
| Medical Diagnosis | Nursing Diagnosis | Collaborative Problem |
|
|
|
| What treatment is needed for each type of diagnosis, and who will complete the task | ||
|
|
|
Treatment Prescribed, Progress Notes, and Patient Education
The next component of the health record includes treatment prescribed by the provider. A treatment plan is used to increase patient outcomes related to specific diseases or conditions. As these prescribed treatments are completed, it is recorded in the progress notes. The EHR can separate each progress note into a specific disciplinary team, such as provider, nurses, respiratory, and other ancillary departments. All interdisciplinary team members document treatments and interventions being done for the patient, as well as the patient’s response to the treatment. This component of the health record is also used for any provided patient education. It is important to remember that the nurse should document exactly what the patient was educated on, how the education was provided, and how the evaluation of knowledge received was determined.
Scenario: Nurse walks into patient room to provide education on their newly prescribed diabetic diet.
Nurse: Hi, my name is Gwen, and I am going to be your nurse today. Do you mind verifying your name and date of birth for me?
Patient: uhm . . . , sure, its Mary Goodwright and 03/17/1972.
Nurse: Hi Ms. Goodwright, I would like to take some time and discuss your new diabetic diet. Is this a good time for you?
Patient: Sure, I am confused about what I can eat now. I think I may need to donate all the food in my house.
Nurse: I understand your concerns, and I am happy to explain your new recommended diet. Let me start by saying that you do not have to donate all the food in your home. I am sure we can come up with a plan.
Patient: Oh, thank you! I also love to bake with my grandchildren, will I still be able to do that?
Nurse: Of course, you can. I have a little video that we show all patients newly diagnosed with diabetes. I also have a brochure I want to go over with you. Would you like to watch the video now and write down any questions you have? I can come back in thirty minutes, and we can discuss it.
Patient: Yes, that sounds like a plan.
Scenario update: Nurse sets video to play on a handheld device and hands the patient paper to write on and a pen. Nurse leaves the room and returns in thirty minutes.
Nurse: Hi, so what did you think of the video?
Patient: I really enjoyed it; I learned a lot of new information, and I do not feel as scared about my diet change now. I think I can do this.
Nurse: That is wonderful! Here is some written information to follow up on the video just in case you get home and cannot remember what was discussed.
Patient: This is very helpful. Thank you.
Scenario update: Gwen answers all the patient’s questions about the video instruction clearly and in terminology she can understand. Gwen goes over written material with the patient.
Nurse: Okay, Ms. Goodwright, I think you have a good understanding of your new diet. If you continue these changes in your diet and eating habits, it will have a positive effect on your health. Do you think you can name a few items from the store that you used to purchase and substitute in a more diabetic-friendly option based on your new knowledge?
Patient: Well, I previously would buy white rice and sweet cereal for breakfast. I have always eaten my white rice with butter and sugar for breakfast. Now, I will buy whole grain rice, and oatmeal for breakfast. Instead of the butter and sugar, I am going to add a few berries or yogurt. I have always loved fried chicken and a baked potato for lunch, but now I am going to switch to sweet potatoes and baked chicken breast. I have always been a big snacker, too. I love chips and cookies, but now I will snack on things like salty nuts or hard-boiled eggs.
Nurse: I am so proud of you! I see positive outcomes in your future.
Sample documentation: “Patient was educated on newly prescribed diabetic diet. Written material and visual instruction were used to educate the patient. Patient returned demonstration by correctly verbalizing several diabetic-friendly meal choices and repeated back information learned from the video presentation.”
Documentation of the Patient Assessment
Documentation of information obtained in the patient assessment is a crucial step in the nursing process. The primary rationale for documentation is to facilitate effective communication between the interdisciplinary health-care team members and guide quality patient care. Documented data are the foundation for formulating clinical reasoning regarding patient concerns, medical diagnosis , nursing diagnosis , and collaborative problems. Making decisions about patient care without the needed supporting documentation can result in poor patient outcomes.
The importance of documentation is addressed by each U.S. state’s Nurse Practice Act, accreditation, and reimbursement agencies, such as The Joint Commission , and the Centers for Medicare and Medicaid Services (CMS). Healthcare facilities have developed policies and procedures that provide the nurse with not only the standards for what should be documented but also assistance in accomplishing the task. The components within the EHR are designed to ensure the nurse gathers all relevant data needed to meet the standards and guidelines for each individual facility.
Nurse:
Enoc, MSN, RN, Chief Nursing Officer
Clinical setting:
Acute care hospital
Years in practice:
23
Facility location:
Rural central Texas
Listening to your patient’s concerns and assessing your patient are important for the overall care in a clinical setting. One time, I had assumed care for a 77-year-old male one evening in the emergency department (ED). He was being admitted for chest pain observation. He had initial complaints of burning on the left side of his chest.
Subjectively his complaints of pain and discomfort were taken as cardiac-related symptoms due to his age and previous cardiac history. On entering the room, I noticed that the patient was fully clothed, had a peripheral IV in his hand, and had a rapid triage done. On further inspection and visiting with the patient on his level of pain and description of “burning-like pain,” I decided to place him in a gown. I did a complete assessment of his chest wall and noticed a linear rash on his left chest wall. The rash was painful to touch, and he described “burning” on the inside.
Using clinical judgment and analyzing the findings on examination, I collaborated with the ED doctor on his case. The patient was originally triaged as “chest pain” when in reality the patient was suffering from shingles. A complete cardiac workup had already been done. This lack of clinical judgment was not cost-effective, nor did it provide quality patient-centered care. As you navigate your nursing career, it is important to consider what the patient tells you (subjective data) and properly assess your patient’s (objective) data to determine appropriate outcomes. Remember, as nurses, we are patient advocates, and much of our care direction is based on our ability to perform accurate clinical assessments.
The patient was discharged home with a prescription for his symptoms of shingles, and it was a learning moment for the team. This case enacted a quality improvement policy change in our ED; from that point on, all patients were to be placed in a gown and a full skin assessment was to be completed and documented.
Corrections or Alterations to Documentation
According to research, EHRs assist in patient diagnosis, reducing errors, improving patient safety, and supporting better patient outcomes (ONC, 2019). To improve risk management and prevent liability, the documentation must be an accurate representation of what took place within a patient interaction. Keeping in mind that a health record is a legal document, if an error is noted, it needs to be corrected immediately. Each healthcare facility will have a specific policy on how to correct an error in documentation. However, most require the nurse to make an entry and note the error, and then document the correct information. Here are some examples of documentation:
- Documentation of expected findings: Patient denies any new onset of symptoms of headaches, dizziness, visual disturbances, numbness, tingling, or weakness. Patient is alert and oriented to person, place, and time. Dress is appropriate, well-groomed, and proper hygiene. Patient is cooperative and appropriately follows instructions during the exam. Speech is clear and facial expressions are symmetrical. Glasgow Coma Scale is 15. Gait is coordinated and erect with good balance. Pupils are 4mm, equal, round and reactive to light and accommodation. Sensation intact in all extremities to light touch. Cranial nerves intact × 12. No deficits demonstrated on Mini-Mental Status Examination. Upper and lower extremity strength and hand grasps are 5/5 (equal with full resistance bilaterally). Follows commands appropriately. Cerebellar function intact as demonstrated through alternating hand movements and finger-to-nose test. Negative Romberg and Pronator drift. Balance is stable during heel-to-toe test. Tolerated exam without difficulty.
- Documentation of unexpected findings: Patient is alert and oriented to person, place, and time. Speech is clear; affect and facial expressions are appropriate to situation. Patient cooperative with exam and exhibits pleasant and calm behavior. Dress is appropriate, well-groomed, and proper hygiene. Posture remains erect in wheelchair, with intermittent drift to left side. History of cerebrovascular accident with left-sided hemiplegia.  Pupils are 4 mm equal and round. Reaction intact right and accommodation intact right eye. Left pupil 2 mm, round nonreactive to light and accommodation. Upper extremity hand grips, nonsymmetrical due to left-sided weakness. Right hand grip and upper extremity strength strong at 4/5. Left lower extremity residual weakness, rated at 1/5, right lower extremity strength 4/5. Sensation intact to light touch bilaterally, R > L. Unable to assess Romberg and Pronator drift.
Communication with the Patient When Completing a Health Record
To be an effective healthcare member, a nurse must first learn to be an effective communicator. Good communication skills can lay the foundation for and maintain an established healthy working environment. Research proves maintaining a healthy work environment is directly linked to improved patient care outcomes and excellence in nursing practice standards (AACN, 2016). These standards include skilled communication: “Nurses must be as proficient in communication skills as they are in clinical skills” (AACN, 2016, p. 2). The same techniques used for communicating with your peers can be used to communicate with your patients.
To minimize data errors during the exchange of information, best practice techniques can be implemented when communicating. Communicate face-to-face with the patient, when possible, and maintain good eye contact (unless not culturally acceptable). After asking a question, allow the patient time to respond. Listen while the patient is talking. It is unprofessional to complete other tasks while the patient is communicating (AACN, 2016).
Verbal Communication
Verbal communication is the manner of exchanging information using oral or written words. A patient’s use of verbal communication can identify important details such as education level, developmental considerations, or geographic or cultural origins. Nurses utilize verbal communication several ways when providing patient care. Examples include interactions between patients and family, giving shift reports to the relieving nurse, communicating with physicians, and evaluating interventions.
The Joint Commission has established the Situation, Background, Assessment, Recommendation ( SBAR ) model of communication as the gold standard for best practice when verbally exchanging information between healthcare team members. SBAR is one of the most widely used and recognized mnemonics within the healthcare industry (Shahid & Thomas, 2018). Figure 4.10 shows a sample SBAR.
Nonverbal Communication
Information exchange without the use of words is called nonverbal communication, or body language. A patient’s body language should be observed by the nurse to reveal nonverbalized issues or problems that need to be addressed. Body language includes one’s posture, gait, level of eye contact, gestures, and facial expressions. For example, if a patient is holding on to the wall as they are walking, this could mean they feel unstable, and nursing intervention is needed. Another example might be if a patient is clenching their fist and pacing the room but states that “nothing is wrong.” The nurse should document nonverbal communication as it is observed and what action was taken to resolve the issue.
If verbal and nonverbal communication misalign, the listener can receive confusing messages. The nurse should ensure their own body language matches what they are verbalizing to the patient. For example, if a patient requests assistance to the bathroom and the nurse makes a disgusted facial expression, instantly the patient perceives unprofessional and disrespectful treatment. The patient might feel ashamed and embarrassed, neither of which is considered positive patient care outcomes. Patients need to feel safe and in a positive environment to heal; this includes positive nonverbal communication from the healthcare team.
Barriers to Communication
As the nurse begins communicating with the patient, assessment of any barriers should be conducted. Common barriers to communication include hearing or cognitive impairments, English as a secondary language, and cultural beliefs/practices. Questioning should be adjusted to adapt for any identified barriers. Using an interpreter, facing the patient when speaking, and not using medical terminology are common techniques to adjust communication for identified barriers. Social determents can also be barriers to communication and learning. Communication regarding healthcare management can be equal among patients, but that does not mean all patient outcomes are equal. The Patient Centered Assessment Method (PCAM) was designed to help identify social determinants that might affect a patient’s ability to adhere or understand the importance of adhering to a health plan (Smeets et al., 2021). The PCAM is an evidence-based tool used to assess patient complexity using the social determinants of health. The tool helps validate why some patients engage in the managing of their health care, and why other patients do not. Many organizations are available to assist with questions regarding social determinants. Examples of social determinants linked to poor health promotion include housing, finances, and employment status which may help the nurse identify other factors that are affecting a patient’s ability to manage their health. For example, if a patient is unemployed and has no transportation, this might explain the lack of seeking routine healthcare visits.
The Centers for Disease Control and Prevention (CDC) published the Ten Essential Public Health Services and How They Can Include Addressing Social Determinants of Health Inequities to help improve health inequities.


9.4: Countable Sets
Recall that if \(A=\emptyset\) , then we say that \(A\) has cardinality 0. Also, if \(card(A)=card([n])\) for \(n\in\mathbb{N}\) , then we say that \(A\) has cardinality \(n\) . We have a special way of describing sets that are in bijection with the natural numbers.
Definition 9.36. If \(A\) is a set such that \(card(A)=card(\mathbb{N})\) , then we say that \(A\) is denumerable and has cardinality \(\mathbf{\aleph_0}\) (read “aleph naught").
Notice if a set \(A\) has cardinality \(1,2,\ldots\) , or \(\aleph_0\) , we can label the elements in \(A\) as “first", “second", and so on. That is, we can “count" the elements in these situations. Certainly, if a set has cardinality 0, counting is not an issue. This idea leads to the following definition.
Definition 9.37. A set \(A\) is called countable if \(A\) is finite or denumerable. A set is called uncountable if it is not countable.
Problem 9.38. Quickly justify that each of the following sets is countable. Feel free to appeal to previous problems. Which sets are denumerable?
- \(\{a,b,c\}\)
- Set of odd natural numbers
- Set of even natural numbers
- \(\{\frac{1}{2^n}\mid n\in \mathbb{N}\}\)
- Set of perfect squares in \(\mathbb{N}\)
- \(\mathbb{Z}\)
- \(\mathbb{N}\times \{a\}\)
Utilize Theorem 9.31 or Corollary 9.33 when proving the next result.
Theorem 9.39. Every infinite set contains a denumerable subset.
Theorem 9.40. Let \(A\) and \(B\) be sets such that \(A\) is countable. If \(f:A\to B\) is a bijection, then \(B\) is countable.
For the next proof, consider the cases when \(A\) is finite versus infinite. The contrapositive of Corollary 9.32 should be useful for the case when \(A\) is finite.
Theorem 9.41. Every subset of a countable set is countable.
Theorem 9.42. A set is countable if and only if it has the same cardinality of some subset of the natural numbers.
Theorem 9.43. If \(f:\mathbb{N}\to A\) is a surjective function, then \(A\) is countable.
Loosely speaking, the next theorem tells us that we can arrange all of the rational numbers then count them “first", “second", “third", etc. Given the fact that between any two distinct rational numbers on the number line, there are an infinite number of other rational numbers (justified by taking repeated midpoints), this may seem counterintuitive.
Here is one possible approach for proving the next theorem. Make a table with column headings \(0, 1, -1, 2,-2,\ldots\) and row headings \(1,2,3,4,5,\ldots\) . If a column has heading \(m\) and a row has heading \(n\) , then the entry in the table corresponds to the fraction \(m/n\) . Find a way to zig-zag through the table making sure to hit every entry in the table (not including column and row headings) exactly once. This justifies that there is a bijection between \(\mathbb{N}\) and the entries in the table. Do you see why? But now notice that every rational number appears an infinite number of times in the table. Resolve this by appealing to Theorem 9.41.
Theorem 9.44. The set of rational numbers \(\mathbb{Q}\) is countable.
Theorem 9.45. If \(A\) and \(B\) are countable sets, then \(A\cup B\) is countable.
We would like to prove a stronger result than the previous theorem. To do so, we need an intermediate result.
Theorem 9.46. Let \(\{A_n\}_{n=1}^{\infty}\) be a collection of sets. Define \(B_1:= A_1\) and for each natural number \(n>1\) , define \[B_n:= A_n\setminus \bigcup_{i=1}^{n-1}A_i.\] Then we we have the following:
- The collection \(\{B_n\}_{n=1}^{\infty}\) is pairwise disjoint.
- \(\displaystyle \bigcup_{n=1}^{\infty}A_n =\bigcup_{n=1}^{\infty}B_n\) .
The next theorem states that the union of a countable collection of countable sets is countable. To prove this, consider two cases:
- The collection of sets is finite.
- The collection of sets is infinite.
To handle the first case, use induction together with Theorem 9.45. The second case is substantially more challenging. First, use Theorem 9.46 to construct a collection \(\{B_n\}\) of pairwise disjoint sets whose union is equal to the union of the original collection. Since each \(B_n\) is a subset of one of the sets in the original collection and each of these sets is countable, each \(B_n\) is also countable by Theorem 9.41. This implies that for each \(n\) , we can write \(B_n=\{b_{n,1}, b_{n,2},b_{n,3},\ldots\}\) , where the set may be finite or infinite. From here, we outline two different approaches for continuing. One approach is to construct a bijection from \(\mathbb{N}\) to \(\bigcup_{n=1}^{\infty}B_n\) using Figure 9.2 as inspiration. One thing you will need to address is what to do when a set in the collection \(\{B_n\}\) is finite. For the second approach, define \(f:\bigcup_{n=1}^{\infty}B_n\to \mathbb{N}\) via \(f(b_{n,m})=2^n3^m\) , verify that this function is injective, and then appeal to Theorem 9.41. Try using both of these approaches when tackling the proof of the following theorem.
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Theorem 9.47. Let \(\Delta\) be equal to either \(\mathbb{N}\) or \([k]\) for some \(k\in\mathbb{N}\) . If \(\{A_n\}_{n\in\Delta}\) is a countable collection of sets such that each \(A_n\) is countable, then \(\bigcup_{n\in \Delta}A_n\) is countable.
Do you use the Axiom of Choice when proving the previous theorem? If so, where?
Theorem 9.48. If \(A\) and \(B\) are countable sets, then \(A\times B\) is countable.
Theorem 9.49. The set of all finite sequences of 0’s and 1’s (e.g., \(0110010\) is a finite sequence consisting of 0’s and 1’) is countable.
Theorem 9.50. The collection of all finite subsets of a countable set is countable.


2.1.5: Cell Biology and Basic Genetics
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When Darwin developed his theory, there are two main things that he did not know: (1)where the variation comes from, and (2) how the traits are passed from one generation to the next. In the decades since On the Origin of Species was published, subsequent generations of biologists, natural scientists, geneticists, paleoanthropologists, etc., have conducted research that has refined our knowledge about evolution. We now can address the things that Darwin did not know. Cell biology and genetics explain where variation comes from and how those traits are inherited.
Cell biology
All living organisms are made of cells . Some organisms are single-celled and spend their life made of just one cell and some organisms, like humans, are multi-cellular (have many cells). Here are some basic points that you always have to remember about cells:
- cells are the structural and functional units of all living things
- all cells come from pre-existing cells
- cells contain hereditary information (which is passed from "parent" cells to "daughter" cells during cell division).
Cell function
Cells are often called "the building blocks of life". Not only are all living things made of cells (the 'building block' part), they also are living functional units themselves. This is because cells do the following:
- obtain nutrients and other molecules through the cellular membrane (or cell wall) and convert that to biologically useful energy and useful molecules for building new cells (such as proteins and nucleic acids)
- make more cells (reproduction)
Types of cells
There are two types of cells: prokaryotic and eukaryotic. Prokaryotes include bacteria and blue-green algae. These are single-celled organisms in which the genetic material is not distinctly separated from the other components of the cell. Eukaryotes include everyone else. These can be single-celled, but are more often multi-celled organisms in which the genetic material is separated from all of the other cellular components (by something called the nucleus ). Because our focus is on humans, the rest of our discussion will center on eukaryotes and eukaryotic cells.
Cell structure
Figure \(\PageIndex{1}\) - A typical animal cell and its parts
Each animal cell consists of a selectively permeable membrane (plants have cell walls) which contains the cytoplasm.
Organelles are like "little organs" within the cytoplasm, each performing a different vital cellular function. These are some of the more important (to us) organelles.
- Mitochondria: These are the "powerhouses" of the cell, responsible for production of the energy-rich molecule, ATP (adenine triphosphate), which powers the activities of the cell. Each cell has hundreds to thousands of mitochondria. Mitochondria have their own DNA, known as mitochondrial DNA (mtDNA), which is different from nuclear DNA (see below). Mitochondrial DNA has become an important tool in evolutionary research.
- Endoplasmic reticulum: This is a transport network for molecules that have specific destinations or require certain modifications. It comes in two types: smooth and rough. Rough endoplasmic reticulum has ribosomes (see below) on the surface.
- Golgi apparatus: Processes and packages large molecules such as proteins and lipids (fats) that are produced by the cell.
- Lysosomes: Breaks down non-usable organelles, food, viruses, and bacteria using enzymes.
- Ribosomes: Protein synthesis
- Nucleus: This is where you'll find the hereditary material, DNA (deoxyribonucleic acid) and RNA (ribonucleic acid).
Exercise \(\PageIndex{1}\)
Explore the cell!!
In this interactive exercise, you can identify the parts of a typical animal cell and review the functions of the various organelles.
Typical animal cell
DNA
DNA (deoxyribonucleic acid) is the genetic blueprint of the cell. The structure of DNA is what's known as a "double helix", which looks much like a twisted ladder. The basic unit of DNA is a molecule called a nucleotide. Nucleotides are made of a sugar (deoxyribonucleic acid), a phosphate group, and a nitrogenous base. There are 4 different types of nucleotides, depending on what type of nitrogenous base they have.
Figure \(\PageIndex{2}\) - DNA structure
- adenine (A)
- thymine (T)
- cytosine (C)
- guanine (G)
These are separated into 2 groups:
- purines: adenine and guanine
- pyrimidines: cytosine and thymine
On the DNA "rungs", the bases must go in pairs: adenine (A) always bonds with thymine (T) and cytosine (C) with guanine (G). So, for base-pairs remember it's always A+T and C+G! (That rule is sometimes called "The base-pair rule".) If you want to explore further, check out what the Human Genome Project has learned about the human DNA sequence
RNA structure
RNA (ribonucleic acid) has a very similar structure to DNA, except that it's single-stranded, has a different sugar (ribonucleic acid, rather than deoxyribonucleic acid), and does not contain thymine (T). Instead, it contains uracil (U), which like thymine, bonds with adenine (A). So, in this case, the base-pair rule is A+U and C+G.
DNA function
For our purposes, we will concern ourselves with two of DNA's functions: replication and protein synthesis.
DNA replication
In this case, the goal is to replicate a DNA double helix to produce more DNA. DNA replication occurs in the nucleus of the cell.
The basic steps of DNA replication are:
1) The DNA is "unzipped" into two single strands.
2) Each unzipped strand acts as a template for reproduction of the complimentary strand.
3) The product is two copies of the original DNA, each containing one strand from the original DNA and one strand that has been added on.
Here's a slightly more thorough illustration of the process: DNA replication - simple.
And for those who are really curious about the details of the process: DNA replication - complex.
And for those who are looking for a little fun with their DNA replication: DNA - the double helix game
Important note: The important thing to remember about DNA replication is that when the complimentary strand is created, sometimes accidents or mistakes happen. For example, a G gets matched up with a T instead of the proper A. This is where some of the variation in hereditary information comes from.
Protein synthesis
Figure \(\PageIndex{3}\) - Protein synthesis
Proteins are the most common large molecules in cells. Bones, muscles, and red blood cells (among lots of other body parts) are made mostly of proteins. Therefore, the production of proteins is obviously a very important process in the body.
Proteins are made of smaller units called amino acids. There are 20 different amino acids, 9 of which are "essential amino acids", which means that they must be consumed through the diet, rather than being synthesized by the body. The sequence of amino acids in a protein determines its structure and function.
The base pair sequence of the DNA molecule is known as the genetic code . The genetic code consists of 3-base sequences called codons .(Remember our friends A, T, G, and C -- the nitrogenous bases? That's what we're talking about here.) Each codon either codes for an amino acid, or signals that the protein chain is starting (an initiation codon) or stopping (a termination codon). The table on the right shows which codons code for which amino acids.
Each DNA molecule contains the information to make up many different proteins. A portion of a DNA molecule responsible for making up a single protein (or sometimes just part of a protein, called a polypeptide) is a gene . Therefore, each DNA molecule consists of many genes, that code for many proteins.
Protein synthesis has two basic steps: (1) transcription and (2) translation.
Definitions
Transcription
1) The gene (DNA) is copied onto RNA. The RNA copy of the gene is called the messenger RNA (or mRNA).
2) The mRNA leaves the nucleus and goes to the rough endoplasmic reticulum. (Remember this organelle from the cell structure section above?)
Translation
1) The mRNA goes into the ribosomes, where tRNA (translation RNA) reads the mRNA.
2) As the tRNA reads the mRNA, it attaches complementary amino acids to the newly synthesized amino acid chain (AKA the growing protein chain).
Note
Important note: The important thing to remember about protein synthesis is that when a protein is synthesized, sometimes accidents or mistakes happen. This is where some of the variation in protein synthesis comes from. Additionally, remember the possible mistakes in DNA replication? If the original DNA strand is changed so that a gene is altered, that can affect protein synthesis as well.
This video has fantastic digital animations illustrating DNA replication and protein synthesis.
Cell replication
To make more cells, we not only have to have replicated DNA, we have to replicate the rest of the cell as well in order to physically divide that replicated DNA. For body cells (AKA somatic cells), this process is called mitosis . For sex cells (AKA sperm and eggs), this process is called meiosis . But first, a little bit about DNA and chromosomes.
Figure \(\PageIndex{4}\) - Chromosome
Chromosomes
Most of the time, DNA in cells is in loosely arranged, dispersed strands. When DNA is like this, it's called chromatin . However, during cell replication DNA condenses into structures called chromosomes .
Each chromosome is made of (the numbers reference the illustration to the right): 1) two "sister" chromatids, 2) a centromere, which is a highly condensed area on the chromosome, 3) the "short arm" of the chromatid, 4) the "long arm" of the chromatid, and the 5) telomere, which is the end of a chromosome and is involved in both replication and chromosomal stability. In this diagram, the different colors represent genes on the chromosome.
Each somatic cell (sex cells are different) has the same number of chromosomes. This number is consistent within species, but often differs between species. The number of chromosomes in each body cell is called the diploid number . Sex cells have half as many chromosomes as do somatic cells. This number is called the haploid number .
Most species have two copies of each chromosome in each somatic cell. The member-chromosomes of each pair are called homologous chromosomes .
Figure \(\PageIndex{5}\) - A typical human (male) karyotype
On the left is a karyotype , which is an image of the full diploid complement of chromosomes from an individual. Above each number is each pair of homologous chromosomes. This is from a human, who typically have 23 pairs of chromosomes in each somatic cell, for a diploid total of 46 chromosomes. For humans, sex cells normally have 23 chromosomes.
Genes are found on each chromosome at a particular location called a locus (sing.) or loci (pl.). Different versions of the same gene are called alleles . If someone has the same version of a gene (the same alleles) on both of their homologous chromosomes, they have homozygous alleles . If, however, someone has two different versions of a gene (different alleles) on their homologous chromosomes, they have heterozygous alleles . For example, if on this person's chromosome 9 pair they have one allele for blood type A and one allele for blood type B, we would say that they are heterozygous for blood type (AB). If they have two alleles for blood type O, we would say that they are homozygous for blood type (OO).
Note
Explore human chromosomes
Mitosis
Somatic cells (AKA body cells) are replicated via the process of mitosis. This is where all new body cells come from. So, when you scratch yourself on something, mitosis is what makes new skin cells for healing that scratch.
Figure \(\PageIndex{6}\) - Mitosis
Mitosis has 4 phases (note that DNA has already replicated when mitosis starts):
1) Prophase:
- the nuclear envelope disappears
- chromatin condenses and forms chromosomes
- the chromosomes move toward the equator of the cell
2) Metaphase
- the chromosomes are lined up along the equator of the cell
3) Anaphase
- the sister chromatids split up and move toward the opposite ends of the cell
4) Telophase
- the cytoplasm divides between the two cells
- the chromosomes turn back into chromatin
The result is two identical "daughter" somatic cells, each of which contains one chromosome from the "parent" cell and one replicated chromosome.
Here's a video that illustrates the process: mitosis in action.
Meiosis
Sex cells (eggs and sperm) are replicated via the process of meiosis. For human females this process occurs and is completed before birth, but for males meiosis begins at puberty and continues throughout their lifespans. Meiosis is more complicated than mitosis in that it has two "cycles".
Figure \(\PageIndex{7}\) - Meiosis
1) Prophase 1
- chromatin condenses and forms chromosomes
- DNA is exchanged between homologous chromosomes. This is called homologus recombination . Recombination is an important source of genetic variation.
- the chromosomes start to line up at the equator of the cell
2) Metaphase 1
- the chromosomes are lined up along the equator of the cell
3) Anaphase 1
- the homologous chromosomes move toward the opposite ends of the cell
4) Telophase 1
- the cytoplasm divides between the two cells
- the chromosomes uncoil and turn back into chromatin
5) Prophase 2
- nuclear envelope disappears
- chromatin condenses into chromosomes
- the chromosomes start to line up at the equator of the cell
6) Metaphase 2
- the chromosomes are lined up along the equator of the cell
7) Anaphase 2
- the sister chromatids split apart at the centromeres and move toward the opposite ends of the cell
8) Telophase 2
- chromosomes turn back into chromatin
- the cytoplasm divides between the two cells
Here's a video that illustrates the process: meiosis in action
Errors during mitosis and meiosis
Figure \(\PageIndex{8}\) - The karyotype of an individual with Down syndrome
The chromosomes separate during meiosis 1 and the sister chromatids separate in meiosis 2 and mitosis. If this proceeds normally, it's called disjunction . If the separations occur abnormally, it's called nondisjunction . What results is an abnormal number of chromosomes (either too many, AKA trisomy OR too few, AKA monosomy).
Some of the more familiar nondisjunctions include:
Down Syndrome - trisomy of chromosome 21 (a karyotype of this is to the right)
Klinefelter Syndrome - extra X chromosomes in males (ex. XXY)
Turner Syndrome - lacking an X chromosome in females (XO)


2.4.2: Interpreting Graphs of Proportional Relationships
Lesson
Let's read stories from the graphs of proportional relationships.
Exercise \(\PageIndex{1}\): What Could the Graph Represent?
Here is a graph that represents a proportional relationship.
- Invent a situation that could be represented by this graph.
- Label the axes with the quantities in your situation.
- Give the graph a title.
- There is a point on the graph. What are its coordinates? What does it represent in your situation?
Exercise \(\PageIndex{2}\): Tyler's Walk
Tyler was at the amusement park. He walked at a steady pace from the ticket booth to the bumper cars.
- The point on the graph shows his arrival at the bumper cars. What do the coordinates of the point tell us about the situation?
- The table representing Tyler's walk shows other values of time and distance. Complete the table. Next, plot the pairs of values on the grid.
- What does the point \((0,0)\) mean in this situation?
- How far away from the ticket booth was Tyler after 1 second? Label the point on the graph that shows this information with its coordinates.
- What is the constant of proportionality for the relationship between time and distance? What does it tell you about Tyler's walk? Where do you see it in the graph?
| time (seconds) | distance (meters) |
|---|---|
| \(0\) | \(0\) |
| \(20\) | \(25\) |
| \(30\) | \(37.5\) |
| \(40\) | \(50\) |
| \(1\) |
Are you ready for more?
If Tyler wanted to get to the bumper cars in half the time, how would the graph representing his walk change? How would the table change? What about the constant of proportionality?
Exercise \(\PageIndex{3}\): Seagulls Eat What?
4 seagulls ate 10 pounds of garbage. Assume this information describes a proportional relationship.
- Plot a point that shows the number of seagulls and the amount of garbage they ate.
- Use a straight edge to draw a line through this point and \((0,0)\).
- Plot the point \((1,k)\) on the line. What is the value of \(k\)? What does the value of \(k\) tell you about this context?
Summary
For the relationship represented in this table, \(y\) is proportional to \(x\). We can see in the table that \(\frac{5}{4}\) is the constant of proportionality because it’s the \(y\) value when \(x\) is 1.
The equation \(y=\frac{5}{4}x\) also represents this relationship.
| \(x\) | \(y\) |
|---|---|
| \(4\) | \(5\) |
| \(5\) | \(\frac{25}{4}\) |
| \(8\) | \(10\) |
| \(1\) | \(\frac{5}{4}\) |
Here is the graph of this relationship.
If \(y\) represents the distance in feet that a snail crawls in \(x\) minutes, then the point \((4,5)\) tells us that the snail can crawl 5 feet in 4 minutes.
If \(y\) represents the cups of yogurt and \(x\) represents the teaspoons of cinnamon in a recipe for fruit dip, then the point \((4,5)\) tells us that you can mix 4 teaspoons of cinnamon with 5 cups of yogurt to make this fruit dip.
We can find the constant of proportionality by looking at the graph, because \(\frac{5}{4}\) is the \(y\)-coordinate of the point on the graph where the \(x\)-coordinate is 1. This could mean the snail is traveling \(\frac{5}{4}\) feet per minute or that the recipe calls for \(1\frac{1}{4}\) cups of yogurt for every teaspoon of cinnamon.
In general, when \(y\) is proportional to \(x\), the corresponding constant of proportionality is the \(y\)-value when \(x=1\).
Glossary Entries
Definition: Coordinate Plane
The coordinate plane is a system for telling where points are. For example. point \(R\) is located at \((3,2)\) on the coordinate plane, because it is three units to the right and two units up.
Definition: Origin
The origin is the point \((0,0)\) in the coordinate plane. This is where the horizontal axis and the vertical axis cross.
Practice
Exercise \(\PageIndex{4}\)
There is a proportional relationship between the number of months a person has had a streaming movie subscription and the total amount of money they have paid for the subscription. The cost for 6 months is $47.94. The point \((6,47.94)\) is shown on the graph below.
- What is the constant of proportionality in this relationship?
- What does the constant of proportionality tell us about the situation?
- Add at least three more points to the graph and label them with their coordinates.
- Write an equation that represents the relationship between \(C\), the total cost of the subscription, and \(m\), the number of months.
Exercise \(\PageIndex{5}\)
The graph shows the amounts of almonds, in grams, for different amounts of oats, in cups, in a granola mix. Label the point \((1,k)\) on the graph, find the value of \(k\), and explain its meaning.
Exercise \(\PageIndex{6}\)
To make a friendship bracelet, some long strings are lined up then taking one string and tying it in a knot with each of the other strings to create a row of knots. A new string is chosen and knotted with the all the other strings to create a second row. This process is repeated until there are enough rows to make a bracelet to fit around your friend's wrist.
Are the number of knots proportional to the number of rows? Explain your reasoning.
(From Unit 2.3.3)
Exercise \(\PageIndex{7}\)
What information do you need to know to write an equation relating two quantities that have a proportional relationship?
(From Unit 2.3.3)


7.2: Quantitative or Qualitative Information
One of the most obvious ways to categorize information is by whether it is quantitative or qualitative. Some sources contain either quantitative information or qualitative information, but sources often contain both.
Many people first think of information as something like what’s in a table or spreadsheet of numbers and words. But information can be conveyed in more ways than textually or numerically.
Quantitative Information – Involves a measurable quantity—numbers are used. Some examples are length, mass, temperature, and time. Quantitative information is often called data, but can also be things other than numbers.
Qualitative Information – Involves a descriptive judgment using concept words instead of numbers. Gender, country name, animal species, and emotional state are examples of qualitative information.
Take a quick look at the example table below. Another way we could display the table’s numerical information is in a graphic format —listing the students’ ages or GPAs on a bar chart, for example, rather than in a list of numbers. Or, all the information in the table could be displayed instead as a video of each student giving those details about themselves.
| Last Name | First Name | Age | Rank | Major | Gender | Current GPA | Photo |
| Adams | Grace | 19 | Sophomore | English | Female | 3.78 | |
| Bloomfield | Erika | 21 | Junior | Physics | Female | 3.89 | |
| Chow | Kimmie | 20 | Senior | Political Science | Female | 3.77 | |
| Crutchfield | Seth | 23 | Senior | Psychology | Male | 3.58 | |
| Fitch | Fredrick | 18 | Freshman | Art | Male | 4.0 | |
| Grover | Oscar | 26 | Junior | Biology | Male | 3.32 |
Increasingly, other formats such as images, sound, and video may be used as information or used to convey information. Some examples:


3.11: Bibliography
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Bourdieu, Pierre. 1970. “The Berber House or the World Reversed.” Social Science Information 9 (2): 151–170.
Douglas, Mary. 1966. Purity and Danger: An Analysis of Concepts of Pollution and Taboo . London: Routledge & Kegan Paul.
Ferguson, James. 2006. Global Shadows: Africa in the Neoliberal World Order . Durham, NC: Duke University Press.
Geertz, Clifford. 1973. The Interpretation of Cultures: Selected Essays . New York: Basic Books.
Gennep, Arnold van. 1960. The Rites of Passage . Translated by Monika B. Vizedom and Gabrielle L. Caffe. Chicago: University of Chicago Press.
Hallowell, A. Irving. 1992. The Ojibwa of Berens River, Manitoba: Ethnography into History . Edited by Jennifer S. H. Brown. Fort Worth, TX: Harcourt Brace Jovanovich.
Kohn, Eduardo. 2013. How Forests Think: Toward an Anthropology beyond the Human . Berkeley: University of California Press.
Li, Ling. 2011. “‘Performing’ Bribery in China: Guanxi-Practice, Corruption with a Human Face.” Journal of Contemporary China 20 (68): 1–20. https://ssrn.com/abstract=1712390 .
Lyons, Harriet D. 2011. “Dame Mary Douglas.” In Fifty Key Anthropologists , edited by Robert J. Gordon, Andrew P. Lyons, and Harriet D. Lyons, 46–51. New York: Routledge.
Peters-Golden, Holly. 2002. Culture Sketches: Case Studies in Anthropology . 3rd ed. Boston: McGraw-Hill.
Scalco, Patricia. 2019. “Weaving Value: Selling Carpets in the Liminal Space of Istanbul’s Grand Bazaar.” Anthropology Today 35 (5): 7–10.
Tylor, Edward Burnett. 1873. Primitive Culture: Researches into the Development of Mythology, Philosophy, Religion, Language, Art, and Custom . 2nd ed. 2 vols. London: John Murray.
Williamson, Kenneth. 2012. “Night Becomes Day: Carnival, Contested Spaces, and the Black Movement in Bahia.” The Journal of Latin American and Caribbean Anthropology 17 (2): 257–278.
Yin, Xiangru. 2017. “An Analysis of Corruption in China: The Guanxi Network of Chinese High Level Officials and Governors.” Master’s thesis, Clark University. International Development, Community, and Environment (IDCE) (140). https://commons.clarku.edu/idce_masters_papers/140 .
Zhang, Lijia. 2013. “Author: In China, ‘Everyone Is Guilty of Corruption.’” CNN. October 24, 2013. http://edition.cnn.com/2013/10/23/op...n-lijia-zhang/ .


8.6: Acknowledgements
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I am grateful to the organisers of the conference — Kathryn Piquette and Ruth Whitehouse — and also to the anonymous reviewers, as well as to Dr A. Karnava, Dr P. Muhly, Dr V. Petrakis, Dr G. Rethemiotakis and Dr J. Smith for their valuable suggestions on the first draft of this paper. Special thanks are due to Dr C. Stray who read the latter and corrected the English. The final draft was written in the fall of 2011 during my S. Seeger Research Fellowship at the Center for Hellenic Studies, Princeton University. Photographs 2–4 and 7–21 are reproduced courtesy of the Heraklion Museum – Archaeological Receipts Fund.


6.2: Age Related Changes to the Skeletal Muscle System
As one ages the mass of skeletal muscle decreases throughout the body. The muscles atrophy resulting in a decreased number of muscle fibers and decrease in the size of the remaining fibers. As skeletal muscles are postmitotic they can not reproduce to produce new fibers. As a result the fibers are replaced with fat tissue. The muscle fiber loss has been reported to be as high as 30% between the ages of 30 and 80. Proper nutrition and exercise can slow the loss muscle cells, but heredity also seems to be a factor.
Along with the loss of skeletal muscle mass comes the loss of skeletal muscle strength. For most individuals there is only a ten to twenty percent reduction in strength up to the age of seventy. After the age of seventy the reduction in strength may increase to fifty percent.
Contributors and Attributions
- Age Related Changes to the Skeletal Muscle System. Authored by : Kelly, Jessica. Provided by : Herkimer College. Located at : NA. Project : AtD Course. License : CC BY: Attribution


17.6: Backwards Martingales
Basic Theory
A backwards martingale is a stochastic process that satisfies the martingale property reversed in time, in a certain sense. In some ways, backward martingales are simpler than their forward counterparts, and in particular, satisfy a convergence theorem similar to the convergence theorem for ordinary martingales. The importance of backward martingales stems from their numerous applications. In particular, some of the fundamental theorems of classical probability can be formulated in terms of backward martingales.
Definitions
As usual, we start with a stochastic process \( \bs{Y} = \{Y_t: t \in T\} \) on an underlying probability space \( (\Omega, \mathscr{F}, \P) \), having state space \( \R \), and where the index set \( T \) (representing time) is either \( \N \) (discrete time) or \( [0, \infty) \) (continuous time). So to review what all this means, \( \Omega \) is the sample space, \( \mathscr{F} \) the \( \sigma \)-algebra of events, \( \P \) the probability measure on \( (\Omega, \mathscr{F}) \), and \( Y_t \) is a random variable with values in \( \R \) for each \( t \in T \). But at this point our formulation diverges. Suppose that \( \mathscr{G}_t \) is a sub \( \sigma \)-algebra of \( \mathscr{F} \) for each \( t \in T \), and that \( \mathfrak G = \{\mathscr{G}_t: t \in T\} \) is decreasing so that if \( s, \, t \in T \) with \( s \le t \) then \( \mathscr{G}_t \subseteq \mathscr{G}_s \). Let \( \mathscr{G}_\infty = \bigcap_{t \in T} \mathscr{G}_t \). We assume that \( Y_t \) is measurable with respect to \( \mathscr{G}_t \) and that \( \E(|Y_t|) \lt \infty \) for each \( t \in T \).
The process \( \bs Y = \{Y_t: t \in T\} \) is a backwards martingale (or reversed martingale ) with respect to \( \mathfrak G = \{\mathscr{G}_t: t \in T\} \) if \(\E(Y_s \mid \mathscr{G}_t) = Y_t\) for all \(s, \, t \in T\) with \(s \le t\).
A backwards martingale can be formulated as an ordinary martingale by using negative times as the indices. Let \( T^- = \{-t: t \in T\} \), so that if \( T = \N \) (the discrete case) then \( T^- \) is the set of non-positive integers, and if \( T = [0, \infty) \) (the continuous case) then \( T^- = (-\infty, 0] \). Recall also that the standard martingale definitions make sense for any totally ordered index set.
Suppose again that \( \bs Y = \{Y_t: t \in T\} \) is a backwards martingale with respect to \( \mathfrak G = \{\mathscr{G}_t: t \in T\} \). Let \( X_t = Y_{-t} \) and \( \mathscr{F}_t = \mathscr{G}_{-t} \) for \( t \in T^- \). Then \( \bs X = \{X_t: t \in T^-\} \) is a martingale with respect to \( \mathfrak F = \{\mathscr{F}_t: t \in T^-\} \).
Proof
Since \( \mathfrak G \) is a decreasing family of sub \( \sigma \)-algebras of \( \mathscr{F} \), the collection \( \mathfrak F \) is an increasing family of sub \( \sigma \)-algebras of \( \scr F \), and hence is a filtration. Next, \( X_t = Y_{-t} \) is measurable with respect to \( \mathscr{G}_{-t} = \mathscr{F}_t \) for \( t \in T^- \), so \( \bs X \) is adapted to \( \mathfrak F \). Finally, if \( s, \, t \in T^- \) with \( s \le t \) then \( -t \le -s \) so \[ \E(X_t \mid \mathscr{F}_s) = \E(Y_{-t} \mid \mathscr{G}_{-s}) = Y_{-s} = X_s \]
Most authors define backwards martingales with negative indices, as above, in the first place. There are good reasons for doing so, since some of the fundamental theorems of martingales apply immediately to backwards martingales. However, for the applications of backwards martingales, this notation is artificial and clunky, so for the most part, we will use our original definition. The next result is another way to view a backwards martingale as an ordinary martingale. This one preserves nonnegative time, but introduces a finite time horizon. For \( t \in T \), let \( T_t = \{s \in T: s \le t\} \), a notation we have used often before.
Suppose again that \( \bs Y = \{Y_t: t \in T\} \) is a backwards martingale with respect to \( \mathfrak G = \{\mathscr{G}_t: t \in T\} \). Fix \( t \in T \) and define \( X^t_s = Y_{t-s} \) and \( \mathscr{F}^t_s = \mathscr{G}_{t-s} \) for \( s \in T_t \). Then \( \bs{X}^t = \{X^t_s: s \in T_t\} \) is a martingale relative to \( \mathfrak{F}^t = \{\mathscr{F}^t_s: s \in T_t\} \).
Proof
The proof is essentially the same as for the previous result. Since \( \mathfrak G \) is a decreasing family of sub \( \sigma \)-algebras of \( \mathscr{F} \), the collection \( \mathfrak{F}^t \) is an increasing family of sub \( \sigma \)-algebras of \( \scr F \), and hence is a filtration. Next, \( X^t_s = Y_{t-s} \) is measurable with respect to \( \mathscr{G}_{t-s} = \mathscr{F}^t_s \) for \( s \in T_t \), so \( \bs{X}^t \) is adapted to \( \mathfrak{F}^t \). Finally, if \( r, \, s \in T_t \) with \( r \le s \) then \( t - s \le t - r \) so \[ \E(X^t_s \mid \mathscr{F}^t_r) = \E(Y_{t-s} \mid \mathscr{G}_{t-r}) = Y_{t-r} = X^t_r \]
Properties
Backwards martingales satisfy a simple and important property.
Suppose that \( \bs Y = \{Y_t: t \in T\} \) is a backwards martingale with repsect to \( \mathfrak G = \{\mathscr{G}_t: t \in T\} \). Then \( Y_t = \E(Y_0 \mid \mathscr{G}_t) \) for \( t \in T \) and hence \( \bs Y \) is uniformly integrable.
Proof
The fact that \( Y_t = \E(Y_0 \mid \mathscr{G}_t) \) for \( t \in T \) follows directly from the definition of a backwards martingale. Since we have assumed that \( \E(|Y_0|) \lt \infty \), it follows from a basic property that \( \bs Y \) is uniformly integrable.
Here is the Doob backwards martingale , analogous to the ordinary Doob martingale, and of course named for Joseph Doob. In a sense, this is the converse to the previous result.
Suppose that \( Y \) is a random variable on our probability space \( (\Omega, \mathscr{F}, \P) \) with \( \E(|Y|) \lt \infty \), and that \(\mathfrak G = \{\mathscr{G}_t: t \in T\} \) is a decreasing family of sub \( \sigma \)-algebras of \( \mathscr{F} \), as above. Let \( Y_t = \E(Y \mid \mathscr{G}_t) \) for \( t \in T \). Then \( \bs Y = \{Y_t: t \in T\} \) is a backwards martingale with respect to \( \mathfrak G \).
Proof
By definition, \( Y_t = \E(Y \mid \mathscr{G}_t) \) is measurable with respect to \( \mathscr{G}_t \). Also, \[ \E(|Y_t|) = \E[|\E(Y \mid \mathscr{G}_t)|] \le \E[\E(|Y| \mid \mathscr{G}_t)] = \E(|Y|) \lt \infty, \quad t \in T \] Next, suppose that \( s, \, t \in T \) with \( s \le t \). Then \( \mathscr{G}_t \subseteq \mathscr{G}_s \) so by the tower property of conditional expected value, \[ \E(Y_s \mid \mathscr{G}_t) = \E[\E(Y \mid \mathscr{G}_s) \mid \mathscr{G}_t] = \E(Y \mid \mathscr{G}_t) = Y_t \]
The convergence theorems are the most important results for the applications of backwards martingales. Recall once again that for \( k \in [1, \infty) \), the k -norm of a real-valued random variable \( X \) is \[ \|X\|_k = \left[\E\left(|X|^k\right)\right]^{1/k} \] and the normed vector space \( \mathscr{L}_k \) consists of all \( X \) with \( \|X\|_k \lt \infty \). Convergence in the space \( \mathscr{L}_1 \) is also referred to as convergence in mean , and convergence in the space \( \mathscr{L}_2 \) is referred to as convergence in mean square . Here is the primary backwards martingale convergence theorem :
Suppose again that \( \bs Y = \{Y_t: t \in T\} \) is a backwards martingale with respect to \( \mathfrak G = \{\mathscr{G}_t: t \in T\} \). Then there exists a random variable \( Y_\infty \) such that
- \( Y_t \to Y_\infty \) as \( t \to \infty \) with probability 1.
- \( Y_t \to Y_\infty \) as \( t \to \infty \) in mean.
- \( Y_\infty = \E(Y_0 \mid \mathscr{G}_\infty) \).
Proof
The proof is essentially the same as the ordinary martingale convergence theorem if we use the martingale constructed from \( \bs Y \) above . So, fix \( t \in T \) and let \( T_t = \{s \in T: s \le t\} \). Let \( X^t_s = Y_{t - s} \) and \( \mathscr{F}^t_s = \mathscr{G}_{t - s} \) for \( s \in T_t \), so that \( \bs{X}^t = \{X^t_s: s \in T_t\} \) is a martingale relative to \( \mathfrak{F}^t = \{\mathscr{F}^t_s: s \in T_t\} \). Now, for \( a, \, b \in \R \) with \( a \lt b \), let \( U_t(a, b) \) denote the number of up-crossings of \( [a, b] \) by \( \bs{X}^t \) on \( T_t \). Note that \( U_t(a, b) \) is also the number of down-crossings of \( [a, b] \) by \( \bs Y \) on \( T_t \). By the up-crossing inequality applied to the martingale \( \bs{X}^t \), \[\E[U_t(a, b)] \le \frac{1}{b - a}[\E(|X_t|) + |a|] = \frac{1}{b - a} [\E(|Y_0|) + |a|] \] Now let \( U_\infty(a, b) \) denote the number of down-crossings of \( [a, b] \) by \( \bs Y \) on all of \( T \). Since \( U_t \uparrow U_\infty \) as \( t \to -\infty \) it follows from the monotone convergence theorem that \[ E[U_\infty(a, b)] \le \frac{1}{b - a} [\E(|Y_0|) + |a|]\] Hence with probability 1, \( U_\infty(a, b) \lt \infty \) for every \( a, \, b \in \Q \) with \( a \lt b \). By the characterization of convergence in terms of down-crossings (completely analogous to the one for up-crossings), there exists a random variable \( Y_{\infty} \) with values in \( \R^* = \R \cup \{-\infty, \infty\} \) such that \( Y_t \to Y_{\infty} \) as \( t \to \infty \). By Fatou's lemma, \[ \E(|Y_\infty|) \le \liminf_{t \to \infty} \E(|Y_t|) \le \E(|Y_0|) \lt \infty \] In particular, \( \P(Y_\infty \in \R) = 1 \). Since \( \bs Y \) is uniformly integrable, and \( Y_\infty \in \mathscr{L}_1 \), it follows that \( Y_t \to Y_\infty \) as \( t \to \infty \) in \( \mathscr{L}_1 \) also.
It remains to show that \( Y_\infty = \E(Y_0 \mid \mathscr{G}_\infty) \). Let \( A \in \mathscr{G}_\infty \). Then \( A \in \mathscr{G}_t \) for every \( t \in T \). Since \( Y_t = \E(Y_0 \mid \mathscr{G}_t) \) it follows by definition that \( \E(Y_t; A) = \E(Y_0; A) \) for every \( t \in T \). Letting \( t \to \infty \) and using the dominated convergence theorem, gives \( \E(Y_\infty ; A) = \E(Y_0; A) \). Hence \( Y_\infty = \E(Y_0 \mid \mathscr{G}_\infty) \).
As a simple extension of the last result, if \( Y_0 \in \mathscr{L}_k \) for some \( k \in [1, \infty) \) then the convergence is in \( \mathscr{L}_k \) also.
Suppose again that \( \bs Y = \{Y_t: t \in T\} \) is a backwards martingale relative to \( \mathfrak G = \{\mathscr{G}_t: t \in T\} \). If \( Y_0 \in \mathscr{L}_k \) for some \( k \in [1, \infty) \) then \( Y_t \to Y_\infty \) as \( t \to \infty \) in \(\mathscr{L}_k\).
Proof
The previous result applies, of course, so we know that there exists a random variable \( Y_\infty \in \mathscr{L}_1 \) such that \( Y_t \to Y_\infty \) as \( t \to \infty \) with probability 1 and in \( \mathscr{L}_1 \). The function \( x \mapsto |x|^k \) is convex on \( \R \) so by Jensen's inequality for conditional expected value, \[ \E(|Y_t|^k) = \E[|\E(Y_0 \mid \mathscr{G}_t)|^k] \le \E[\E(|Y_0|^k \mid \mathscr{G}_t] = \E(|Y_0|^k) \lt \infty \] so \( Y_t \in \mathscr{L}_k \) for every \( t \in T \). By Fatou's lemma, \[ \E(|Y_\infty|^k) \le \liminf_{t \to \infty} \E(|Y_t|^k) \le \E(|Y_0|^k) \lt \infty \] so \( Y_\infty \in \mathscr{L}_k \) also. Next, since \( Y_t = \E(Y_0 \mid \mathscr{G}_t) \) and \( Y_\infty \) is measurable with respect to \( \mathscr{G}_t \), we can use Jensen's inequality again to get \[ |Y_t - Y_\infty|^k = |\E(Y_0 - Y_\infty \mid \mathscr{G}_t)|^k \le \E(|Y_0 - Y_\infty|^k \mid \mathscr{G}_t) \] It follows that the family of random variables \( \{|Y_t - Y_\infty|^k: t \in T\} \) is uniformly integrable, and hence \( \E(|Y_t - Y_\infty|^k) \to 0 \) as \( t \to \infty \).
Applications
The Strong Law of Large Numbers
The strong law of large numbers is one of the fundamental theorems of classical probability. Our previous proof required that the underlying distribution have finite variance. Here we present an elegant proof using backwards martingales that does not require this extra assumption. So, suppose that \( \bs X = \{X_n: n \in \N_+\} \) is a sequence of independent, identically distributed random variables with common mean \( \mu \in \R \). In statistical terms, \( \bs X \) corresponds to sampling from the underlying distribution. Next let \[ Y_n = \sum_{i=1}^n X_i, \quad n \in \N \] so that \( \bs Y = \{Y_n: n \in \N\} \) is the partial sum process associated with \( \bs X \). Recall that the sequence \( \bs Y \) is also a discrete-time random walk. Finally, let \( M_n = Y_n / n \) for \( n \in \N_+ \) so that \( \bs M = \{M_n: n \in \N_+\} \) is the sequence of sample means .
The law of large numbers
- \( M_n \to \mu \) as \( n \to \infty \) with probability 1.
- \( M_n \to \mu \) as \( n \to \infty \) in mean.
Proof
As usual, let \( (\Omega, \mathscr{F}, \P) \) denote the underlying probability space. Also, equalities involving random variables (and particularly conditional expected values) are assumed to hold with probability 1. Now, for \( n \in \N \), let \[ \mathscr{G}_n = \sigma\{Y_n, Y_{n+1}, Y_{n+2}, \ldots\} = \sigma\{Y_n, X_{n+1}, X_{n+2} \ldots\}\] so that \( \mathfrak G = \{\mathscr{G}_n: n \in \N\} \) is a decreasing family of sub \( \sigma \)-algebras of \( \mathscr{F} \). The core of the proof is to show that \( \bs M \) is a backwards martingale relative to \( \mathfrak G \). Let \( n \in \N_+ \). Clearly \( M_n \) is measurable with respect to \( \mathscr{G}_n \). By independence, \( \E(X_i \mid \mathscr{G}_n) = \E(X_i \mid Y_n) \) for \( i \in \{1, 2, \ldots, n\} \). By symmetry (the sequence \( \bs X \) is exchangeable), \( \E(X_i \mid Y_n) = \E(X_j \mid Y_n) \) for \( i, \, j \in \{1, 2, \ldots, n\} \). Hence for \( i \in \{1, 2, \ldots, n\} \) \[ Y_n = \E(Y_n \mid \mathscr{G}_n) = \sum_{j=1}^n E(X_j \mid \mathscr{G}_n) = \sum_{j=1}^n \E(X_i \mid \mathscr{G}_n) = n \E(X_i \mid \mathscr{G}_n) \] so that \( \E(X_i \mid \mathscr{G}_n) = Y_n / n = M_n \) for each \( i \in \{1, 2, \ldots, n\} \). Next, \[ \E(Y_n \mid \mathscr{G}_{n+1}) = \E(Y_{n+1} - X_{n+1} \mid \mathscr{G}_{n+1}) = Y_{n+1} - \E(X_{n+1} \mid \mathscr{G}_{n+1}) = Y_{n+1} - \frac{1}{n+1} Y_{n+1} = \frac{n}{n + 1} Y_{n+1} \] Dividing by \( n \) gives \( \E(M_n \mid \mathscr{G}_{n+1}) = M_{n+1} \) and hence \( \bs M \) is a backwards martingale with respect to \( \mathfrak G \). From the backwards martingale convergence theorem, there exists \( M_\infty \) such that \( M_n \to M_\infty \) as \( n \to \infty \) with probability 1 and in mean. Next, for \( n, \, k \in \N_+ \) simple algebra gives \[ M_{n+k} = \frac{1}{n + k} \sum_{i=1}^k X_i + \frac{n}{n + k} \frac{1}{n} \sum_{i = k + 1}^{k + n} X_i \] Letting \( n \to \infty \) then shows that \[ M_\infty = \lim_{n \to \infty} \frac{1}{n} \sum_{i = k + 1}^{k + n} X_i \] for every \( k \in \N_+ \). Hence \( M_\infty \) is a tail random variable for the IID sequence \( \bs X \). From the Kolmogorov 0-1 law, \( M_\infty \) must be a constant. Finally, convergence in mean implies that the means converge, and since \( \E(M_n) = \mu \) for each \( n \), it follows that \( M_\infty = \mu \).
Exchangeable Variables
We start with a probability space \( (\Omega, \mathscr F, \P) \) and another measurable space \( (S, \mathscr S) \). Suppose that \( \bs X = (X_1, X_2, \ldots) \) is a sequence of random variables each taking values in \( S \). Recall that \( \bs X \) is exchangeable if for every \( n \in \N \), every permutation of \( (X_1, X_2, \ldots, X_n) \) has the same distribution on \( (S^n, \mathscr{S}^n) \) (where \( \mathscr{S}^n \) is the \( n \)-fold product \( \sigma \)-algebra). Clearly if \( \bs X \) is a sequence of independent, identically distributed variables, then \( \bs X \) is exchangeable. Conversely, if \( \bs X \) is exchangeable then the variables are identically distributed (by definition), but are not necessarily independent. The most famous example of a sequence that is exchangeable but not independent is Pólya's urn process, named for George Pólya. On the other hand, conditionally independent and identically distributed sequences are exchangeable. Thus suppose that \( (T, \mathscr{T}) \) is another measurable space and that \( \Theta \) is a random variable taking values in \( T \).
If \( \bs X \) is conditionally independent and identically distributed given \( \Theta \), then \( \bs X \) is exchangeable.
Proof
Implicit in the statement is that the variables in the sequence have a regular conditional distribution \( \mu_\Theta \) given \( \Theta \). Then for every \( n \in \N_+ \), the conditional distribution of every permutation of \( (X_1, X_2, \ldots, X_n) \), given \( \Theta \), is \( \mu^n_\Theta \) on \( (S^n, \mathscr{S}^n) \), where \( \mu_\Theta^n \) is the \( n \)-fold product measure. Unconditionally, the distribution of any permutation is \( B \mapsto \E[\mu^n_\Theta(B)] \) for \( B \in \mathscr{S}^n \).
Often the setting of this theorem arises when we start with a sequence of independent, identically distributed random variables that are governed by a parametric distribution, and then randomize one of the parameters. In a sense, we can always think of the setting in this way: Imagine that \( \theta \in T \) is a parameter for a distribution on \( S \). A special case is the beta-Bernoulli process, in which the success parameter \( p \) in sequence of Bernoulli trials is randomized with the beta distribution. On the other hand, Pólya's urn process is an example of an exchangeable sequence that does not at first seem to have anything to do with randomizing parameters. But in fact, we know that Pólya's urn process is a special case of the beta-Bernoulli process. This connection gives a hint of de Finetti's theorem , named for Bruno de Finetti, which we consider next. This theorem states any exchangeable sequence of indicator random variables corresponds to randomizing the success parameter in a sequence of Bernoulli trials.
de Finetti's Theorem . Suppose that \( \bs X = (X_1, X_2, \ldots) \) is an exchangeable sequence of random variables, each taking values in \( \{0, 1\} \). Then there exists a random variable \( P \) with values in \( [0, 1] \), such that given \( P = p \in [0, 1] \), \( \bs X \) is a sequence of Bernoulli trials with success parameter \( p \).
Proof
As usual, we need some notation. First recall the falling power notation \( r^{(j)} = r (r - 1) \cdots (r - j + 1) \) for \( r \in \R \) and \( j \in \N \). Next for \( n \in \N_+ \) and \( k \in \{0, 1, \ldots, n\} \), let \[ B^n_k = \left\{(x_1, x_2, \ldots, x_n) \in \{0, 1\}^n: \sum_{i=0}^n x_i = k\right\} \] That is, \( B^n_k \) is the set of bit strings of length \( n \) with 1 occurring exactly \( k \) times. Of course, \( \#(B^n_k) = \binom{n}{k} = n^{(k)} / k! \).
Suppose now that \( \bs X = (X_1, X_2, \ldots) \) is an exchangeable sequence of variables with values in \( \{0, 1\} \). For \( n \in \N_+ \) let \( Y_n = \sum_{i=1}^n X_i \) and \( M_n = Y_n / n \). So \( \bs Y = \{Y_n: n \in \N_+\} \) is the partial sum process associated with \( \bs X \) and \( \bs M = \{M_n: n \in \N_+\} \) the sequence of sample means. Let \( \mathscr{G}_n = \sigma\{Y_n, Y_{n+1}, \ldots\} \) and \( \mathscr{G}_\infty = \bigcap_{n=0}^\infty \mathscr{G}_n \). The family of \( \sigma \)-algebras \( \mathfrak G = \{\mathscr{G}_n: n \in \N_+\} \) is decreasing. The key to the proof is to find two backwards martingales and use the backwards martingale convergence theorem .
Let \( m \in \N_+ \) and \( k \in \{0, 1, \ldots m\} \) The crucial insight is that by exchangeability, given \( Y_m = k \), the random vector \( (X_1, X_2, \ldots, X_m) \) is uniformly distributed on \( B^m_k \). So if \( n \in \N_+ \) and \( n \le m \), the random vector \( (X_1, X_2, \ldots, X_n) \), again given \( Y_m = k \), fits the hypergeometric model: a sample of size \( n \) chosen at random and without replacement from a population of \( m \) objects of which \( k \) are type 1 and \( m - k \) are type 0. Thus, if \( j \in \{0, 1, \ldots, n\} \) and \( (x_1, x_2, \ldots, x_n) \in B^n_j \) then \[ \P(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n \mid Y_m = k) = \frac{k^{(j)} (m - k)^{(n - j)}}{m^{(n)}} \] Equivalently, \[ \P(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n \mid Y_m) = \frac{Y_m^{(j)} (m - Y_m)^{(n - j)}}{m^{(n)}} \] Given \( Y_m \), the variables \( (Y_{m+1}, Y_{m+2}, \ldots) \) give no additional information about the distribution of \( (X_1, X_2, \ldots, X_n) \) and hence \[ \P(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n \mid \mathscr{G}_m) =\E[\bs{1}(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n) \mid \mathscr{G}_n] = \frac{Y_m^{(j)} (m - Y_m)^{(n - j)}}{m^{(n)}} \] For fixed \( n \), \( j \), and \( (x_1, x_2, \ldots, x_n) \in B^n_j \), the conditional expected value in the middle of the displayed equation, as a function of \( m \), is a Doob backward martingale with respect to \( \mathfrak G \) and hence converges to \( \P(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n \mid \mathscr{G}_\infty) \) as \( m \to \infty \).
Next we show that \( \bs M \) is a backwards martingale with respect to \( \mathfrak G \). Trivially \( M_n \) is measurable with respect to \( \mathscr{G}_n \) and \( \E(M_n) \le 1 \) for each \( n \in \N \). Thus we need to show that \( \E(M_n \mid \mathscr{G}_m) = M_m \) for \( m, \, n \in \N_+ \) with \( n \le m \). From our previous work with \( (X_1, X_2, \ldots, X_n) \) we know that the conditional distribution of \( Y_n \) given \( Y_m = k \) is hypergeometric with parameters \( m \), \( k \), and \( n \): \[\P(Y_n = j \mid Y_m = k) = \binom{n}{j} \frac{k^{(j)} (m - k)^{(n - j)}}{m^{(n)}}, \quad j \in \{0, 1, \ldots, n\}\] Recall that the mean of the hypergeometric distribution is the sample size times the proportion of type 1 objects in the population. Thus, \[ \E(M_n \mid Y_m = k) = \frac{1}{n} \E(Y_n \mid Y_m = k) = \frac{1}{n} n \frac{k}{m} = \frac{k}{m} \] Or equivalently, \( \E(M_n \mid Y_m) = Y_m / m = M_m \). Once again, given \( Y_m \), the variables \( Y_{m+1}, Y_{m+2} \) give no additional information and so \( \E(Y_n \mid \mathscr{G}_m) = Y_m \). Hence \( \bs M \) is a backwards martingale with respect to \( \mathfrak G \). From the backwards martingale convergence theorem , there exists a random variable \( P \) such that \( M_n \to P \) as \( n \to \infty \) with probability 1.
It just remains to connect the dots. Suppose now that \( n \in \N_+ \) and \( j \in \{0, 1, \ldots n\} \) and that \( m \in \N_+ \) and \( k_m \in \{0, 1, \ldots, m\} \). From simple calculus, if \( n \) and \( j \) are fixed and \( k_m / m \to p \in [0, 1] \) as \( m \to \infty \) then \[ \frac{k_m^{(j)} (m - k_m)^{(n - j)}}{m^{(n)}} \to p^j (1 - p)^{n - j} \text{ as } m \to \infty \] (You may recall that this computation is used in the proof of the convergence of the hypergeometric distribution to the binomial.) Returning to the joint distribution, recall that if \( (x_1, x_2, \ldots, x_n) \in B^n_j \) then \[ \P(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n \mid \mathscr{G}_m) = \frac{Y_m^{(j)} (m - Y_m)^{(n - j)}}{m^{(n)}} \] Let \( m \to \infty \). Since \( Y_m / m \to P \) as \( m \to \infty \) we get \[ \P(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n \mid \mathscr{G}_\infty) = P^j (1 - P)^{n - j} \] Random variable \( P \) is measurable with respect to \( \mathscr{G}_\infty \) so \[ \P(X_1 = x_1, X_2 = x_2, \ldots, X_n = x_n \mid P) = P^j (1 - P)^{n - j} \text{ as } m \to \infty \] Given \( P = p \in [0, 1] \), \( \bs X \) is a sequence of Bernoulli trials with success parameter \( p \).
De Finetti's theorem has been extended to much more general sequences of exchangeable variables. Basically, if \( \bs X = (X_1, X_2, \ldots) \) is an exchangeable sequence of random variables, each taking values in a significantly nice measurable space \( (S, \mathscr{S}) \) then there exists a random variable \( \Theta \) such that \( \bs X \) is independent and identically distributed given \( \Theta \). In the proof, the result that \( M_n \to P \) as \( n \to \infty \) with probability 1, where \( M_n = \frac{1}{n} \sum_{i=1}^n X_i \), is known as de Finetti's strong law of large numbers . De Finetti's theorem, and it's generalizations are important in Bayesian statistical inference. For an exchangeable sequence of random variables (our observations in a statistical experiment), there is a hidden, random parameter \( \Theta \). Given \( \Theta = \theta \), the variables are independent and identically distributed. We gain information about \( \Theta \) by imposing a prior distribution on \( \Theta \) and then updating this, based on our observations and using Baye's theorem, to a posterior distribution .
Stated more in terms of distributions, de Finetti's theorem states that the distribution of \( n \) distinct variables in the exchangeable sequence is a mixture of product measures. That is, if \( \mu_\theta \) is the distribution of a generic \( X \) on \( (S, \mathscr{S}) \) given \( \Theta = \theta \), and \( \nu \) is the distribution of \( \Theta \) on \( (T, \mathscr{T}) \), then the distribution of \( n \) of the variables on \( (S^n \mathscr{S}^n) \) is \[ B \mapsto \int_T \mu_\theta^n(B) \, d\nu(\theta) \]


1.12: Why Working Retail Is Tough
- List some challenges of working in the retail field
It is important to understand the key challenges retailers face as well as how today’s retailer is working to overcome these challenges.
Inventory Levels and Assortment
Adequate inventory levels are essential for today’s retailer. Having too little or too much can have a dramatic affect on the retailer’s reputation as well as the consumer’s perception. Today’s retailers must ensure the right amount of product is available at the right times and in the right places. Mark & Spencer’s sales losses in 2012 are a key example of how low inventory levels can affect the reputation of a retailer. First quarter clothing sales were down 6.8% compared to the prior year. There were several reasons but, one was that they didn’t order enough inventory for popular items which caused many customers to leave their store without purchasing what they wanted. While having too little inventory is a significant issue, you can also have too much inventory that leads to unprofitable markdowns.
A lack of the right type of inventory also affects customer perception which could, in turn, affect sales. For example, a customer wouldn’t expect to see Christmas trees in August in their favorite retailer. It would signal to the consumer that the retailer doesn’t understand the trends or the season. Therefore, the consumer might be influenced to shop elsewhere. Also consider the other potential impacts of inadequate inventory levels as well as not having the right type of assortment for the customer.
Mobile Experience and Engagement
In 2013 mobile was almost 25% ($60 billion) of e-commerce revenue. For any retailer that has online operations this is a significant part of the business. Consumers use their mobile phones for everything from shopping for different prices for products and even shipping costs. More and more are doing this inside of the store as well as outside of the store. In fact, 90% of consumers use smart phones while shopping in the store. So how can retailers push the envelope here to engage the consumer even more? By 2021 mobile revenue is expected to be $420 billion in the United States. An example of a company capitalizing on this trend is Walgreens. More than 60% of the retailer’s online traffic is from smartphone devices. Their mobile app includes services such as prescription transfers and Quick Print. In addition, they are partnering with a service called MDLive to provide customers with medical doctors 24/7 via smartphone cameras.
Digital Disruption
Technology is continuously changing the way consumers shop. There are five stages consumers go through whenever purchasing a product. These are:
- Need recognition
- Information search
- Evaluating alternatives
- Decision to purchase
- Post-purchase evaluation.
You have probably experienced this yourself in your own consumption experiences. First there is a recognized need for an item. Then you look for products that fulfill your need. Social media sites such as Facebook, and Instagram have revolutionized how fast and to what extent consumers can receive information about a product which, in turn, helps fulfill the information search stage. Consumers can rate products and share with their friend base on these sites allowing for more information search as well as evaluation of alternatives. They are also able to communicate directly with the company via these social platforms creating an opportunity for them to offer post-purchase evaluation. According to a Deloitte report consumers who utilize social media are four times more likely to make a purchase than those who don’t. In addition, 29% are more inclined to make a purchase the same day.
Lastly, it is important to note that there are generational differences in how technology is utilized and received by today’s consumer. For example, 47% of millennials are influenced by social media as opposed to 19% of other age groups. Retailers use this information to adapt their marketing strategies based on the target demographics of the products they are selling.
The Socially Conscious Consumer
Consumers have tremendous power to drive trends of which products are available and how those goods are produced. More and more consumers are questioning the political and environmental impact of their purchases which has lead to the rise of the socially conscious consumer. For evidence of this phenomenon, one need look no further than the recent explosion of organic products on the market. Over the last 15 years, organic products have gone from being a little-known niche market to one of the fastest growing, increasingly mainstream sectors in American Agriculture. According to a United States Department of Agriculture (USDA) 2016 press release “USDA Reports Record Growth in U.S. Organic Producers” , the number of certified organic operations in the United States has increased by almost 300% since the agency began keeping records in 2002 with most years seeing double-digit growth in the organics sector. Significantly, growth in demand for organic products continued throughout the economic recession, suggesting that more and more shoppers are convinced that the health and environmental benefits associated with organic products outweigh the higher cost.
Similar growth has been observed in demand for “eco-friendly” or “green” products, particularly among millennials. Citing a 2015 Nielsen study, Michael Hozik of the Georgetown Environmental Law Review states, “Just over the span of one year, millennials willing to pay more for products and services from companies committed to positive environmental and social change increased from 55% in 2014 to 72% in 2015.” In response to increased demand, the number of applications for eco-friendly labels doubled between 2006 and 2007 and the selection of eco-friendly products made available by retailers increased by 73% from 2009 to 2010. Household names like Walmart, Nestle, Proctor & Gamble, Sony, and many more have created ambitious targets for reducing greenhouse gas emissions and pledged to increase the sustainability of their supply chains. Big name brands are increasingly recognizing the necessity of wooing the growing number of eco-conscious shoppers while also recognizing this trend may cut into their profit margins.
Summary
Above are a few key challenges in detail. However, as I stated before there are numerous others as well. Watch the video below to gain an understanding of a few other challenges that have a significant impact on today’s retailer. Consider how these are impacting you now as a consumer and an industry professional. What else do you believe could be some other notable challenges in the future and what actions do you believe today’s retailer could take to work through these challenges and stay more competitive in today’s environment as well as that of the future?
You can view the transcript for “What is the #1 Challenge Facing Retailers in 2018?” (opens in new window) .
 Authored by : Leslie McIntyre. Located at : http://www.okologie.org/blogs/2017/4/2/the-eco-conscious-consumer-part-i-the-case-for-voting-with-your-dollars . License : CC BY-SA: Attribution-ShareAlike
- What Is The #1 Challenge Facing Retailers In 2018?. Authored by : Samsung Business USA. Located at : https://youtu.be/liXy2gGQ_x8 . License : All Rights Reserved . License Terms : Standard YouTube License