31.3: The Opioid Epidemic and Substance Use Disorders
By the end of this section, you should be able to:
- Describe the United States opioid epidemic.
- Explain how the opioid epidemic arose in the United States.
- Identify how stigma impacts people with opioid use disorder and health outcomes.
- Apply the role of the nurse upholding nursing’s mission to society in addressing the opioid epidemic.
Substance use disorders (SUDs) are chronic, relapsing, potentially deadly conditions that occur when the recurrent problematic use of substances impairs an individual’s health and ability to function to meet vocational, academic, social, or personal responsibilities (CDC, 2022e; Stone et al., 2021). SUDs range in severity, duration, and complexity and are classified as mild, moderate, or severe based on the criteria defined in the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5) (Stone et al., 2021). SUDs and other mental health disorders can be co-occuring (NIMH, 2023). The National Institute of Mental Health has identified three possibilities to explain why SUDs and other mental disorders may occur together:
- Common risk factors (e.g., genetics, stress, trauma, social and environmental factors)
- Brain changes in people with mental disorders, which may heighten the rewarding effects of substance use
- Substance use may trigger changes in brain structure and function, making an individual more likely to develop a mental disorder (NIMH, 2023)
Accidental drug overdose is now the leading cause of accidental deaths in the U.S, largely due to the opioid epidemic (CDC, 2022d). Opioids accounted for 75 percent of all drug overdoses deaths in 2020 (CDC, 2022a). Synthetic opioids have caused overdose deaths, primarily linked to fentanyl manufactured by illicit means, increasing the risk of overdose (CDC, 2020a). The risk of overdose is high with any use of illicitly manufactured fentanyl, as fentanyl is up to 50 times stronger than heroin (Congressional Research Service [CRS], 2022). This risk is increased among individuals who are opioid naïve , meaning they are not receiving opioids on a daily basis, or those whose tolerance to opioids has decreased following periods of abstinence (Baldwin et al., 2021).
Roots of the Opioid Epidemic
The CDC describes the rise of the opioid epidemic as a series of three waves corresponding with overlapping factors that accelerated surges in deaths from substance use in the United States (Figure 31.2) (CDC, 2022f).
- First wave (1990s): Health care professionals’ increased prescribing of prescription opioids, including natural opioids, semi-synthetic opioids, and methadone, resulted in overdose deaths which steadily grew through the year 2016 (CDC, 2022f).
- Second wave (2007): Rapid increases in Mexican production led to greater availability of low-cost heroin to the United States (CRS, 2022). In 2015, overdose deaths involving heroin surpassed the number of deaths related to opioid pills (CDC, 2022f).
- Third wave (2013): Illicitly manufactured fentanyl became available (CDC, 2022f; CRS, 2022). By 2016, fentanyl and tramadol overdose deaths in the United States surpassed those from heroin and prescription drug misuse (CRS, 2022). Illicitly manufactured fentanyl continues to evolve, with its widespread availability in combined formulations including heroin, counterfeit pills, and cocaine (CDC, 2022f).
U.S. drug overdose deaths accelerated at the height of the COVID-19 pandemic (2020 to 2022), as social isolation, unemployment, and reduced access to drug treatment and recovery support services damaged mental health (CDC, 2020). The stress and social isolation of the pandemic led to increased substance use (Figure 31.3) (Baldwin et al., 2021; CDC, 2020). Synthetic opioid deaths continue to climb, while opioid pill and heroin overdose deaths have slackened but remain at high levels (CDC, 2022f; Ciccarone, 2019).
Recognizing that SUDs are linked to multiple health problems and result in overdose and death, Healthy People 2030 provides evidence-based resources and strategies to prevent SUDs at the school, family, and community levels.
The Impact of Stigma on SUDs
Addressing stigma is critical to combating the opioid crisis, as stigma is attached to SUDs and addiction (McGinty & Barry, 2020). Research demonstrates a pervasive stigma in U.S. culture that addiction is a personal choice, reflecting a lack of willpower and a moral failing (McGinty & Barry, 2020). Research has demonstrated that people identified with SUDs are perceived as more blameworthy and dangerous compared to individuals labeled with mental illness (van Boekel et al., 2013). Such stereotypes often lead to less helping behavior and more avoidance of people with drug addiction than of those with mental illness (van Boekel et al., 2013). Stigmatizing beliefs predispose negative attitudes about addiction and target certain races and socioeconomic classes (McGinty & Barry, 2020). Bias may assume an individual’s poor personal choices led to their SUD instead of social factors such as poverty, a history of trauma, or structural barriers to accessing effective treatments (McGinty & Barry, 2020).
Studies show people with SUDs face negative labeling and stereotyping, status loss, and discrimination (McGinty & Barry, 2020; National Institute on Drug Abuse [NIDA], 2023b). This stigma can discourage people who need help from seeking care (NIDA, 2023b). The language that people, including health care professionals, use to describe SUDs can contribute to stigma and discrimination against people with these conditions (NIDA, 2023b). Changing the culture, attitudes, and practices around substance use is essential to lasting health care reform (HHS, 2016). This includes creating a society where people who need help feel comfortable seeking it and where health care professionals and population health nurses treat clients with SUDs with the same level of compassion and care as they would clients with any other chronic disease. It also means facilitating a mindset that everyone can offer the care and support needed to ensure a meaningful difference in someone’s recovery (HHS, 2016).
This American Heart Association video discusses the need to destigmatize drug use to minimize the negative effects of discriminatory and inaccurate perceptions on individuals with SUDs and related conditions.
Watch the video, and then respond to the following questions.
- What are some factors that influence the stigma of SUDs?
- How can nurses decrease barriers to care for individuals with SUDs?
- What innovative strategies can nurses develop to eliminate stigma in their organizations or personal interactions with clients with SUDs?
- Reflect on your feelings, attitudes, and experiences with individuals with SUDs and describe how you can promote change in destigmatizing SUDs.
Addressing the Opioid Crisis
The opioid crisis affects Americans of all ages, ethnicities, socioeconomic classes, and geographic areas. With the notable increase in Americans dying from opioid-involved overdoses, multiple legal, social, and public health efforts have ensued to curb opioid misuse and drug-related overdose deaths (CRS, 2022). Collaborative multidisciplinary strategies must address barriers to care, effective legislation and regulation, equitable evidence-based interventions, and awareness and education initiatives among communities nationwide.
SUDs Prevention
Prevention strategies at the school, family, and community levels are key to reducing SUDs, with interventions to accelerate treatment to reduce opioid-related deaths (ODPHP, n.d.-a). The CDC outlines six principles and five strategic priorities rigorously applied to research and evaluation projects to reduce overdoses and substance use–related harms. Nurses can apply these guidelines by promoting evidence-based strategies to ensure the delivery and implementation of effective methods to prevent and reduce overdose and substance use–related harms for diverse audiences and settings (CDC, 2022c). To strengthen efforts to reduce drug overdoses, public health nurses can build multidisciplinary partnerships to collaborate with public safety and community organizations at national, state, and local levels (CDC, 2022c). See Table 31.2.
| The CDC’s Six Guiding Principles to Address the Overdose Crisis | |
|---|---|
| 1. Promote Health Equity | Ensure equitable opportunity to prevent overdose and substance use–related harms by resolving health disparities related to the overdose crisis. Promote interventions that advance health equity in all communities. |
| 2. Address Underlying Factors | Identify harmful and protective factors to better design interventions to address the overdose crisis, while attending to health disparities and inequities. |
| 3. Partner Broadly | Form broad and diverse partnerships as a foundation of preventing overdose and substance use–related harms that include opportunities to develop, coordinate, and implement targeted strategies to prevent harm. |
| 4. Take Evidence-Based Action | Promote evidence-based action to ensure the delivery of effective methods for preventing and reducing overdose and substance use–related harms that are translated and adapted for diverse populations. |
| 5. Advance Science |
Build the evidence base for what is most effective to end the overdose crisis by:
|
| 6. Drive Innovation | Promote the generation, implementation, evaluation, and widespread adoption of new and innovative ideas to address the overdose crisis. |
The CDC’s prevention framework includes five strategic priorities in response to the overdose crisis:
- Monitor, Analyze, and Communicate Trends
- Build State, Tribal, Local, and Territorial Capacity
- Support Providers, Health Systems, Payors, and Employers
- Partner with Public Safety and Community Organizations
- Raise Public Awareness and Reduce Stigma
Government Efforts
Federal, state, and local governments have mandated legal and policy initiatives to curb opioid misuse and drug-related overdose deaths in the United States (CRS, 2022). Congress has enacted laws that prevent the overprescribing and misuse of opioids, reduce capabilities for domestic diversion and illicit trafficking, and curtail foreign supply with sanction efforts (CRS, 2022). The federal government has also increased the appropriation of funds to expand the availability of substance use prevention, treatment, and recovery services (CRS, 2022). Federal funding supports evidence-based initiatives to reduce opioid use, such as medication-assisted treatment (MAT), peer recovery networks, and harm-reduction strategies, such as needle exchange programs and the distribution of naloxone to reverse opioid overdoses (CRS, 2022). Community efforts are discussed further in the next section.
Community Efforts
The community plays a crucial role in the public health response to the opioid epidemic (Worcester County Health Department, 2019). The SAMHSA publication (2022) Community Engagement: An Essential Component of an Effective and Equitable Substance Use Prevention System outlines the community’s role in reducing substance misuse. Communities can increase participation in these efforts by schools and community agencies and provide educational programs targeting high-risk, vulnerable individuals and populations using community resources (Figure 31.4). Local boards of education have piloted multidisciplinary instructional units on prescription opioids and heroin with middle-school students, teaching them how opioids and heroin affect the brain, its chemistry, and the societal impacts of addiction (Worcester County Health Department, 2019). Such approaches are promising prevention strategies to maximize harm reduction (HHS, 2016).
Other examples of community efforts include prescription drug monitoring programs (PDMPs), which have reduced opioid prescribing, multiple provider episodes (“doctor shopping”), and opioid-related overdose deaths in states where they are mandated. Free sterile needle and syringe programs and other support services, such as counseling and testing for sexually transmitted infections (STIs), have dramatically reduced disease transmission. For example, studies have demonstrated that needle/syringe exchange programs effectively reduce HIV transmission and do not increase rates of community drug use (HHS, 2016).
Naloxone is an FDA-approved opioid antagonist medication that effectively reverses opioid overdose (HHS, 2016). Education and the distribution of naloxone to at-risk individuals and their families, public health nurses, emergency medical technicians, police officers, other first responders, and community-based opioid overdose prevention programs has saved countless lives (HHS, 2016).
In addition to financially investing in community-based efforts, urban areas are improving access to treatment and housing by engaging and supporting impacted communities (Syed, 2023). For example, Philadelphia has increased the availability of permanent housing opportunities to support people experiencing homelessness at different stages of recovery and piloted the first licensed mobile wound-care van in Pennsylvania by expanding mobile outreach teams citywide (Syed, 2023).
In contrast, rural communities experience a disproportionately high burden from the opioid crisis (Palombi et al., 2019). An effective grassroots approach to rural community engagement involves collaborative teams composed of community members, public health professionals, university faculty, law enforcement, and medical professionals, among others, that convene from the bottom up through community coalitions and forums (Palombi et al., 2019). Coalitions in Minnesota have raised awareness and galvanized community efforts around diverse topics including substance misuse, syringe exchanges, and naloxone distribution (Palombi et al., 2019). As depicted in the video, America Addicted , the ongoing challenges of the opioid crisis are overwhelming community efforts to curb opioid overdoses. A collaborative government and community effort is needed to create substance-free communities that address social determinants of health (Palombi et al., 2019).
Research is ongoing to explore how the interaction of biological and social factors may contribute to an individual’s increased vulnerability to be at risk for developing SUDs (Amaro et al., 2021). Chronic exposure to stressors such as poverty, food and housing insecurity, racism, and health inequities may underpin a maladaptive stress response, resulting in physical dependence and SUDs (Amaro et al., 2021). This article from the journal Neuropharmacology explores how the stress of these social determinants of health play a critical role in creating vulnerability to substance use (Amaro et al., 2021).
(See Amaro et al., 2021.)
SUDs Treatment
Individuals with SUDs have historically received treatment through mental health services or substance use disorder treatment programs (HHS, 2016). As discussed, the demand in the United States for mental health services often exceeds the supply (Kuntz, 2022). Increasingly, primary or general health care practices are involved in delivery of treatment (HHS, 2016). Increasing nonopioid pain management of minor procedures, like dental rehabilitation, and of health conditions, like muscle strains, can minimize opioid exposure and dependency. The most effective treatment for opioid use disorders is medication-assisted treatment (MAT), which can combine pharmacologic agents and counseling or behavioral therapy and other support services (HHS, 2016). Successful MAT programs for opioid addiction have been shown to decrease overdose deaths, be cost-effective, reduce transmissions of HIV and hepatitis C related to IV drug use, and moderate associated criminal activity (HHS, 2016).
This National Institute on Drug Abuse video describes medications for opioid use disorder (MOUD) and how they work. These medications can support recovery and improve health by reducing cravings, easing withdrawal symptoms, and possibly reversing an opioid overdose (NIDA, 2021). Pharmacotherapy for SUDs can be used alone or as part of a larger treatment plan (NIDA, 2021). These medicines are safe and effective and can save lives (NIDA, 2021).
Watch the video, and then respond to the following questions.
- Which medication is commonly used in treatment programs for substance use disorders to produce a noneuphoric state and to replace opioid use?
- Which medication is most appropriate for the treatment of a client with SUDs after the opioids have been completely cleared from their body?
- Why must a community health or public health nurse be familiar with the different medication options to treat opioid use disorder?
The Nurse’s Role in Addressing the Opioid Epidemic and Substance Use Disorders
Because nurses practice in various direct-care, care-coordination, leadership, and executive roles, they are uniquely situated to help clients and their families understand the treatment options for SUDs (ANA, 2018). Nurses can screen for early identification of risk factors surrounding SUDs and assist with nonopioid pain management and alternative medication modalities, regional anesthetic interventions, surgery, psychological therapies, rehabilitative/physical therapy, and complementary and alternative medicine (CAM) (ANA, 2018). As educators and client advocates, nurses can champion sustainable harm-reduction programs linked to positive outcomes and lobby for health policies and regulations that promote equitable distribution and availability of high-quality health care services. Through awareness, education, early identification, evidence-based interventions, research, interdisciplinary collaboration, and political advocacy, nurses possess the leadership skills, knowledge, care, and capabilities to address this public health crisis.
It may be hard to tell whether a person is high or experiencing an overdose. If you are unsure, treat it like an overdose—you could save a life.
- Call 911 immediately.*
- Administer naloxone, if available.
- Try to keep the person awake and breathing.
- Lie the person on their side to prevent choking.
- Stay with the person until emergency assistance arrives.
*Most states have laws that may protect a person who is overdosing, or the person who called for help, from legal trouble.
(See CDC, 2023c.)


5.8: The Gamma Distribution
In this section we will study a family of distributions that has special importance in probability and statistics. In particular, the arrival times in the Poisson process have gamma distributions, and the chi-square distribution in statistics is a special case of the gamma distribution. Also, the gamma distribution is widely used to model physical quantities that take positive values.
The Gamma Function
Before we can study the gamma distribution , we need to introduce the gamma function , a special function whose values will play the role of the normalizing constants.
Definition
The gamma function \( \Gamma \) is defined as follows \[ \Gamma(k) = \int_0^\infty x^{k-1} e^{-x} \, dx, \quad k \in (0, \infty) \] The function is well defined, that is, the integral converges for any \(k \gt 0\). On the other hand, the integral diverges to \( \infty \) for \( k \le 0 \).
Proof
Note that \[ \int_0^\infty x^{k-1} e^{-x} \, dx = \int_0^1 x^{k-1} e^{-x} \, dx + \int_1^\infty x^{k-1} e^{-x} \, dx \] For the first integral on the right, \[\int_0^1 x^{k-1} e^{-x} \, dx \le \int_0^1 x^{k-1} \, dx = \frac{1}{k}\] For the second integral, let \( n = \lceil k \rceil \). Then \[ \int_1^\infty x^{k-1} e^{-x} \, dx \le \int_1^\infty x^{n-1} e^{-x} \, dx \] The last integral can be evaluated explicitly by integrating by parts, and is finite for every \( n \in \N_+ \).
Finally, if \( k \le 0 \), note that \[ \int_0^1 x^{k-1} e^{-x}, \, dx \ge e^{-1} \int_0^1 x^{k-1} \, dx = \infty \]
The gamma function was first introduced by Leonhard Euler.
The (lower) incomplete gamma function is defined by \[ \Gamma(k, x) = \int_0^x t^{k-1} e^{-t} \, dt, \quad k, x \in (0, \infty) \]
Properties
Here are a few of the essential properties of the gamma function. The first is the fundamental identity.
\(\Gamma(k + 1) = k \, \Gamma(k)\) for \(k \in (0, \infty)\).
Proof
This follows from integrating by parts, with \( u = x^k \) and \( dv = e^{-x} \, dx \): \[ \Gamma(k + 1) = \int_0^\infty x^k e^{-x} \, dx = \left(-x^k e^{-x}\right)_0^\infty + \int_0^\infty k x^{k-1} e^{-x} \, dx = 0 + k \, \Gamma(k) \]
Applying this result repeatedly gives \[ \Gamma(k + n) = k (k + 1) \cdots (k + n - 1) \Gamma(k), \quad n \in \N_+ \] It's clear that the gamma function is a continuous extension of the factorial function.
\(\Gamma(k + 1) = k!\) for \(k \in \N\).
Proof
This follows from the fundmental identity and the fact that \(\Gamma(1) = 1\).
The values of the gamma function for non-integer arguments generally cannot be expressed in simple, closed forms. However, there are exceptions.
\(\Gamma\left(\frac{1}{2}\right) = \sqrt{\pi}\).
Proof
By definition, \[ \Gamma\left(\frac{1}{2}\right) = \int_0^\infty x^{-1/2} e^{-x} \, dx \] Substituting \( x = z^2 / 2 \) gives \[ \Gamma\left(\frac{1}{2}\right) = \int_0^\infty \sqrt{2} e^{-z^2/2} \, dz = 2 \sqrt{\pi} \int_0^\infty \frac{1}{\sqrt{2 \pi}} e^{-z^2/2} \, dz\] But the last integrand is the PDF of the standard normal distribution, and so the integral evaluates to \( \frac{1}{2} \)
We can generalize the last result to odd multiples of \( \frac{1}{2} \).
For \( n \in \N \), \[ \Gamma\left(\frac{2 n + 1}{2}\right) = \frac{1 \cdot 3 \cdots (2 n - 1)}{2^n} \sqrt{\pi} = \frac{(2 n)!}{4^n n!} \sqrt{\pi} \]
Proof
This follows from the previous result and the fundamental identity .
Stirling's Approximation
One of the most famous asymptotic formulas for the gamma function is Stirling's formula , named for James Stirling. First we need to recall a definition.
Suppose that \( f, \, g: D \to (0, \infty) \) where \( D = (0, \infty) \) or \( D = \N_+ \). Then \( f(x) \approx g(x) \) as \( x \to \infty \) means that \[ \frac{f(x)}{g(x)} \to 1 \text{ as } x \to \infty \]
Stirling's formula \[ \Gamma(x + 1) \approx \left( \frac{x}{e} \right)^x \sqrt{2 \pi x} \text{ as } x \to \infty \]
As a special case, Stirling's result gives an asymptotic formula for the factorial function: \[ n! \approx \left( \frac{n}{e} \right)^n \sqrt{2 \pi n} \text{ as } n \to \infty \]
The Standard Gamma Distribution
Distribution Functions
The standard gamma distribution with shape parameter \( k \in (0, \infty) \) is a continuous distribution on \( (0, \infty) \) with probability density function \(f\) given by \[ f(x) = \frac{1}{\Gamma(k)} x^{k-1} e^{-x}, \quad x \in (0, \infty) \]
Clearly \( f \) is a valid probability density function, since \( f(x) \gt 0 \) for \( x \gt 0 \), and by definition, \( \Gamma(k) \) is the normalizing constant for the function \( x \mapsto x^{k-1} e^{-x} \) on \( (0, \infty) \). The following theorem shows that the gamma density has a rich variety of shapes, and shows why \(k\) is called the shape parameter.
The gamma probability density function \( f \) with shape parameter \( k \in (0, \infty) \) satisfies the following properties:
- If \(0 \lt k \lt 1\), \(f\) is decreasing with \(f(x) \to \infty\) as \(x \downarrow 0\).
- If \(k = 1\), \(f\) is decreasing with \(f(0) = 1\).
- If \(k \gt 1\), \(f\) increases and then decreases, with mode at \( k - 1 \).
- If \( 0 \lt k \le 1 \), \( f \) is concave upward.
- If \( 1 \lt k \le 2 \), \( f \) is concave downward and then upward, with inflection point at \( k - 1 + \sqrt{k - 1} \).
- If \( k \gt 2 \), \( f \) is concave upward, then downward, then upward again, with inflection points at \( k - 1 \pm \sqrt{k - 1} \).
Proof
These results follow from standard calculus. For \( x \gt 0 \), \begin{align*} f^\prime(x) &= \frac{1}{\Gamma(k)} x^{k-2} e^{-x}[(k - 1) - x] \\ f^{\prime \prime}(x) &= \frac{1}{\Gamma(k)} x^{k-3} e^{-x} \left[(k - 1)(k - 2) - 2 (k - 1) x + x^2\right] \end{align*}
The special case \(k = 1\) gives the standard exponential distribuiton . When \(k \ge 1\), the distribution is unimodal.
In the simulation of the special distribution simulator, select the gamma distribution. Vary the shape parameter and note the shape of the density function. For various values of \(k\), run the simulation 1000 times and compare the empirical density function to the true probability density function.
The distribution function and the quantile function do not have simple, closed representations for most values of the shape parameter. However, the distribution function has a trivial representation in terms of the incomplete and complete gamma functions.
The distribution function \( F \) of the standard gamma distribution with shape parameter \( k \in (0, \infty) \) is given by \[ F(x) = \frac{\Gamma(k, x)}{\Gamma(k)}, \quad x \in (0, \infty) \]
Approximate values of the distribution and quantile functions can be obtained from special distribution calculator, and from most mathematical and statistical software packages.
Using the special distribution calculator, find the median, the first and third quartiles, and the interquartile range in each of the following cases:
- \(k = 1\)
- \(k = 2\)
- \(k = 3\)
Moments
Suppose that \( X \) has the standard gamma distribution with shape parameter \( k \in (0, \infty) \). The mean and variance are both simply the shape parameter.
The mean and variance of \( X \) are
- \(\E(X) = k\)
- \(\var(X) = k\)
Proof
- From the fundamental identity , \[ \E(X) = \int_0^\infty x \frac{1}{\Gamma(k)} x^{k-1} e^{-x} \, dx = \frac{\Gamma(k + 1)}{\Gamma(k)} = k\]
- From the fundamental identity again \[ \E\left(X^2\right) = \int_0^\infty x^2 \frac{1}{\Gamma(k)} x^{k-1} e^{-x} \, dx = \frac{\Gamma(k + 2)}{\Gamma(k)} = (k + 1) k\] and hence \( \var(X) = \E\left(X^2\right) - [\E(X)]^2 = k \)
In the simulation of the special distribution simulator, select the gamma distribution. Vary the shape parameter and note the size and location of the mean \( \pm \) standard deviation bar. For selected values of \(k\), run the simulation 1000 times and compare the empirical mean and standard deviation to the distribution mean and standard deviation.
More generally, the moments can be expressed easily in terms of the gamma function:
The moments of \( X \) are
- \(\E(X^a) = \Gamma(a + k) \big/ \Gamma(k)\) if \(a \gt -k\)
- \(\E(X^n) = k^{[n]} = k (k + 1) \cdots (k + n - 1)\) if \(n \in \N\)
Proof
- For \( a \gt -k \), \[ \E(X^a) = \int_0^\infty x^a \frac{1}{\Gamma(k)} x^{k-1} e^{-x} \, dx = \frac{1}{\Gamma(k)} \int_0^\infty x^{a + k} e^{-x} \, dx = \frac{\Gamma(a + k)}{\Gamma(k)} \]
- If \( n \in \N \), then by the fundamental identity , \( \Gamma(k + n) = k (k + 1) \cdots (k + n - 1) \Gamma(k) \), so the result follows from (a).
Note also that \( \E(X^a) = \infty \) if \( a \le -k \). We can now also compute the skewness and the kurtosis.
The skewness and kurtosis of \( X \) are
- \( \skw(X) = \frac{2}{\sqrt{k}} \)
- \( \kur(X) = 3 + \frac{6}{k} \)
Proof
These results follows from the previous moment results and the computational formulas for skewness and kurtosis.
In particular, note that \( \skw(X) \to 0 \) and \( \kur(X) \to 3 \) as \( k \to \infty \). Note also that the excess kurtosis \( \kur(X) - 3 \to 0 \) as \( k \to \infty \).
In the simulation of the special distribution simulator, select the gamma distribution. Increase the shape parameter and note the shape of the density function in light of the previous results on skewness and kurtosis. For various values of \(k\), run the simulation 1000 times and compare the empirical density function to the true probability density function.
The following theorem gives the moment generating function.
The moment generating function of \( X \) is given by \[ \E\left(e^{t X}\right) = \frac{1}{(1 - t)^k}, \quad t \lt 1 \]
Proof
For \( t \lt 1 \), \[ \E\left(e^{t X}\right) = \int_0^\infty e^{t x} \frac{1}{\Gamma(k)} x^{k-1} e^{-x} \, dx = \int_0^\infty \frac{1}{\Gamma(k)} x^{k-1} e^{-x(1 - t)} \, dx \] Substituting \( u = x(1 - t) \) so that \( x = u \big/ (1 - t) \) and \( dx = du \big/ (1 - t) \) gives \[ \E\left(e^{t X}\right) = \frac{1}{(1 - t)^k} \int_0^\infty \frac{1}{\Gamma(k)} u^{k-1} e^{-u} \, du = \frac{1}{(1 - t)^k} \]
The General Gamma Distribution
The gamma distribution is usually generalized by adding a scale parameter.
If \(Z\) has the standard gamma distribution with shape parameter \(k \in (0, \infty)\) and if \( b \in (0, \infty) \), then \(X = b Z\) has the gamma distribution with shape parameter \(k\) and scale parameter \(b\).
The reciprocal of the scale parameter, \(r = 1 / b\) is known as the rate parameter , particularly in the context of the Poisson process. The gamma distribution with parameters \(k = 1\) and \(b\) is called the exponential distribution with scale parameter \(b\) (or rate parameter \(r = 1 / b\)). More generally, when the shape parameter \(k\) is a positive integer, the gamma distribution is known as the Erlang distribution , named for the Danish mathematician Agner Erlang. The exponential distribution governs the time between arrivals in the Poisson model, while the Erlang distribution governs the actual arrival times.
Basic properties of the general gamma distribution follow easily from corresponding properties of the standard distribution and basic results for scale transformations.
Distribution Functions
Suppose that \( X \) has the gamma distribution with shape parameter \( k \in (0, \infty) \) and scale parameter \( b \in (0, \infty) \).
\(X\) has probability density function \( f \) given by \[ f(x) = \frac{1}{\Gamma(k) b^k} x^{k-1} e^{-x/b}, \quad x \in (0, \infty) \]
Proof
Recall that if \( g \) is the PDF of the standard gamma distribution with shape parameter \( k \) then \( f(x) = \frac{1}{b} g\left(\frac{x}{b}\right) \) for \( x \gt 0 \).
Recall that the inclusion of a scale parameter does not change the shape of the density function, but simply scales the graph horizontally and vertically. In particular, we have the same basic shapes as for the standard gamma density function .
The probability density function \( f \) of \( X \) satisfies the following properties:
- If \(0 \lt k \lt 1\), \(f\) is decreasing with \(f(x) \to \infty\) as \(x \downarrow 0\).
- If \(k = 1\), \(f\) is decreasing with \(f(0) = 1\).
- If \(k \gt 1\), \(f\) increases and then decreases, with mode at \( (k - 1) b \).
- If \( 0 \lt k \le 1 \), \( f \) is concave upward.
- If \( 1 \lt k \le 2 \), \( f \) is concave downward and then upward, with inflection point at \( b \left(k - 1 + \sqrt{k - 1}\right) \).
- If \( k \gt 2 \), \( f \) is concave upward, then downward, then upward again, with inflection points at \( b \left(k - 1 \pm \sqrt{k - 1}\right) \).
In the simulation of the special distribution simulator, select the gamma distribution. Vary the shape and scale parameters and note the shape and location of the probability density function. For various values of the parameters, run the simulation 1000 times and compare the empirical density function to the true probability density function.
Once again, the distribution function and the quantile function do not have simple, closed representations for most values of the shape parameter. However, the distribution function has a simple representation in terms of the incomplete and complete gamma functions.
The distribution function \( F \) of \( X \) is given by \[ F(x) = \frac{\Gamma(k, x/b)}{\Gamma(k)}, \quad x \in (0, \infty) \]
Proof
From the defintion we can take \( X = b Z \) where \( Z \) has the standard gamma distribution with shape parameter \( k \). Then \( \P(X \le x) = \P(Z \le x/b) \) for \( x \in (0, \infty) \), so the result follows from the distribution function of \( Z \) .
Approximate values of the distribution and quanitle functions can be obtained from special distribution calculator, and from most mathematical and statistical software packages.
Open the special distribution calculator. Vary the shape and scale parameters and note the shape and location of the distribution and quantile functions. For selected values of the parameters, find the median and the first and third quartiles.
Moments
Suppose again that \( X \) has the gamma distribution with shape parameter \( k \in (0, \infty) \) and scale parameter \( b \in (0, \infty) \).
The mean and variance of \( X \) are
- \(\E(X) = b k\)
- \(\var(X) = b^2 k\)
Proof
From the definition , we can take \( X = b Z \) where \( Z \) has the standard gamma distribution with shape parameter \( k \). Then using the mean and variance of \( Z \) ,
- \( \E(X) = b \E(Z) = b k \)
- \( \var(X) = b^2 \var(Z) = b^2 k \)
In the special distribution simulator, select the gamma distribution. Vary the parameters and note the shape and location of the mean \( \pm \) standard deviation bar. For selected values of the parameters, run the simulation 1000 times and compare the empirical mean and standard deviation to the distribution mean and standard deviation.
The moments of \( X \) are
- \(\E(X^a) = b^a \Gamma(a + k) \big/ \Gamma(k)\) for \(a \gt -k\)
- \(\E(X^n) = b^n k^{[n]} = b^n k (k + 1) \cdots (k + n - 1)\) if \(n \in \N\)
Proof
Again, from the definition , we can take \( X = b Z \) where \( Z \) has the standard gamma distribution with shape parameter \( k \). The results follow from the moment results for \( Z \) , since \( E(X^a) = b^a \E(Z^a) \).
Note also that \( \E(X^a) = \infty \) if \( a \le -k \). Recall that skewness and kurtosis are defined in terms of the standard score, and hence are unchanged by the addition of a scale parameter.
The skewness and kurtosis of \( X \) are
- \( \skw(X) = \frac{2}{\sqrt{k}} \)
- \( \kur(X) = 3 + \frac{6}{k} \)
The moment generating function of \(X\) is given by \[ \E\left(e^{t X}\right) = \frac{1}{(1 - b t)^k}, \quad t \lt \frac{1}{b} \]
Proof
From the definition , we can take \( X = b Z\) where \( Z \) has the standard gamma distribution with shape parameter \( k \). Then \( \E\left(e^{t X}\right) = \E\left[e^{(t b )Z}\right] \), so the result follows from the moment generating function of \( Z \) .
Relations
Our first result is simply a restatement of the meaning of the term scale parameter .
Suppose that \(X\) has the gamma distribution with shape parameter \(k \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\). If \(c \in (0, \infty)\), then \(c X\) has the gamma distribution with shape parameter \(k\) and scale parameter \(b c\).
Proof
From the definition , we can take \( X = b Z \) where \( Z \) has the standard gamma distribution with shape parameter \( k \). Then \( c X = c b Z \).
More importantly, if the scale parameter is fixed, the gamma family is closed with respect to sums of independent variables.
Suppose that \(X_1\) and \(X_2\) are independent random variables, and that \(X_i\) has the gamma distribution with shape parameter \(k_i \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\) for \(i \in \{1, 2\}\). Then \(X_1 + X_2\) has the gamma distribution with shape parameter \(k_1 + k_2\) and scale parameter \(b\).
Proof
Recall that the MGF of \( X = X_1 + X_2 \) is the product of the MGFs of \( X_1 \) and \( X_2 \), so \[ \E\left(e^{t X}\right) = \frac{1}{(1 - b t)^{k_1}} \frac{1}{(1 - b t)^{k_2}} = \frac{1}{(1 - b t)^{k_1 + k_2}}, \quad t \lt \frac{1}{b} \]
From the previous result, it follows that the gamma distribution is infinitely divisible:
Suppose that \(X\) has the gamma distribution with shape parameter \(k \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\). For \( n \in \N_+ \), \( X \) has the same distribution as \( \sum_{i=1}^n X_i \), where \((X_1, X_2, \ldots, X_n)\) is a sequence of independent random variables, each with the gamma distribution with with shape parameter \( k / n \) and scale parameter \( b \).
From the sum result and the central limit theorem, it follows that if \(k\) is large, the gamma distribution with shape parameter \( k \) and scale parameter \( b \) can be approximated by the normal distribution with mean \(k b\) and variance \(k b^2\). Here is the precise statement:
Suppose that \( X_k \) has the gamma distribution with shape parameter \( k \in (0, \infty) \) and fixed scale parameter \( b \in (0, \infty) \). Then the distribution of the standardized variable below converges to the standard normal distribution as \(k \to \infty\): \[ Z_k = \frac{X_k - k b}{\sqrt{k} b} \]
In the special distribution simulator, select the gamma distribution. For various values of the scale parameter, increase the shape parameter and note the increasingly
normal
shape of the density function. For selected values of the parameters, run the experiment 1000 times and compare the empirical density function to the true probability density function.
The gamma distribution is a member of the general exponential family of distributions:
The gamma distribution with shape parameter \(k \in (0, \infty)\) and scale parameter \(b \in (0, \infty)\) is a two-parameter exponential family with natural parameters \((k - 1, -1 / b)\), and natural statistics \((\ln X, X)\).
Proof
This follows from the definition of the general exponential family. The gamma PDF can be written as \[ f(x) = \frac{1}{b^k \Gamma(k)} \exp\left[(k - 1) \ln x - \frac{1}{b} x\right], \quad x \in (0, \infty) \]
For \( n \in (0, \infty) \), the gamma distribution with shape parameter \( n/2 \) and scale parameter 2 is known as the chi-square distribution with \( n \) degrees of freedom. The chi-square distribution is important enough to deserve a separate section.
Computational Exercise
Suppose that the lifetime of a device (in hours) has the gamma distribution with shape parameter \(k = 4\) and scale parameter \(b = 100\).
- Find the probability that the device will last more than 300 hours.
- Find the mean and standard deviation of the lifetime.
Answer
Let \(X\) denote the lifetime in hours.
- \(\P(X \gt 300) = 13 e^{-3} \approx 0.6472\)
- \(\E(X) = 400\), \(\sd(X) = 200\)
Suppose that \(Y\) has the gamma distribution with parameters \(k = 10\) and \(b = 2\). For each of the following, compute the true value using the special distribution calculator and then compute the normal approximation. Compare the results.
- \(\P(18 \lt X \lt 25)\)
- The 80th percentile of \(Y\)
Answer
- \(\P(18 \lt X \lt 25) = 0.3860\), \(\P(18 \lt X \lt 25) \approx 0.4095\)
- \(y_{0.8} = 25.038\), \(y_{0.8} \approx 25.325\)


1.9.3: Entropy and Spontaneous Reaction
It is often stated that the entropy of a system is a maximum at equilibrium. This is not generally true and is certainly not the case for closed systems at either (a) fixed \(\mathrm{T}\) and \(\mathrm{p}\), or (b) fixed \(\mathrm{T}\) and \(\mathrm{V}\).
We rewrite the Master Equation in the following way:
\[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dU}+(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Temperature \(\mathrm{T}\) is positive and non-zero. At constant energy and constant volume (i.e. isoenergetic and isochoric), spontaneous processes are accompanied by an increase in entropy. This statement is important in statistical thermodynamics where the condition, ‘constant \(\mathrm{U}\) and constant \(\mathrm{V}\)’ is important.
The following equation defines the enthalpy \(\mathrm{H}\) of a closed system.
\[\mathrm{H}=\mathrm{U}+\mathrm{p} \, \mathrm{V} \nonumber \]
Then
\[\mathrm{dU}=\mathrm{dH}-\mathrm{p} \, \mathrm{dV}-\mathrm{V} \, \mathrm{dp} \nonumber \]
From equation (a),
\[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dH}-(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}-(\mathrm{V} / \mathrm{T}) \, \mathrm{dp}+(\mathrm{p} / \mathrm{T}) \, \mathrm{dV}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi \text { with } \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Hence,
\[\mathrm{dS}=(1 / \mathrm{T}) \, \mathrm{dH}-(\mathrm{V} / \mathrm{T}) \, \mathrm{dp}+(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi ; \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Temperature \(\mathrm{T}\) is always positive. Hence at constant enthalpy and pressure (i.e. iso-enthalpic and isobaric) all spontaneous processes produce an increase in entropy.
We have identified two sets of conditions under which an increase in entropy accompanies a spontaneous process. If we follow through a similar argument with respect to the Gibbs energy, the outcome is not straightforward. By definition,
\[\mathrm{G}=\mathrm{H}-\mathrm{T} \, \mathrm{S} \nonumber \]
Then
\[\mathrm{dG}=\mathrm{dH}-\mathrm{T} \, \mathrm{dS}-\mathrm{S} \, \mathrm{dT} \nonumber \]
Or,
\[\mathrm{S}=-\mathrm{dG} / \mathrm{dT}+\mathrm{dH} / \mathrm{dT}-\mathrm{T} \, \mathrm{dS} / \mathrm{dT} \nonumber \]
But from equation (e)
\[\mathrm{dH} / \mathrm{dT}=\mathrm{T} \, \mathrm{dS} / \mathrm{dT}+(\mathrm{V} / \mathrm{T}) \, \mathrm{dp} / \mathrm{dT}-(\mathrm{A} / \mathrm{T}) \, \mathrm{d} \xi / \mathrm{dT} \nonumber \]
Hence,
\[\mathrm{S}=-(\mathrm{dG} / \mathrm{dT})+\mathrm{V} \,(\mathrm{dp} / \mathrm{dT})-\mathrm{A} \,(\mathrm{d} \xi / \mathrm{dT}) \text { with } \mathrm{A} \, \mathrm{d} \xi \geq \text { zero } \nonumber \]
Clearly no definite conclusions can be drawn about changes in entropy S under isobaric - isothermal conditions. We stress these points because again it is often tempting to link, misguidedly, entropies to the degree of ‘muddled-up-ness’. This is the basis of many explanations of entropy. For example, neither the volume nor energy of a deck of cards change on shuffling. Whether what actually happens on shuffling a new well-ordered deck of cards clarifies the meaning of entropy seems doubtful.


7.1: Conclusion
As you become more proficient in measuring vital signs and interpreting the findings, you should remember a few key points.
There are many methods to take vitals signs. The correct technique is essential to obtaining an accurate measurement.
Vital sign measurements have very little meaning on their own. Healthcare providers engage in critical thinking and correlate these measurements with subjective and other objective data. Thinking critically about these measurements will best inform clinical decision-making. Healthcare providers look holistically at the person and their health and wellness state to determine whether vital sign measurements are within normal limits for this individual person.
It is also essential to acknowledge that clients may have additional information that can provide insight into their body, which may influence the technique and location for measuring vital signs and the significance of the findings. Sharing findings with clients is also a good opportunity for health promotion teaching.
Points to Consider
It is important to document vital signs in a timely manner. The healthcare provider reports any abnormal and unexpected findings to the most responsible provider. For example, students should share the findings with their preceptor or clinical instructor in a timely manner.
Figure 7.1: Cells (Illustration credit: Hilary Tang)
Artist Statement – Cells
The human body is a messy phenomenon. From the organ to the cell, the brain to the neurotransmitter, like clockwork, everything is constantly happening. Down to our very core, the tiniest components are working in conjunction, harmoniously, to let us eat, breathe, and move. Without our constant consciousness, we are living. What an organized chaos our bodies are.


11.7E: Limbic System
- Describe the functions of the limbic system
The limbic system, or paleomammalian brain, is a set of brain structures in the precortex and subcortex of the brain. It includes the hippocampus, amygdala, anterior thalamic nuclei, septum, limbic cortex, and fornix, and supports a variety of functions including emotion, behavior, motivation, long-term memory, and olfaction. The term “limbic” comes from the Latin limbus, for “border” or “edge,” because the limbic system forms the inner border of the cortex.
The limbic system consists of various structures that each support distinctive brain functions.
The Limbic System: This diagram of the limbic system delineates components of the diencephalon and cerebrum.
Hippocampus and Associated Structures
- Hippocampus: Required for the formation of long-term memories and implicated in maintenance of cognitive maps for navigation.
- Amygdala: Involved in signaling the cortex of motivationally-significant stimuli, such as those related to reward and fear, and in social functions, such as mating.
- Fornix: A white matter structure that carries signals from the hippocampus to the mammillary bodies and septal nuclei.
- Mammillary body: Important for the formation of memory.
Septal Nuclei
- These lie below the rostrum of the corpus callosum and anterior to the lamina terminalis. The septal nuclei receive reciprocal connections from the olfactory bulb, hippocampus, amygdala, hypothalamus, midbrain, habenula, cingulate gyrus, and thalamus.
Limbic Lobe
A phylogenetically old structure consisting of the following structures:
- Parahippocampal gyrus: Plays a role in the formation of spatial memory
- Cingulate gyrus: Conducts autonomic functions regulating heart rate, blood pressure, and cognitive and attentional processing
- Dentate gyrus: Thought to contribute to the formation of new memories
Additional Structures
- Entorhinal cortex: Important memory and associative components
- Piriform cortex: Processes olfactory information
- Fornicate gyrus: Region encompassing the cingulate and parahippocampal gyrus
- Nucleus accumbens: Involved in reward, pleasure, and addiction
- Orbitofrontal cortex: Involved in cognitive processing during decision-making
Limbic System Function
The limbic system operates by influencing the endocrine system and the autonomic nervous system. It is highly interconnected with the nucleus accumbens, the brain’s pleasure center, which plays a role in sexual arousal and the “high” derived from certain recreational drugs. The structures of the limbic system are involved in motivation, emotion, learning, and memory.
The limbic system is also tightly connected to the prefrontal cortex. Some scientists contend that this connection is related to the pleasure obtained from solving problems. To cure severe emotional disorders, this connection was sometimes surgically severed, a procedure of psychosurgery called a prefrontal lobotomy. Patients who underwent this procedure often became passive and lacked motivation.
Key Points
- The lymbic system includes the hippocampus, amygdala, anterior thalamic nuclei, septum, limbic cortex, and fornix, which together support a variety of functions including emotion, behavior, motivation, long-term memory, and olfaction.
- The limbic system operates by influencing the endocrine system and the autonomic nervous system. It is highly interconnected with the nucleus accumbens, the brain’s pleasure center, which plays a role in sexual arousal and the “high” derived from certain recreational drugs.
- The limbic system is also tightly connected to the prefrontal cortex. Some scientists contend that this connection is related to the pleasure obtained from solving problems.
Key Terms
- nucleus accumbens : A collection of neurons that forms the main part of the ventral striatum. It is thought to play an important role in reward, pleasure, laughter, addiction, aggression, fear, and the placebo effect.
- hippocampus : A part of the brain located inside the temporal lobe consisting mainly of gray matter. It is a component of the limbic system and plays a role in memory and emotion.
- mammillary bodies : Act as a relay for impulses coming from the amygdalae and hippocampi.
- prefrontal cortex : The anterior part of the frontal lobes of the brain, lying in front of the motor and premotor areas. This brain region has been implicated in planning complex cognitive behavior, personality expression, decision making, and moderating social behavior, but its primary function is the orchestration of thoughts and actions in accordance with internal goals.
- septal nucle : Play a role in reward and reinforcement along with the nucleus accumbens.
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1.1: Algebraic Operations With Integers
The set \(\mathbb{Z}\) of all integers, which this book is all about, consists of all positive and negative integers as well as 0. Thus \(\mathbb{Z}\) is the set given by \[\mathbb{Z}=\{...,-4,-3,-2,-1,0,1,2,3,4,...\}.\] While the set of all positive integers, denoted by \(\mathbb{N}\), is defined by \[\mathbb{N}=\{1,2,3,4,...\}.\]
On \(\mathbb{Z}\), there are two basic binary operations, namely addition (denoted by \(+\)) and multiplication (denoted by \(\cdot\)), that satisfy some basic properties from which every other property for \(\mathbb{Z}\) emerges.
- The Commutativity property for addition and multiplication \[\begin{aligned} a+b=b+a\\ a\cdot b=b\cdot a\end{aligned}\]
- Associativity property for addition and multiplication \[\begin{aligned} (a+b)+c&=&a+(b+c)\\ (a\cdot b)\cdot c&=& a\cdot (b\cdot c)\end{aligned}\]
- The distributivity property of multiplication over addition \[\begin{aligned} a\cdot (b+c)&=&a\cdot b+a\cdot c.\end{aligned}\]
In the set \(\mathbb{Z}\) there are "identity elements" for the two operations \(+\) and \(\cdot\), and these are the elements \(0\) and \(1\) respectively, that satisfy the basic properties \[\begin{aligned} a + 0 =0+a=a\\ a\cdot 1 = 1\cdot a=a\end{aligned}\] for every \(a\in\mathbb{Z}\).
The set \(\mathbb{Z}\) allows
additive inverses
for its elements, in the sense that for every \(a\in\mathbb{Z}\) there exists another integer in \(\mathbb{Z}\), denoted by \(-a\), such that \[a+(-a)=0.\] While for multiplication, only the integer 1 has a
multiplicative inverse
in the sense that 1 is the only integer \(a\) such that there exists another integer, denoted by \(a^{-1}\) or by \(1/a\), (namely 1 itself in this case) such that \[a\cdot a^{-1}=1.\]
From the operations of addition and multiplication one can define two other operations on \(\mathbb{Z}\), namely subtraction (denoted by \(-\)) and division (denoted by \(/\)). Subtraction is a binary operation on \(\mathbb{Z}\), i.e. defined for any two integers in \(\mathbb{Z}\), while division is not a binary operation and thus is defined only for some specific couple of integers in \(\mathbb{Z}\). Subtraction and division are defined as follows:
- \(a-b\) is defined by \(a+(-b)\), i.e. \(a-b=a+(-b)\) for every \(a,b\in\mathbb{Z}\)
- \(a/b\) is defined by the integer \(c\) if and only if \(a=b\cdot c\).
Contributors and Attributions
-
Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution ( CC BY ) license.


10.1: Basic Concepts
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Treatment and control groups. In experimental research, some subjects are administered one or more experimental stimulus called a treatment (the treatment group ) while other subjects are not given such a stimulus (the control group ). The treatment may be considered successful if subjects in the treatment group rate more favourably on outcome variables than control group subjects. Multiple levels of experimental stimulus may be administered, in which case, there may be more than one treatment group. For example, in order to test the effects of a new drug intended to treat a certain medical condition like dementia, if a sample of dementia patients is randomly divided into three groups, with the first group receiving a high dosage of the drug, the second group receiving a low dosage, and the third group receiving a placebo such as a sugar pill (control group), then the first two groups are experimental groups and the third group is a control group. After administering the drug for a period of time, if the condition of the experimental group subjects improved significantly more than the control group subjects, we can say that the drug is effective. We can also compare the conditions of the high and low dosage experimental groups to determine if the high dose is more effective than the low dose.
Treatment manipulation. Treatments are the unique feature of experimental research that sets this design apart from all other research methods. Treatment manipulation helps control for the ‘cause’ in cause-effect relationships. Naturally, the validity of experimental research depends on how well the treatment was manipulated. Treatment manipulation must be checked using pretests and pilot tests prior to the experimental study. Any measurements conducted before the treatment is administered are called pretest measures , while those conducted after the treatment are posttest measures .
Random selection and assignment. Random selection is the process of randomly drawing a sample from a population or a sampling frame. This approach is typically employed in survey research, and ensures that each unit in the population has a positive chance of being selected into the sample. Random assignment, however, is a process of randomly assigning subjects to experimental or control groups. This is a standard practice in true experimental research to ensure that treatment groups are similar (equivalent) to each other and to the control group prior to treatment administration. Random selection is related to sampling, and is therefore more closely related to the external validity (generalisability) of findings. However, random assignment is related to design, and is therefore most related to internal validity. It is possible to have both random selection and random assignment in well-designed experimental research, but quasi-experimental research involves neither random selection nor random assignment.
Threats to internal validity. Although experimental designs are considered more rigorous than other research methods in terms of the internal validity of their inferences (by virtue of their ability to control causes through treatment manipulation), they are not immune to internal validity threats. Some of these threats to internal validity are described below, within the context of a study of the impact of a special remedial math tutoring program for improving the math abilities of high school students.
History threat is the possibility that the observed effects (dependent variables) are caused by extraneous or historical events rather than by the experimental treatment. For instance, students’ post-remedial math score improvement may have been caused by their preparation for a math exam at their school, rather than the remedial math program.
Maturation threat refers to the possibility that observed effects are caused by natural maturation of subjects (e.g., a general improvement in their intellectual ability to understand complex concepts) rather than the experimental treatment.
Testing threat is a threat in pre-post designs where subjects’ posttest responses are conditioned by their pretest responses. For instance, if students remember their answers from the pretest evaluation, they may tend to repeat them in the posttest exam.
Not conducting a pretest can help avoid this threat.
Instrumentation threat , which also occurs in pre-post designs, refers to the possibility that the difference between pretest and posttest scores is not due to the remedial math program, but due to changes in the administered test, such as the posttest having a higher or lower degree of difficulty than the pretest.
Mortality threat refers to the possibility that subjects may be dropping out of the study at differential rates between the treatment and control groups due to a systematic reason, such that the dropouts were mostly students who scored low on the pretest. If the low-performing students drop out, the results of the posttest will be artificially inflated by the preponderance of high-performing students.
Regression threat —also called a regression to the mean—refers to the statistical tendency of a group’s overall performance to regress toward the mean during a posttest rather than in the anticipated direction. For instance, if subjects scored high on a pretest, they will have a tendency to score lower on the posttest (closer to the mean) because their high scores (away from the mean) during the pretest were possibly a statistical aberration. This problem tends to be more prevalent in non-random samples and when the two measures are imperfectly correlated.


10.3: Types of Contracts
Bilateral and Unilateral Contracts
In a bilateral contract , both parties make a promise of performance. These contracts are also called mutual or reciprocal contracts. Bilateral contracts are the most common form of contracts. They include ordering food in a restaurant, buying gas for vehicles, purchasing goods and services, etc.
A unilateral contract , on the other hand, is a contract where one party makes a promise that the other party can accept only by doing something. For example, a business offers a reward for information leading to the arrest of a thief. A person cannot collect the reward money by promising to give information–he or she must perform under the contract by providing the information.
Express and Implied Contracts
An express contract is a contract in words (orally or in writing) in which the terms are spelled out directly. The parties to an express contract, whether written or oral, clearly intend to make a legally enforceable agreement. For example, an agreement to buy a car for $1,000 and to take title next Monday is an express contract.
An implied contract is a contract that is inferred from the parties’ actions. Although no discussion of terms took place, an implied contract exists if it is clear from the conduct of the parties that they have an agreement. A delicatessen patron who asks for a “turkey sandwich to go” has made a contract and is obligated to pay when the sandwich is made. By ordering the food, the patron is implicitly agreeing to the price, whether posted or not.
Quasi-contract: Contract Implied in Law
Both express and implied contracts embody an actual agreement of the parties. A quasi-contract , by contrast, is an obligation imposed by law to avoid unjust enrichment of one person at the expense of another. In fact, a quasi-contract is not a contract at all. It is a judicial remedy in which the court decides what a contract should look like between the parties to prevent injustice. For example, a carpenter mistakenly believes a homeowner hired him to repair her porch, when it was actually the neighbor who hired him. One morning the carpenter arrives and begins work. Rather than stop him, the homeowner lets him proceed, excited to get her porch fixed for free. Although no contract exists because there was no offer, acceptance or consideration, the law will imply a contract between the carpenter and homeowner for the value of the work.
Enforceability
A contract that is fully enforceable and reflects the parties’ intent is valid . Conversely, an unenforceable contract is a contract where the parties intend to form a valid bargain but the court declares that it cannot be enforced for legal reasons. For example, Ramesh owes Jai money, but Jai has waited too long to collect it and the statute of limitations has run out. The contract for repayment is unenforceable and Jai is out of luck unless Ramesh makes a new promise to pay or actually pays part of the debt.
An agreement that is lacking one of the legal elements of a contract is void because it never was a contract. In other words, it is not legally enforceable because it is not a contract at all. An agreement that is illegal is also void. For example, a promise to commit a crime in return for payment is void because neither side can enforce the agreement in court.
By contrast, a voidable contract is a contract that can be annulled. It is a contract that is unenforceable by one party but enforceable by the other. For example, a minor may “avoid” a contract with an adult; meaning the adult may not enforce the contract against the minor if the minor refuses to carry out the bargain. The adult must comply if the minor wishes the contract to be performed. A contract may be voidable by both parties if they are both minors. Usually, the parties to a voidable contract are entitled to be restored to their original position.
A voidable contract remains a valid contract until it is voided. Thus, a contract with a minor remains in force unless the minor decides he does not wish to be bound by it. When minors become adults, they have two choices:
- Ratify the contract–that is, agree to be bound by it; or
- Disaffirm the contract–that is, disavow or avoid it.
Ratification may be explicit or implicit. For example, by continuing to make payments or retaining goods for an unreasonable period of time, a party may ratify the contract. If a party has not disaffirmed the contract while still a minor, she may do so within a reasonable time after becoming an adult.
Degree of Completion
An executory contract is a contract that has yet to be completed. Most executory contracts are enforceable. If some, but not all, of the terms of the contract have been performed, the contract is called partially executed . A contract that has been completed or carried out fully by both parties is called an executed contract .


17.4B: Electrocardiogram and Correlation of ECG Waves with Systole
An electrocardiogram, or ECG, is a recording of the heart’s electrical activity as a graph over a period of time.
- Describe electrocardiograms and their correlation with systole
Key Points
- An ECG is used to measure the rate and regularity of heartbeats as well as the size and position of the chambers, the presence of damage to the heart, and the effects of drugs or devices used to regulate the heart, such as a pacemaker.
- The ECG device detects and amplifies the tiny electrical changes on the skin that are caused when the heart muscle depolarizes during each heartbeat, and then translates the electrical pulses of the heart into a graphic representation.
- A typical ECG tracing of the cardiac cycle (heartbeat) consists of a P wave (atrial depolarization ), a QRS complex (ventricular depolarization), and a T wave (ventricular repolarization). An additional wave, the U wave ( Purkinje repolarization), is often visible, but not always.
- The ST complex is usually elevated during a myocardial infarction.
- Atrial fibrillation occurs when the P wave is missing and represents irregular, rapid, and inefficient atrial contraction, but is generally not fatal on its own.
- Ventricular fibrillation occurs when all normal waves of an ECG are missing, represents rapid and irregular heartbeats, and will quickly cause sudden cardiac death.
Key Terms
- fibrillation : A condition in which parts of the ECG do not appear normally, representing irregular, rapid, disorganized, and inefficient contractions of the atria or ventricles.
- ST segment : The line between the QRS complex and the T wave, representing the time when the ventricles are depolarized before repolarization begins.
An electrocardiogram (ECG or EKG) is a recording of the heart’s electrical activity as a graph over a period of time, as detected by electrodes attached to the outer surface of the skin and recorded by a device external to the body. The graph can show the heart’s rate and rhythm. It can also detect enlargement of the heart, decreased blood flow, or the presence of current or past heart attacks. ECGs are the primary clinical tool to measure electrical and mechanical performance of the heart.
The ECG works by detecting and amplifying tiny electrical changes on the skin that occur during heart muscle depolarization. The output for the ECG forms a graph that shows several different waves, each corresponding to a different electrical and mechanical event within the heart. Changes in these waves are used to identify problems with the different phases of heart activity.
ECG : Illustration of a patient undergoing a 12-lead ECG.
The P Wave
Normal Systole ECG : The U wave is not visible in all ECGs.
The first wave on an ECG is the P wave, indicating atrial depolarization in which the atria contract (atrial systole ). The P wave is the first wave on the ECG because the action potential for the heart is generated in the sinoatrial (SA) node, located on the atria, which sends action potentials directly through Bachmann’s bundle to depolarize the atrial muscle cells.
Increased or decreased P waves can indicate problems with the potassium ion concentration in the body that will alter nerve activity. A missing P wave indicates atrial fibrillation, a cardiac arrhythmia in which the heart beats irregularly, preventing efficient ventricular diastole. This is generally not fatal on its own.
The QRS Complex
The QRS complex refers to the combination of the Q, R, and S waves, and indicates ventricular depolarization and contraction (ventricular systole). The Q and S waves are downward waves while the R wave, an upward wave, is the most prominent feature of an ECG. The QRS complex represents action potentials moving from the AV node, through the bundle of His and left and right branches and Purkinje fibers into the ventricular muscle tissue. Abnormalities in the QRS complex may indicate cardiac hypertrophy or myocardial infarctions.
The T Wave and ST Segment
Animation of a Normal ECG Wave : The red lines represent the movement of the electrical signal through the heart.
The T Wave indicates ventricular repolarization, in which the ventricles relax following depolarization and contraction. The ST segment refers to the gap (flat or slightly upcurved line) between the S wave and the T wave, and represents the time between ventricular depolarization and repolarization. An elevated ST segment is the classic indicator for myocardial infarctions, though missing or downward sloping sloping ST segments may indicate myocardial ischemia.
Following the T wave is the U wave, which represents repolarization of the Purkinje fibers. It is not always visible on an ECG because it is a very small wave in comparison to the others.
Ventricular Fibrillation
When ECG output shows no identifiable P waves, QRS complexes, or T waves, it imdicates ventricular fibrillation, a severe arrhythmia. During ventricular fibrillation, the heart beats extremely fast and irregularly and can no longer pump blood, acting as a mass of quivering, disorganized muscle movements. Ventricular fibrillation will cause sudden cardiac death within minutes unless electrical resuscitation (with an AED) is performed immediately. It generally occurs with myocardial infarcations and heart failure, and is thought to be caused by action potentials that re-enter the AV nodes from the muscle tissue and induce rapid, irregular, weak contractions of the heart that fail to pump blood.


21.5: Cesarean Birth Discomforts
By the end of this section, you will be able to:
- Describe the postpartum recovery after a cesarean birth
- Plan the nursing education provided to a postpartum person who experienced a cesarean birth before being discharged home
Postpartum recovery after a cesarean birth includes many of the same assessments as a vaginal birth and postoperative surgeries. Vital signs, breasts, location and tone of the fundus , lochia , extremities, and level of pain are all assessed. Additional assessments of the postpartum person after a cesarean birth include the dressing over the incision for the first 24 hours and the incision itself after that, nausea and vomiting and abdomen, deep vein thrombosis ( DVT ), mobility, and the lungs for signs of infiltrates if general anesthesia was used. Pain and discomfort will be increased in persons after a cesarean birth.
Discomforts Related to Cesarean Births
The nurse recognizes certain discomforts related to cesarean birth , such as abdominal pain, incision pain, increased constipation and gas, and difficulty with ambulation . Education provided to the postpartum person includes education on and remedies for these discomforts. The nurse encourages ambulation, pain management prior to the pain becoming extreme, eating a healthy diet, and asking for help with activities of daily living.
Culture, Breast-Feeding, and Cesarean Birth
The breast-feeding rates in China show a 6-month exclusive rate of only 27.8 percent. Chinese people with a history of cesarean birth have even lower rates of breast-feeding at 6 months. Hu et al. (2020) found that people in China having a cesarean birth had a decreased willingness to breast-feed and more difficulty in breast-feeding than those having a vaginal birth. Because the cesarean rate in China increased to 35.9 percent, educating people about the importance of breast-feeding is essential. In this study, patients scheduled for cesarean section were provided additional breast-feeding education during their postpartum stay. Results showed the intervention group had higher breast-feeding rates up to 4 months postpartum, fewer cracked nipples, and fewer reports of insufficient milk supply. Education made a significant difference to the persons and infants in this study.
(Hu et al., 2020)
Gas Pain
Gas pain is common after a cesarean birth. Anesthetics used during surgery slow the bowels and decrease peristalsis. This leads to constipation and difficulty in passing gas. Also, opening of the peritoneum during surgery allows air to become trapped in the abdomen. That air must be absorbed and released as flatulence. The nurse encourages early and frequent ambulation and avoidance of extremely cold liquids to help release gas. Simethicone is often prescribed to relieve gas pain by aiding in gas release (Elsevier, 2024).
Incisional Pain
Pain related to cesarean birth is mostly incisional pain . As in any surgery, postpartum persons will feel increased incisional pain with movement, coughing, and ambulation. When coughing, the person is encouraged to splint the incision by placing a pillow over the lower abdomen to decrease pain (Figure 21.8). The nurse also encourages the person to ambulate slowly and to splint the incision when getting in and out of bed. Some health-care providers will order an abdominal binder to act as a splint.
Pain medication will be ordered by the health-care provider. The nurse will assess pain frequently and administer medications as needed. A morphine injection (Duramorph) may be used if spinal anesthesia was administered for a cesarean birth. A patient-controlled analgesia pump may be used, which slowly administers pain medication and allows the patient to give themselves a bolus when pain is intense. Some health-care providers will insert a catheter into the incision. The catheter is connected to a pain ball pump that administers a steady flow of pain medication directly to the incision to control pain. Figure 21.9 shows a pain pump used after cesarean birth.
Sleep Difficulties
Sleep can be difficult after a cesarean birth because of incisional and gas pain. The postpartum person experiences pain at the incision while turning or moving. This can disrupt sleep. Breast-feeding or holding the newborn can also cause incisional pain and difficulty in sleeping. The nurse encourages relaxation techniques, deep breathing, and relaxation imagery to help the person release stress prior to sleeping. The nurse assesses pain and provides pain medications prior to pain becoming severe and interfering with sleep (Elsevier, 2024). The nurse can cluster assessments and interventions to decrease the number of sleep interruptions. They can also encourage the patient to take naps and ask the family to limit interruptions.
Incisional Healing
Incisions from cesarean births heal differently, depending on the type of incision. The majority of cesarean births involve low, transverse incisions. These incisions heal faster with less pain than vertical incisions. For incisions closed with staples, the staples will be removed several days after the surgery. Other incisions are closed with suture that is dissolvable and will not need removal of stitches. Still other incisions are closed using surgical glue. Almost all cesarean birth incisions are covered with a dressing immediately after surgery. The dressing most often is removed the next day to allow for airflow to the incision. The nurse assesses the incision and instructs the postpartum patient to keep the incision clean and dry.
Restrictions Related to Surgery
After surgery, ambulation is encouraged. Exercise, however, is restricted until the patient is seen by the health-care provider at the 6-week postpartum checkup. The nurse discourages lots of stair climbing and housework. Sexual intercourse is restricted with a cesarean birth until the patient is seen for follow-up. While persons are taking pain medications, driving a car is restricted (Elsevier, 2024).
When to Follow Up
Postpartum follow-up will be determined by the health-care provider, if complications occur, and if staples need removing. For staple removal, the person is usually seen within the first few days after discharge. Otherwise, persons are usually seen in the office between 2 and 6 weeks to evaluate the incision, screen for postpartum depression, discuss contraception, and perform a postpartum assessment. In addition, some health-care providers see the person for follow-up again between 10 and 12 weeks for an annual gynecologic exam.
Nursing Education for the Birthing Person Experiencing a Cesarean Birth
For persons experiencing a cesarean birth, the nurse provides routine postpartum education, education on pain medications, along with education on restrictions and recommendations related to the procedure.
Opioid Medication
Opioid medication causes drowsiness and sleepiness. The nurse encourages the postpartum patient to rest and focus on caring for themselves and the newborn. The nurse explains that while taking opioid pain medications, the postpartum person should not drive a car. Driving is safe once the person has transitioned to only acetaminophen or ibuprofen .
Lifting Restrictions
Lifting is restricted to the weight of the newborn for the first 6 weeks postpartum. Persons are discouraged from lifting older children. The nurse encourages the family to assist the older child to sit next to the postpartum person in a chair or on the couch.
Importance of Fluids
Increased water intake helps to increase breast milk and prevent dehydration. Persons experiencing a cesarean birth are at higher risk for blood clots. Dehydration can increase that risk even more.
Pillow on the Lap
The person can use a pillow or an abdominal binder to splint the incision while moving or getting out of bed. The pillow can also be used to assist in holding the newborn. When the person coughs or sneezes, the pillow is also used for splinting to prevent pain (see Figure 21.8).
Football Hold for Breast-Feeding
Breast-feeding can be painful if the newborn is held against the incision. The nurse will demonstrate different ways to hold the newborn while nursing. Football hold allows the newborn to avoid contact with the incision but still nurse with a good latch. The newborn is held to the person’s side supported by pillows. The hand supports the baby’s head and turns the baby to face the breast (Figure 21.10).
Importance of Ambulating
Ambulation after cesarean section is important for several reasons. Ambulation keeps the blood moving in the extremities helping to prevent venous stasis and DVT s.  Ambulation also aids in recovery by increasing circulation and helping to decrease edema.


2.2: What is the purpose of education?
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By: Karen Herndon
Introduction
Have you ever asked yourself what the purpose of education is? It is believed to be a question that is highly thought about but in reality it is not. We tend to focus on the tasks at hand rather than the overall goal. Many will find that when asked specifically what education’s purpose is, the answer in return is nothing outside of the course curriculum (Bass, 1997). Well you may be questioning why this is. In order to provide an answer we need to consider both sides of the topic. On one hand the government controls the educational system which enforces the use of mandatory testing to evaluate each student as well as the educational institution. In other words, emphasis is placed on providing students with the skill that they will need to succeed. On the other hand, there is the belief that children should not only be presented with the abilities to learn but they should be able to expand what is given through individual growth and development beyond materials obtained from the classroom. Their inner talents need to be brought out and polished (Minor, 2007).
Note
“Do not then train youth to learn by force and harshness; but direct them to it by what amuses their minds so that you may be better able to discover with accuracy the peculiar bent of the genius of each." ~Plato (Minor, 2007)
Government Effects on Education’s Purpose
If one looks closely at the government’s heavy involvement in what to teach and what not to teach children it’s the administering of assessment tests. These tests do not allow a child to form opinions on subjects which in turn impedes social growth and development. The government forces school districts to meet certain minimal requirements so unfortunately the focal point has become mainly to teach material that students will need to know in order to obtain passing scores on standardized tests (Bass, 1997). This is attributed to the government’s implementation of laws such as the No Child Left Behind Act (NCLB) (A Firsthand Look at NCLB, 2006). Students are expected to follow certain guidelines and curriculum; however teachers are pressed for time. In order to ensure they achieve all of the mandated guidelines and curriculum they teach only what they are required to. This leaves little time to introduce material to students that may be beneficial to them in the long run (Bass, 1997). For example the NCLB has received strict criticism for focusing too much on testing and not enough on actually teaching and allowing the student to further their knowledge. It has been brought up that the NCLB’s ignorance towards equity has caused problems. For instance one school system may possess funding which would make it easier to obtain the minimum passing score versus a school system that lacked funding. (A Firsthand Look at NCLB, 2006).
Social Development’s Role in Education
Education is not only being presented with material to learn but to also expand one’s knowledge of themselves and their surroundings. It appears that the growing trend today in school systems is to teach students particular course work with little to no regard to instruction on how this material could relate to life. Since most subjects utilize textbooks they are heavily relied upon and do not allow the student much freedom to think outside the box (Lim, 2005). Children will naturally form opinions from the material that is presented and being able to express these opinions will only help them to grow socially and eventually fit in with the rest of the world. Social growth is very important because when it is time for the child to become independent if they have not developed that part of their life then interaction with other people and situations could be quite daunting. Most parents and parental guardians do want their child to follow a structured curriculum; however, they still want their child to have time for recreation and family. Childhood is an important part of everyone’s life. It encourages social interaction and development as well as teaching them to be independent. People that are deprived of their childhood regret it when they grow up which could negatively impact their lives and the lives of others (Lim, 2005).
Note
“The only purpose of education is to teach a student how to live his life-by developing his mind and equipping him to deal with reality. The training he needs is theoretical, i.e., conceptual. He has to be taught to think, to understand, to integrate, to prove. He has to be taught the essentials of the knowledge discovered in the past-and he has to be equipped to acquire further knowledge by his own effort. ~Ayn Rand
(Yero, 2001-2002. p. 1)
Thomas Jefferson’s View
With regard to the purpose of education Thomas Jefferson was one of the biggest advocates of the principle that ignorance and political liberty could not co-exist. Jefferson believed the purpose of education was to properly prepare young minds so they would be able to make educated decisions and uphold the integrity of the country. He insisted on providing four main subjects to elementary school students. Geography, arithmetic along with reading and writing made up these subjects. These subjects were deemed crucial for the proper development of children in order to function later in life. One example of this is that Jefferson believed that children needed to be given proper education in order to become informed voters. He supported free education through taxation as well as equal opportunity education. Jefferson believed the purpose of education was not to segregate but to educate (Jewett, 1997).
Evaluating Student Assessment
Getting back to assessment and student evaluation drives one to consider if whether we are going about education wrong as a country by placing so much importance upon standardized tests. As previously stated, it has been debated whether The No Child Left Behind Act does little to expand the constantly developing minds of our youth (A Firsthand Look at NCLB, 2006). A child’s mind can be compared to a sponge, in that they need to soak up information and substance in order for it to expand, otherwise, it dries out. If you observe the examinations and how students are generally evaluated in today’s school systems, you will see that the vast majority of them are made up of multiple choice questions. Since there is typically only one correct answer, it is almost impossible for the child to reflect upon the question and develop a view that they can grow from or share with others if they so choose (Yero, 2001-2002).
In Summation
When considering both sides of the purpose of education you may want to reflect on the words of Plato when he said: “Do not then train youth to learn by force and harshness; but direct them to it by what amuses their minds so that you may be better able to discover with accuracy the peculiar bent of the genius of each (Minor, 2007).” However, the government’s involvement in education has had positive impacts on students since the NCLB has forced teachers and other school officials to focus on all children including those with special needs (A Firsthand Look at NCLB, 2006). No matter what side of the river you are on with regard to your opinion on this topic, it is hard to discredit the fact that one purpose (if not the main purpose) of education is to prepare the youth for their future and ours.
Exercise \(\PageIndex{1}\)
1. Thomas Jefferson believed that
A. Awareness and political liberty could not co-exist.
B. Awareness and tyranny could not co-exist.
C. Ignorance and political liberty could not co-exist.
D. Ignorance and tyranny could not co-exist.
2. What does NCLB stand for?
A. No Child Left Backward
B. No Child Left Behind
C. No Children Left Backward
D. No Children Left Behind
3. Thomas Jefferson affirmed that the purpose of education was not to segregate but to?
A. Educate
B. Estimate
C. Intimidate
D. Procrastinate
4. What plays an integral part in everyone's life?
A. Childhood
B. Hapiness
C. Upbringing
D. Wealth
5. Generally student evaluations in today's school systems are primarily made up of what?
A. Essay questions or topics
B. Multiple choice questions
C. Short answer questions
D. True and false questions
- Answer
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1. C
2. B
3. A
4. A
5. B
References
A First Hand Look at NCLB. (2006). Educational Leadership , (ERIC Document Reproduction Service No. EJ766269) Retrieved February 2, 2008, from ERIC database.
Bass, Randall V. (1997). The Purpose of Education. The Educational Forum 61. 128-32. Retrieved February 1, 2008 from http://vnweb.hwwilsonweb.com.proxy.lib.odu.edu/hww/jumpstart.jhtml?recid=0bc05f7a67b1790e1e9c442f93fe94fd41479b00e75d7c18b3344ea852044009af47f7afa8bcd3df&fmt=H
Jewett, Thomas O. (1997). Thomas Jefferson and the Purposes of Education. The Educational Forum 61. 110-113. Retrieved February 1, 2008 from http://vnweb.hwwilsonweb.com.proxy.lib.odu.edu/hww/jumpstart.jhtml?recid=0bc05f7a67b1790e1e9c442f93fe94fd41479b00e75d7c18b3344ea852044009ceb909ea41cb5644&fmt=H
Lim, Mike. (2005). What is Education? The All I Need . Retrieved February 2, 2008 from http://www.theallineed.com/family/05032602.htm
Minor, Summer. (2007). The Purpose of Education is… Mom is Teaching Blog . Retrieved February 2, 2008 from http://www.momisteaching.com/the-purpose-of-education-is/+purpose+of+education+summer+minor&hl=en&ct=clnk&cd=1&gl=us
Yero, Judith Lloyd. (2001-2002). The Meaning of Education. Teacher’s Mind Resources . 1-3. Retrieved February 2, 2008 from www.teachersmind.com/pdfdirectory/Education.PDF