5.2: Layers of the Skin
By the end of this section, you will be able to:
- Identify the components of the integumentary system
- Describe the layers of the skin and the functions of each layer
- Identify and describe the hypodermis and deep fascia
- Describe the role of keratinocytes and their life cycle
- Describe the role of melanocytes in skin pigmentation
Although you may not typically think of the skin as an organ, it is in fact made of tissues that work together as a single structure to perform unique and critical functions. The skin and its accessory structures make up the integumentary system , which provides the body with overall protection. The skin is made of multiple layers of cells and tissues, which are held to underlying structures by connective tissue (Figure \(\PageIndex{1}\)). The deeper layer of skin is well vascularized (has numerous blood vessels). It also has numerous sensory, and autonomic and sympathetic nerve fibers ensuring communication to and from the brain.
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The skin consists of two main layers and a closely associated layer. View this animation to learn more about layers of the skin. What are the basic functions of each of these layers?
The Epidermis
The epidermis is composed of keratinized, stratified squamous epithelium. It is made of four or five layers of epithelial cells, depending on its location in the body. It does not have any blood vessels within it (i.e., it is avascular). Skin that has four layers of cells is referred to as “thin skin.” From deep to superficial, these layers are the stratum basale, stratum spinosum, stratum granulosum, and stratum corneum. Most of the skin can be classified as thin skin. “Thick skin” is found only on the palms of the hands and the soles of the feet. It has a fifth layer, called the stratum lucidum, located between the stratum corneum and the stratum granulosum (Figure \(\PageIndex{2}\)).
The cells in all of the layers except the stratum basale are called keratinocytes. A keratinocyte is a cell that manufactures and stores the protein keratin. Keratin is an intracellular fibrous protein that gives hair, nails, and skin their hardness and water-resistant properties. The keratinocytes in the stratum corneum are dead and regularly slough away, being replaced by cells from the deeper layers (Figure \(\PageIndex{3}\)).
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View the University of Michigan WebScope to explore the tissue sample in greater detail. If you zoom on the cells at the outermost layer of this section of skin, what do you notice about the cells?
Stratum Basale
The stratum basale (also called the stratum germinativum) is the deepest epidermal layer and attaches the epidermis to the basal lamina, below which lie the layers of the dermis. The cells in the stratum basale bond to the dermis via intertwining collagen fibers, referred to as the basement membrane. A finger-like projection, or fold, known as the dermal papilla (plural = dermal papillae) is found in the superficial portion of the dermis. Dermal papillae increase the strength of the connection between the epidermis and dermis; the greater the folding, the stronger the connections made (Figure \(\PageIndex{4}\)).
The stratum basale is a single layer of cells primarily made of basal cells. A basal cell is a cuboidal-shaped stem cell that is a precursor of the keratinocytes of the epidermis. All of the keratinocytes are produced from this single layer of cells, which are constantly going through mitosis to produce new cells. As new cells are formed, the existing cells are pushed superficially away from the stratum basale. Two other cell types are found dispersed among the basal cells in the stratum basale. The first is a Merkel cell , which functions as a receptor and is responsible for stimulating sensory nerves that the brain perceives as touch. These cells are especially abundant on the surfaces of the hands and feet. The second is a melanocyte , a cell that produces the pigment melanin. Melanin gives hair and skin its color, and also helps protect the living cells of the epidermis from ultraviolet (UV) radiation damage.
In a growing fetus, fingerprints form where the cells of the stratum basale meet the papillae of the underlying dermal layer (papillary layer), resulting in the formation of the ridges on your fingers that you recognize as fingerprints. Fingerprints are unique to each individual and are used for forensic analyses because the patterns do not change with the growth and aging processes.
Stratum Spinosum
As the name suggests, the stratum spinosum is spiny in appearance due to the protruding cell processes that join the cells via a structure called a desmosome . The desmosomes interlock with each other and strengthen the bond between the cells. It is interesting to note that the “spiny” nature of this layer is an artifact of the staining process. Unstained epidermis samples do not exhibit this characteristic appearance. The stratum spinosum is composed of eight to 10 layers of keratinocytes, formed as a result of cell division in the stratum basale (Figure \(\PageIndex{5}\)). Interspersed among the keratinocytes of this layer is a type of dendritic cell called the Langerhans cell , which functions as a macrophage by engulfing bacteria, foreign particles, and damaged cells that occur in this layer.
Interactive Link
View the University of Michigan WebScope to explore the tissue sample in greater detail. If you zoom on the cells at the outermost layer of this section of skin, what do you notice about the cells?
The keratinocytes in the stratum spinosum begin the synthesis of keratin and release a water-repelling glycolipid that helps prevent water loss from the body, making the skin relatively waterproof. As new keratinocytes are produced atop the stratum basale, the keratinocytes of the stratum spinosum are pushed into the stratum granulosum.
Stratum Granulosum
The stratum granulosum has a grainy appearance due to further changes to the keratinocytes as they are pushed from the stratum spinosum. The cells (three to five layers deep) become flatter, their cell membranes thicken, and they generate large amounts of the proteins keratin, which is fibrous, and keratohyalin , which accumulates as lamellar granules within the cells (see Figure 5.5). These two proteins make up the bulk of the keratinocyte mass in the stratum granulosum and give the layer its grainy appearance. The nuclei and other cell organelles disintegrate as the cells die, leaving behind the keratin, keratohyalin, and cell membranes that will form the stratum lucidum, the stratum corneum, and the accessory structures of hair and nails.
Stratum Lucidum
The stratum lucidum is a smooth, seemingly translucent layer of the epidermis located just above the stratum granulosum and below the stratum corneum. This thin layer of cells is found only in the thick skin of the palms, soles, and digits. The keratinocytes that compose the stratum lucidum are dead and flattened (see Figure 5.5). These cells are densely packed with eleidin , a clear protein, derived from keratohyalin, which gives these cells their transparent (i.e., lucid) appearance.
Stratum Corneum
The stratum corneum is the most superficial layer of the epidermis and is the layer exposed to the outside environment (see Figure 5.5). The increased keratinization (also called cornification) of the cells in this layer gives it its name. There are usually 15 to 30 layers of cells in the stratum corneum. This dry, dead layer helps prevent the penetration of microbes and the dehydration of underlying tissues, and provides a mechanical protection against abrasion for the more delicate, underlying layers. Cells in this layer are shed periodically and are replaced by cells pushed up from the stratum granulosum (or stratum lucidum in the case of the palms and soles of feet). The entire layer is replaced during a period of about 4 weeks. Cosmetic procedures, such as microdermabrasion, help remove some of the dry, upper layer and aim to keep the skin looking “fresh” and healthy.
Dermis
The dermis might be considered the “core” of the integumentary system (derma- = “skin”), as distinct from the epidermis (epi- = “upon” or “over”) and hypodermis (hypo- = “below”). It contains blood and lymph vessels, nerves, and other structures, such as hair follicles and sweat glands. The dermis is made of two layers of connective tissue that compose an interconnected mesh of elastin and collagenous fibers, produced by fibroblasts (Figure \(\PageIndex{6}\)).
Papillary Layer
The papillary layer is made of loose, areolar connective tissue, which means the collagen and elastin fibers of this layer form a loose mesh. This superficial layer of the dermis projects into the stratum basale of the epidermis to form finger-like dermal papillae (see Figure 5.7).  In addition, the papillary layer contains phagocytes, defensive cells that help fight bacteria or other infections that have breached the skin. This layer also contains lymphatic capillaries, nerve fibers, and touch receptors called the Meissner corpuscles.
Reticular Layer
Underlying the papillary layer is the much thicker reticular layer , composed of dense, irregular connective tissue. This layer is well vascularized and has a rich sensory and sympathetic nerve supply. The reticular layer appears reticulated (net-like) due to a tight meshwork of fibers. Elastin fibers provide some elasticity to the skin, enabling movement. Collagen fibers provide structure and tensile strength, with strands of collagen extending into both the papillary layer and the hypodermis. In addition, collagen binds water to keep the skin hydrated. Collagen injections and Retin-A creams help restore skin turgor by either introducing collagen externally or stimulating blood flow and repair of the dermis, respectively.
Hypodermis
The hypodermis (also called the subcutaneous layer or superficial fascia) is a layer directly below the dermis and serves to connect the skin to the underlying fascia (fibrous tissue) of the bones and muscles. It is not strictly a part of the skin, although the border between the hypodermis and dermis can be difficult to distinguish. The hypodermis consists of well-vascularized, loose, areolar connective tissue and adipose tissue, which functions as a mode of fat storage and provides insulation and cushioning for the integument.
Everyday Connection
Lipid Storage
The hypodermis is home to most of the fat that concerns people when they are trying to keep their weight under control. Adipose tissue present in the hypodermis consists of fat-storing cells called adipocytes. This stored fat can serve as an energy reserve, insulate the body to prevent heat loss, and act as a cushion to protect underlying structures from trauma.
Where the fat is deposited and accumulates within the hypodermis depends on hormones (testosterone, estrogen, insulin, glucagon, leptin, and others), as well as genetic factors. Fat distribution changes as our bodies mature and age. Males tend to accumulate fat in different areas (neck, arms, lower back, and abdomen) than do females (breasts, hips, thighs, and buttocks). The body mass index (BMI) is often used as a measure of fat, although this measure is, in fact, derived from a mathematical formula that compares body weight (mass) to height. Therefore, its accuracy as a health indicator can be called into question in individuals who are extremely physically fit.
In many animals, there is a pattern of storing excess calories as fat to be used in times when food is not readily available. In much of the developed world, insufficient exercise coupled with the ready availability and consumption of high-calorie foods have resulted in unwanted accumulations of adipose tissue in many people. Although periodic accumulation of excess fat may have provided an evolutionary advantage to our ancestors, who experienced unpredictable bouts of famine, it is now becoming chronic and considered a major health threat. Recent studies indicate that a distressing percentage of our population is overweight and/or clinically obese. Not only is this a problem for the individuals affected, but it also has a severe impact on our healthcare system. Changes in lifestyle, specifically in diet and exercise, are the best ways to control body fat accumulation, especially when it reaches levels that increase the risk of heart disease and diabetes.
Pigmentation
The color of skin is influenced by a number of pigments, including melanin, carotene, and hemoglobin. Recall that melanin is produced by cells called melanocytes, which are found scattered throughout the stratum basale of the epidermis. The melanin is transferred into the keratinocytes via a cellular vesicle called a melanosome (Figure \(\PageIndex{7}\)).
Melanin occurs in two primary forms. Eumelanin exists as black and brown, whereas pheomelanin provides a red color. Dark-skinned individuals produce more melanin than those with pale skin. Exposure to the UV rays of the sun or a tanning salon causes melanin to be manufactured and built up in keratinocytes, as sun exposure stimulates keratinocytes to secrete chemicals that stimulate melanocytes. The accumulation of melanin in keratinocytes results in the darkening of the skin, or a tan. This increased melanin accumulation protects the DNA of epidermal cells from UV ray damage and the breakdown of folic acid, a nutrient necessary for our health and well-being. In contrast, too much melanin can interfere with the production of vitamin D, an important nutrient involved in calcium absorption. Thus, the amount of melanin present in our skin is dependent on a balance between available sunlight and folic acid destruction, and protection from UV radiation and vitamin D production.
It requires about 10 days after initial sun exposure for melanin synthesis to peak, which is why pale-skinned individuals tend to suffer sunburns of the epidermis initially. Dark-skinned individuals can also get sunburns, but are more protected than are pale-skinned individuals. Melanosomes are temporary structures that are eventually destroyed by fusion with lysosomes; this fact, along with melanin-filled keratinocytes in the stratum corneum sloughing off, makes tanning impermanent.
Too much sun exposure can eventually lead to wrinkling due to the destruction of the cellular structure of the skin, and in severe cases, can cause sufficient DNA damage to result in skin cancer. When there is an irregular accumulation of melanocytes in the skin, freckles appear. Moles are larger masses of melanocytes, and although most are benign, they should be monitored for changes that might indicate the presence of cancer (Figure \(\PageIndex{8}\)).
Disorders of the...
Integumentary System
The first thing a clinician sees is the skin, and so the examination of the skin should be part of any thorough physical examination. Most skin disorders are relatively benign, but a few, including melanomas, can be fatal if untreated. A couple of the more noticeable disorders, albinism and vitiligo, affect the appearance of the skin and its accessory organs. Although neither is fatal, it would be hard to claim that they are benign, at least to the individuals so afflicted.
Albinism is a genetic disorder that affects (completely or partially) the coloring of skin, hair, and eyes. The defect is primarily due to the inability of melanocytes to produce melanin. Individuals with albinism tend to appear white or very pale due to the lack of melanin in their skin and hair. Recall that melanin helps protect the skin from the harmful effects of UV radiation. Individuals with albinism tend to need more protection from UV radiation, as they are more prone to sunburns and skin cancer. They also tend to be more sensitive to light and have vision problems due to the lack of pigmentation on the retinal wall. Treatment of this disorder usually involves addressing the symptoms, such as limiting UV light exposure to the skin and eyes. In vitiligo , the melanocytes in certain areas lose their ability to produce melanin, possibly due to an autoimmune reaction. This leads to a loss of color in patches (Figure \(\PageIndex{9}\)). Neither albinism nor vitiligo directly affects the lifespan of an individual.
Other changes in the appearance of skin coloration can be indicative of diseases associated with other body systems. Liver disease or liver cancer can cause the accumulation of bile and the yellow pigment bilirubin, leading to the skin appearing yellow or jaundiced ( jaune is the French word for “yellow”). Tumors of the pituitary gland can result in the secretion of large amounts of melanocyte-stimulating hormone (MSH), which results in a darkening of the skin. Similarly, Addison’s disease can stimulate the release of excess amounts of adrenocorticotropic hormone (ACTH), which can give the skin a deep bronze color. A sudden drop in oxygenation can affect skin color, causing the skin to initially turn ashen (white). With a prolonged reduction in oxygen levels, dark red deoxyhemoglobin becomes dominant in the blood, making the skin appear blue, a condition referred to as cyanosis ( kyanos is the Greek word for “blue”). This happens when the oxygen supply is restricted, as when someone is experiencing difficulty in breathing because of asthma or a heart attack. However, in these cases the effect on skin color has nothing do with the skin’s pigmentation.
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This ABC video follows the story of a pair of fraternal African-American twins, one of whom is albino. Watch this video to learn about the challenges these children and their family face. Which ethnicities do you think are exempt from the possibility of albinism?


22.1: The Biot-Savart Law
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The Biot-Savart law allows us to determine the magnetic field at some position in space that is due to an electric current. More precisely, the Biot-Savart law allows us to calculate the infinitesimal magnetic field, \(d\vec B\), that is produced by a small section of wire, \(d\vec l\), carrying current, \(I\), such that \(d\vec l\) is co-linear with the wire and points in the direction of the electric current:
\[d\vec B=\frac{\mu_{0}I}{4\pi}\frac{d\vec l\times\hat r}{r^{2}}\]
where, \(\vec r\), is the vector from the element of wire, \(d\vec l\), to the point where we would like to determine the magnetic field, as illustrated in Figure \(\PageIndex{1}\). \(µ_{0}\) is a constant of proportionality called the “permeability of free space”, and has the value \(µ_{0} = 4π × 10^{−7}\text{T} ·\text{m/A}\).
The Biot-Savart Law has some similarities with the Coulomb Law to calculate the electric field, as the magnitude of the magnetic field decreases as the inverse of the square distance between the source and the field. However, it can only be expressed in differential form (i.e. as an infinitesimal), and it requires working in three dimensions, because of the cross product. It is usually more convenient to use the Biot-Savart Law in the form:
\[\begin{aligned}d\vec B = \frac{\mu_{0}I}{4\pi}\frac{d\vec l\times \vec r}{r^{3}} \end{aligned}\]
where the unit vector \(\hat r\) was replaced by \(\vec r/r\).
The procedure for applying the Biot-Savart Law is as follows
HowTo: Applying the Biot-Savart Law
- Make a really good diagram, as you will have to include some 3D aspects.
- Choose an infinitesimal section of wire, \(d\vec l\).
- Determine the vector \(\vec r\).
- Determine the cross-product, \(d\vec l ×\vec r\), which will point in the direction of the magnetic field from that infinitesimal section of wire.
- Write out the infinitesimal vector \(d\vec B\), and determine its components.
- Think about symmetry! As you sum the \(d\vec B\), will some components cancel? If yes, you do not need to do those integrals.
- Determine the total magnetic field, component by component, by summing (integrating) the components of \(d\vec B\) over the wire.
Magnetic field from a straight current-carrying wire
In this section, we use the Biot-Savart Law to determine the magnetic field a distance, \(h\), from the center of a finite straight wire of length, \(L\), carrying current, \(I\), as illustrated in Figure \(\PageIndex{2}\).
We start by choosing an infinitesimal element of wire, \(d\vec l\), a distance y above the center of the wire, as shown (we choose the origin to be located at the center of the wire). The vector \(d\vec l\) is thus given by:
\[\begin{aligned}d\vec l=dl\hat y \end{aligned}\]
The vector, \(\vec r\), from \(d\vec l\) to the point at which we would like to know the magnetic field is given by:
\[\begin{aligned} \vec r&=r\cos\theta\hat x-r\sin\theta\hat y \\[4pt] r&=\sqrt{h^{2}+y^{2}}=\frac{h}{\cos\theta} \end{aligned}\]
The cross-product between \(d\vec l\) and \(\vec r\) is easily found with the right-hand rule to point into the page (corresponding to the negative \(z\) direction). The magnitude of the cross-product is given by:
\[\begin{aligned} ||d\vec l ×\vec r|| = dlr \sin φ \end{aligned}\]
where \(φ = π/2 + θ\) is the angle between \(d\vec l\) and \(\vec r\), so that \(\sin φ = \cos θ\). The cross-product can thus be written in terms of \(θ\) as:
\[\begin{aligned} d\vec l ×\vec r = −dlr \cos θ\hat z \end{aligned}\]
Note that we can also determine the cross-product algebraically instead of using the righthand rule and the magnitude:
\[\begin{aligned} d\vec l\times\vec r &=(dl\vec y)\times (r\cos\theta\hat x-r\sin\theta\hat y) \\[4pt] &=dlr\cos\theta(\hat y\times\hat x)-rdl\sin\theta(\hat y\times\hat y) \\[4pt] &=-dlr\cos\theta\hat z \end{aligned}\]
The infinitesimal magnetic field element, \(d\vec B\), is given by:
\[\begin{aligned} d\vec B=\frac{\mu_{0}I}{4\pi}\frac{d\vec l\times\vec r}{r^{3}}=-\frac{\mu_{0}I}{4\pi}\frac{dl\cos\theta}{r^{2}}\hat z \end{aligned}\]
Any segment along the wire will result in a magnetic field that is into the page (negative \(z\) direction), thus there will be no cancellations due to any symmetries. We can now proceed to perform the integral.
We can use either \(θ\) or \(y\) to label the wire elements and carry out the integration. We will choose to integrate over \(θ\), requiring us to express \(dl\) and \(r\) in terms of \(θ\) (and constants), as those are the only quantities in \(d\vec B\) that depend on the position of \(d\vec l\). In order to express \(dl\) in terms of \(dθ\), we first relate \(θ\) to \(y\), the position of the wire element:
\[\begin{aligned}y=h\tan\theta \rightarrow dl=dy=\frac{dy}{d\theta}d\theta = \frac{h}{\cos ^{2}\theta}d\theta \end{aligned}\]
and \(r\) is given by:
\[\begin{aligned} r=\frac{h}{\cos\theta}\rightarrow\frac{1}{r^{2}}=\frac{\cos ^{2}\theta}{h^{2}} \end{aligned}\]
Putting this altogether into \(d\vec B\):
\[\begin{aligned} d\vec B=-\frac{\mu_{0}I}{4\pi}\frac{dl\cos\theta}{r^{2}}\hat z =-\frac{\mu_{0}I}{4\pi}\left( \frac{h}{\cos ^{2}\theta}d\theta\right)\left(\frac{\cos ^{2}\theta}{h^{2}}\right)\cos\theta\hat z=-\frac{\mu_{0}I}{4\pi h}\cos\theta d\theta\hat z=dB_{z}\hat z \end{aligned}\]
We define the angle, \(θ_{0}\), to be the maximum amplitude of the angle \(θ\) when integrating over the wire (see Figure \(\PageIndex{2}\)), so that we integrate \(θ\) from \(−θ_{0}\) to \(+θ_{0}\):
\[\begin{aligned} B_{z}=\int_{-\theta_{0}}^{+\theta_{0}} dB_{z}=-\frac{\mu_{0}I}{4\pi h}\int_{-\theta_{0}}^{+\theta_{0}}\cos\theta d\theta =-\frac{\mu_{0}I}{4\pi h}(2\sin\theta_{0})=-\frac{\mu_{0}I}{2\pi h}\sin\theta_{0} \end{aligned}\]
Using the given dimensions:
\[\begin{aligned} \sin\theta_{0}=\frac{L/2}{\sqrt{h^{2}+\frac{L^{2}}{4}}} \end{aligned}\]
Thus, the magnetic field, \(\vec B\), a distance, \(h\), from the center of a wire of length, \(L\), carrying current, \(I\), in the positive \(y\) direction is given by:
\[\vec B=-\frac{\mu_{0}I}{2\pi h}\frac{L/2}{\sqrt{h^{2}+\frac{L^{2}}{4}}}\quad\text{(finite wire)}\]
The magnetic field must be rotationally symmetric; that is, if the wire is vertical, the magnetic field at a distance h must look the same regardless of the angle from which we view the vertical wire (we should always see the magnetic field going into the page at the point that we use in Figure \(\PageIndex{2}\)). Thus, the magnetic field lines must form circles around the wire, as illustrated in Figure \(\PageIndex{3}\). Note that the direction of the magnetic field is given by the right-hand rule for axial vectors; when you align your thumb with the current, your fingers curl in the direction of the magnetic field.
It is of particular interest to investigate the limiting case of an infinitely long wire, in the limit of \(L → ∞\), or equivalently, \(θ_{0} → \frac{π}{2}\). The latter is easiest to evaluate, since \(\sin θ_{0} → 1\). The magnitude of the magnetic field, \(\vec B\), a distance, \(h\), from an infinite wire carrying current, \(I\), is given by:
\[B=\frac{\mu_{0}I}{2\pi h}\quad\text{(infinite wire)}\]
One can often make the approximation that the wire is infinite in length, when the distance, \(h\), is small compared to the length, \(L\), of the wire.
Magnetic field from a circular current-carrying wire
In this section, we examine the magnetic field that is created by a circular current-carrying loop of wire. We can determine the shape of the magnetic field, by considering small sections as straight wires, with circular magnetic field lines around them. As we move closer to the center of the ring, those fields sum together, as illustrated in Figure \(\PageIndex{4}\). Note that the magnetic field from a loop of current is identical to that from a bar magnet (as a bar magnet is, of course, a collection of current loops).
Below, we use the Biot-Savart Law to derive an expression for the magnitude of the magnetic field at a distance, \(h\), from the center of a ring of radius, \(R\), along its axis of symmetry, when there is a current, \(I\), in the ring. While the mathematics are much easier than the case for the straight wire, the challenge in this case is to visualize the calculation in three dimensions! Figure \(\PageIndex{5}\) shows the loop of current, as well as our choice of the coordinate system (with the origin at the center of the ring). In particular, we wish to calculate the magnetic field at a distance, \(h\), along the \(z\) axis. The \(x\) axis goes into the page.
In order to apply the Biot-Savart Law, we choose an element, \(d\vec l\), of wire at the top of the ring, as illustrated. At this position, the element, \(d\vec l\), points in the positive \(x\) direction (into the page):
\[\begin{aligned} d\vec l=dl\hat x \end{aligned}\]
The vector, \(\vec r\), from the wire element to the point where we wish to determine the magnetic field is given by:
\[\begin{aligned}\vec r=-r\sin\theta\hat y+r\cos\theta\hat z \end{aligned}\]
and the angle \(θ\) will be the same for all wire elements along the ring. The cross-product, \(d\vec l ×\vec r\), can be evaluated algebraically:
\[\begin{aligned} d\vec l\times\vec r&=(dl\hat x)\times(-r\sin\theta\hat y+r\cos\theta\hat z) \\[4pt] &=-rdl\sin\theta(\hat x\times\hat y)+rdl\cos\theta (\hat x\times\hat z) \\[4pt] &=-rdl\sin\theta\hat z +rdl\cos\theta (-\hat y) \\[4pt] &=-rdl\sin\theta\hat z-rdl\cos\theta\hat y \end{aligned}\]
so that the element of magnetic field, \(d\vec B\), corresponding to that choice of \(d\vec l\), will lie in the \(y − z\) plane, as illustrated in Figure \(\PageIndex{5}\). Note that the vector \(d\vec B\) is perpendicular to the vector \(\vec r\) (since it is the cross-product of \(\vec r\) with another vector). The magnetic field element, \(d\vec B\), is given by:
\[\begin{aligned} d\vec B&=\frac{\mu_{0}I}{4\pi}\frac{d\vec l\times\vec r}{r^{3}}=\frac{\mu_{0}I}{4\pi r^{3}}(-rdl\sin\theta\hat z-rdl\cos\theta\hat y) \\[4pt] &=\frac{\mu_{0}I}{4\pi r^{2}}(-dl\sin\theta\hat z-dl\cos\theta\hat y)=dB_{z}\hat z+dB_{y}\hat y \end{aligned}\]
As the wire element, \(d\vec l\), moves around the circle, the tip of the resulting magnetic field vector element traces a circle centered on the \(z\) axis, as illustrated in Figure \(\PageIndex{6}\). Note that, in general, \(d\vec B\) will also have an \(x\) component. Thus, only the \(z\) component of the magnetic field will not be canceled when we sum together the magnetic field elements that come from the different wire elements.
The total magnetic field will be in the negative \(z\) direction, as anticipated from Figure \(\PageIndex{4}\). Summing together the \(z\) components of the infinitesimal magnetic fields:
\[\begin{aligned} dB_{z}&=-\frac{\mu_{0}I}{4\pi r^{2}}dl\sin\theta \\[4pt] B_{z}&=\int dB_{z}=-\int \frac{\mu_{0}I}{4\pi r^{2}}dl\sin\theta \end{aligned}\]
Note that in this case, both \(r\) and \(θ\) are constant for all of the \(d\vec l\), allowing us to take them out of the integral. The integral is then just a sum of the dl elements, which must add up to the circumference of the ring:
\[\begin{aligned} B_{z}=\int dB_{z}=-\frac{\mu_{0}I}{4\pi r^{2}}\sin\theta \int_{0}^{2\pi R}dl =-\frac{\mu_{0}I}{4\pi r^{2}}\sin\theta (2\pi R)=-\frac{\mu_{0}I}{2r^{2}}R\sin\theta \end{aligned}\]
In terms of the variables that we are given:
\[\begin{aligned} r&=\sqrt{R^{2}+h^{2}} \\[4pt] \sin\theta &=\frac{R}{r}=\frac{R}{\sqrt{R^{2}+h^{2}}} \end{aligned}\]
\[\therefore\vec B=-\frac{\mu_{0}I}{2}\frac{R^{2}}{(R^{2}+h^{2})^{\frac{3}{2}}}\hat z\quad\text{(field from a loop of current)}\]
In this case, the math was relatively straightforward (no substitutions to evaluate the integral), however it is challenging to visualize the problem in three dimensions.
Exercise \(\PageIndex{1}\)
A coil is made of \(N\) loops of current-carrying wire packed closely together. What is the magnetic field at the center of the coil?
- \(\frac{µ_{0}I}{2R}\)
- \(\frac{Nµ_{0}I}{2R}\)
- \(\frac{Nµ_{0}I}{2R^{2}}\)
- \(\frac{µ_{0}I}{R}\)
- Answer


8.4: Chapter 4
Case Study #1
1. Patient education for Autumn regarding her PICC line:
- Explanation of the PICC line: A peripherally inserted central catheter (PICC) line is a long, thin tube inserted into a vein in the arm and passed through the vein until the tip lies in a larger vein near the heart.
- Importance of hand hygiene.
- Avoidance of heavy lifting and vigorous activities that may put pressure on the PICC line or dislodge it.
- Signs and symptoms of PICC line complications such as redness, swelling, pain, and drainage.
- Promptly reporting any concerns or changes to the health care provider.
2. Maintenance care priorities for a PICC line:
- Monitoring for signs of infection at the insertion site.
- Keeping the PICC line and dressing dry and secure. Monitor length of catheter visible at insertion site and arm circumference.
- Changing the dressing as directed.
- Avoiding tugging or pulling the PICC line.
3. Specific concerns related to Autumn’s need for a PICC line:
- Assessing for phlebitis (inflammation of the vein) related to the PICC line.
- Monitoring for complications related to the cellulitis.
4. Purpose of the PICC line: The PICC line is used to provide long-term, continuous access to the bloodstream for administering medications or drawing blood.
5. Frequency of PICC line assessment: The PICC line should be assessed every shift or as directed by the health care provider.
6. Changing the dressing for a PICC line: The dressing should be changed every 48-72 hours or as directed by the health care provider. A sterile dressing should be used, and aseptic technique should be followed to minimize the risk of infection.
7. Difference between PICC line and a peripheral IV: A peripheral intravenous (IV) line is a short-term solution for administering medications or fluids, whereas a PICC line is a long-term solution. A peripheral IV is inserted into a vein in the hand or arm, whereas a PICC line is inserted into a vein in the arm and passed through a large central vein until the tip lies near the heart.
Difference between PICC line and an implanted port: A port is a device similar to a PICC line that is surgically implanted under the skin and has a small reservoir that can be accessed with a needle. Unlike a PICC line, a port does not need to be reinserted, as the port can remain in place for months or years.
All of the answers to the interactive element are found within the element.


1.28: Sex education- counseling patients from various cultural backgrounds
At the end of this case, students will be able to:
- Consider cultural and social perspectives regarding sex education
- Describe ways to educate multi-generational patients from different cultures about sex education
- Illustrate how to use culturally sensitive communication in high stakes conversations
- Create a communication plan using personal knowledge and community resources to provide sex education
Introduction
Education on sexual practice varies widely in different countries, nations, cultures, and religions. Sexuality is often tied to morals and personal values, in addition to its status as a health topic. Whether a person sees sexuality as natural versus sacred may determine how they view sex education. Under a natural framework, sexuality is a factor of the human experience equal to any other bodily function and should be taught with the same emphasis. When seen as a sacred function, sexuality is placed differently compared to the education of other aspects of bodily function. While this sacredness is widely respected, it may render the subject taboo or limited in educational discussions.
Comprehensive sexual education includes the physical, emotional, intellectual, and social aspects of sexuality and interpersonal relationships/connections, and not just the physical act of sex. 1 Government leaders and parents, in some cultures, believe that early sexual education will lead to earlier sexual activity. However, it is more common that countries with more structured sexual education teaching have lower rates of teenage pregnancy. 2,3 In some countries, sexual education is taught extensively in schools, whereas in others, the subject is not allowed in schools. For instance, in the United Kingdom, sexual education is mandatory, but each school varies in how they approach the subject. The emergence of human immunodeficiency virus (HIV) and acquired immunodeficiency syndrome (AIDS) in the 1980s in the United States increased demand for sex education beyond abstinence; however, there is still not a federal mandate for sexual education. 4 ,5 In Belgium, there is a sexual education website curated for youth that children as young as seven years old can view and understand, 6 however, there are other countries where sexual education isn’t allowed in schools. In India, sex education is not mandatory in schools, but the Youth Parliament Foundation established in 2002 is increasing information availability, including through one of their campaigns, “know your body, know your rights” (KYBKYR). 7,8
There are many socio-cultural challenges to sexual education around the world. One important concern is that in countries where sex education is not taught thoroughly or is not the same between different schools, some young people may not be able to recognize signs of sexual abuse.  The United Nations Population Fund (UNFPA) has noted that the COVID-19 pandemic has the potential to increase the prevalence of these practices, resulting from the lack of in-person school and extra-curricular activities, economic hardships, and lack of access/funding for health programs. They also state that while legislation and guidelines are helpful, changing social norms is the key to achieving gender equality and eliminating these harmful practices. 10
The Demographic and Health Surveys (DHS) Program under the US Agency for International Development (USAID) conducts surveys and analyzes data on health, HIV, and nutrition in countries worldwide. 1 1 Based on their estimates, as of 2014, an estimated 225 million women in developing regions had an unmet need for modern contraception. 12 This unmet needed is defined per the DHS to include woman of reproductive age (15-49 years old) who are: (1) married or unmarried and sexually active, (2) not using any method of contraception, either modern or traditional, (3) fecund (fertile), or (4) those who do not want to have a child (or another child) in the next two years or at all. 11,12
The overarching goal of sexual education should be to present sexuality and relationships as a natural and healthy part of life, and to reduce negative outcomes such as unwanted/unintended pregnancy, sexually transmitted infections, or abuse, in addition to ensuring that people have the knowledge to make healthy sexual decisions.  Learning how to communicate this vital information in a culturally sensitive manner is important for healthcare providers to be able to move forward.
Case and Questions (part 1)
You are part of a medical mission trip to Jaipur, India. A multigenerational family visits your health clinic where you are providing health education. The family consists of GM (the grandmother, 50 years old), MM (the mother, 34 years old), and two daughters (DM and EM, 18 and 16 years old). MM has an obstetric history of G9-P7-A2. During your session, she mentions that she no longer wants any more children. Upon further discussion, you learn that the patient does not currently use any form of contraception, and she mentions that her family is traditionally against using contraceptives for religious reasons.
- How would you plan to educate MM on contraceptive options, including any relevant pharmacologic and non-pharmacologic options?
- How would you better your own understanding about the patient and her family’s belief systems regarding contraception?
- How would you use culturally-appropriate methods to communicate with the patient?
Case and Questions (part 2)
After discussing contraceptive options, DM shares that she has also encountered issues with her menstrual cycle. She expresses frustration at the limitations she experiences when she is on her period. Her mother does not let her enter the kitchen for fear of bringing bad luck. Additionally, EM shares that she does not have a consistent supply of hygiene products. She asks if the clinic can provide any sanitary pads.
- What are some resources you can use to help EM access menstrual hygiene products?
- What are some cultural beliefs and practices that exist regarding menstrual cycles in different cultures?
Case and Questions (part 3)
You take the time to address both DM and EM’s questions about sex education and her menstrual needs. In the middle of your discussion, GM, the grandmother cuts you off and begins to accuse you of “westernizing her grandchildren.”  Since she does not speak English, your interpreter, a local resident, quickly steps in to translate.
- What methods can you use to prepare your interpreter to act as a bridge for effective communication as you quickly respond to this question? How will you prepare for the session with the translator?
- Describe ways you can respectfully respond to GM’s concerns, simultaneously showing respect for the patient’s family culture while sharing evidence-based sex education.
Author Commentary
Sex education is crucial as it allow individuals to knowledgeably care for themselves and make decisions about their healthcare. Pharmacists have a powerful opportunity to combine medication knowledge with communication tools to open an avenue for quality communication that meets the support and resource needs of menstruating individuals. Intercultural patient care can be fraught with miscommunication pitfalls. But a respectful and curious attitude can help healthcare providers create a safe and open space for honest discussion. Additionally, taking the time to understand cultural and religious beliefs can help providers understand why patients have certain beliefs about menstruation and sex education.
Healthcare providers have a unique opportunity to address systemic issues of inadequate sex education. They are also given the ability to speak to beliefs that are not based on fact. Gently correcting myths or adding factual background to existing beliefs can empower women to fully manage their personal health. This type of education can help address larger issues related to inequity in healthcare decision making, sexual based mistreatment, and a lack of access to menstrual hygiene products.
Patient Approaches and Opportunities
Communication across cultures can be challenging to navigate. Discussion of subjects that may be considered sensitive or even taboo can create tension early in the conversation. Care must be taken to communicate clearly. Unfortunately, certain societies’ cultural and social norms have restricted essential education in subjects like sex, feminine hygiene, and reproduction. Individuals living with these societal norms often believe that sex education may lead younger generations to be promiscuous. However, the opposite is more often true. 5 When young people are educated about sexuality in a healthy way and understand safe sexual practices, they are less likely to get that information from friends or inaccurate sources.
Pharmacists have a unique ability to address the challenges stated above because they are often accessible to people even in rural areas. Counseling patients on their medications can often open the door to crucial conversations about taboo medical subjects. Incorporating culturally sensitive tools of communication can create a high level of impact and improve the dialogue between the pharmacist and patient. Patients will be able to find a safe and open environment to address medication related questions but also extremely important sex education queries. Pharmacists have a pivotal role in global health and patient communication. Using pharmacist-specific communication skills may be the answer to the disparities seen in sex education globally.
Related chapters of interest:
- Saying what you mean doesn’t always mean what you say: cross-cultural communication
- Ethical decision-making in global health: when cultures clash
- Hormonal contraception: from emergency coverage to long-term therapy
- From belly to baby: preparing for a healthy pregnancy
- Digging deeper: improving health communication with patients
- Uncrossed wires: working with non-English speaking patient populations
External resources:
-
Websites:
- World Health Organization. Sexual and Reproductive Health and research (SRH), including the Human Reproduction Programme (HRP). https://www.who.int/teams/sexual-and-reproductive-health-and-research
- UN Women. https://www.unwomen.org/en
- UN Population Fund. International Technical and Programmatic Guidance on Out-of-School Comprehensive Sexuality Education (CSE). https://www.unfpa.org/featured-publication/international-technical-and-programmatic-guidance-out-school-comprehensive
- World Health Organization. 2016 WHO medical eligibility criteria for contraceptive use. https://www.fhi360.org/sites/default/files/media/documents/resource-chart-medical-eligibility-contraceptives-english.pdf
-
Journal articles:
- Morales A, Garcia-Montaño E, Barrios-Ortega C, Niebles-Charris J, Garcia-Roncallo P, Abello-Luque D, et al. Adaptation of an effective school-based sexual health promotion program for youth in Colombia. Soc Sci Med 2019;222:207-215.
- Mwaria M, Chen C, Coppola N, Maurice I, Phifer M. A culturally responsive approach to improving replication of a youth sexual health program. Health Promot Pract 2016;17(6):781-792.
- Juckett G, Unger K. Appropriate use of medical interpreters. Am Fam Physician 2014;90(7):476-80.
- Hadziabdic E, Hjelm K. Working with interpreters: practical advice for use of an interpreter in healthcare . Int J Evid Based Healthc 2013;11(1):69-76.
References
- Latifnejad Roudsari R, Javadnoori M, Hasanpour M, Hazavehei SMM, Taghipour A. Socio-cultural challenges to sexual health education for female adolescents in Iran. Iran J Reprod Med 2013;11(2):101-10.
- Saewyc EM, Poon CS, Homma Y, Skay CL. Stigma management? The links between enacted stigma and teen pregnancy trends among gay, lesbian, and bisexual students in British Columbia. Can J Hum Sex 2008;17(3):123-39.
- Guttmacher Institute. Review of key findings of “Emerging Answers 2007” report on sex education programs. February 2016. https://www.guttmacher.org/article/2007/11/review-key-findings-emerging-answers-2007-report-sex-education-programs . Accessed January 29, 2021.
- Donovan P. School-based sexuality education: the issues and challenges. Fam Plann Perspect 1998; 30(4):188-93.
- Landry DJ, Kaeser L, Richards CL. Abstinence promotion and the provision of information about contraception in public school district sexuality education policies. Fam Plann Perspect 1999;31(6):280-6.
- Study International. Sex education around the world: how were you taught? November 2017. https://www.studyinternational.com/news/sex-education/ . Accessed January 29, 2021.
- Advocates for Youth. The future of sex education. https://www.futureofsexed.org/ . Accessed January 29, 2021.
- Know Your Body, Know Your Rights. The YP Foundation. 2020. http://theypfoundation.org/know-your-body-know-your-rights/ . Accessed January 29, 2021.
- United Nations Population Fund. Sexual and reproductive health. 2020. https://www.unfpa.org/sexual-reproductive-health . Accessed January 29, 2021.
- United Nations Population Fund. Comprehensive sexuality education. https://www.unfpa.org/comprehensive-sexuality-education . Accessed January 29, 2021.
- The DHS Program, Demographic and health surveys. http://www.dhsprogram.com . Accessed January 29, 2021.
- Singh S, Darroch JE and Ashford LS. Guttmacher Institute. Adding it up: the costs and benefits of investing in sexual and reproductive health, 2014. https://www.guttmacher.org/report/adding-it-costs-and-benefits-investing-sexual-and-reproductive-health-2014 . Accessed January 29, 2021.


12.9: The Secretary Problem
In this section we will study a nice problem known variously as the secretary problem or the marriage problem . It is simple to state and not difficult to solve, but the solution is interesting and a bit surprising. Also, the problem serves as a nice introduction to the general area of statistical decision making.
Statement of the Problem
As always, we must start with a clear statement of the problem.
We have \(n\) candidates (perhaps applicants for a job or possible marriage partners). The assumptions are
- The candidates are totally ordered from best to worst with no ties.
- The candidates arrive sequentially in random order.
- We can only determine the relative ranks of the candidates as they arrive. We cannot observe the absolute ranks .
- Our goal is choose the very best candidate; no one less will do.
- Once a candidate is rejected, she is gone forever and cannot be recalled.
- The number of candidates \(n\) is known.
The assumptions, of course, are not entirely reasonable in real applications. The last assumption, for example, that \(n\) is known, is more appropriate for the secretary interpretation than for the marriage interpretation.
What is an optimal strategy? What is the probability of success with this strategy? What happens to the strategy and the probability of success as \(n\) increases? In particular, when \(n\) is large, is there any reasonable hope of finding the best candidate?
Strategies
Play the secretary game several times with \(n = 10\) candidates. See if you can find a good strategy just by trial and error.
After playing the secretary game a few times, it should be clear that the only reasonable type of strategy is to let a certain number \(k - 1\) of the candidates go by, and then select the first candidate we see who is better than all of the previous candidates (if she exists). If she does not exist (that is, if no candidate better than all previous candidates appears), we will agree to accept the last candidate, even though this means failure. The parameter \(k\) must be between 1 and \(n\); if \(k = 1\), we select the first candidate; if \(k = n\), we select the last candidate; for any other value of \(k\), the selected candidate is random, distributed on \(\{k, k + 1, \ldots, n\}\). We will refer to this
let \(k - 1\) go by
strategy as
strategy \(k\)
.
Thus, we need to compute the probability of success \(p_n(k)\) using strategy \(k\) with \(n\) candidates. Then we can maximize the probability over \(k\) to find the optimal strategy, and then take the limit over \(n\) to study the asymptotic behavior.
Analysis
First, let's do some basic computations.
For the case \(n = 3\), list the 6 permutations of \(\{1, 2, 3\}\) and verify the probabilities in the table below. Note that \(k = 2\) is optimal.
| \(k\) | 1 | 2 | 3 |
|---|---|---|---|
| \(p_3(k)\) | \(\frac{2}{6}\) | \(\frac{3}{6}\) | \(\frac{2}{6}\) |
Answer
The following table gives the \( 3! = 6 \) permutations of the candidates \( (1, 2, 3) \), and the candidate selected by each strategy. The last row gives the total number of successes for each strategy.
| Permutation | \(k = 1\) | \(k = 2\) | \(k = 3\) |
|---|---|---|---|
| \((1, 2, 3)\) | 1 | 3 | 3 |
| \((1, 3, 2)\) | 1 | 2 | 2 |
| \((2, 1, 3)\) | 2 | 1 | 3 |
| \((2, 3, 1)\) | 2 | 1 | 1 |
| \((3, 1, 2)\) | 3 | 1 | 2 |
| \((3, 2, 1)\) | 3 | 2 | 1 |
| Total | 2 | 3 | 2 |
In the secretary experiment, set the number of candidates to \(n = 3\). Run the experiment 1000 times with each strategy \( k \in \{1, 2, 3\} \)
For the case \(n = 4\), list the 24 permutations of \(\{1, 2, 3, 4\}\) and verify the probabilities in the table below. Note that \(k = 2\) is optimal. The last row gives the total number of successes for each strategy.
| \(k\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \(p_4(k)\) | \(\frac{6}{24}\) | \(\frac{11}{24}\) | \(\frac{10}{24}\) | \(\frac{6}{24}\) |
Answer
The following table gives the \(4! = 24\) permutations of the candidates \((1, 2, 3, 4)\), and the candidate selected by each strategy.
| Permutation | \(k = 1\) | \(k = 2\) | \(k = 3\) | \(k = 4\) |
|---|---|---|---|---|
| \((1, 2, 3, 4)\) | 1 | 4 | 4 | 4 |
| \((1, 2, 4, 3)\) | 1 | 3 | 3 | 3 |
| \((1, 3, 2, 4)\) | 1 | 4 | 4 | 4 |
| \((1, 3, 4, 2)\) | 1 | 3 | 2 | 2 |
| \((1, 4, 2, 3)\) | 1 | 3 | 3 | 3 |
| \((1, 4, 3, 2)\) | 1 | 2 | 2 | 2 |
| \((2, 1, 3, 4)\) | 2 | 1 | 4 | 4 |
| \((2, 1, 4, 3)\) | 2 | 1 | 3 | 3 |
| \((2, 3, 1, 4)\) | 2 | 1 | 1 | 4 |
| \((2, 3, 4, 1)\) | 2 | 1 | 1 | 1 |
| \((2, 4, 1, 3)\) | 2 | 1 | 1 | 3 |
| \((2, 4, 3, 1)\) | 2 | 1 | 1 | 1 |
| \((3, 1, 2, 4)\) | 3 | 1 | 4 | 4 |
| \((3, 1, 4, 2)\) | 3 | 1 | 2 | 2 |
| \((3, 2, 1, 4)\) | 3 | 2 | 1 | 4 |
| \((3, 2, 4, 1)\) | 3 | 2 | 1 | 1 |
| \((3, 4, 1, 2)\) | 3 | 1 | 1 | 2 |
| \((3, 4, 2, 1)\) | 3 | 2 | 2 | 1 |
| \((4, 1, 2, 3)\) | 4 | 1 | 3 | 3 |
| \((4, 1, 3, 2)\) | 4 | 1 | 2 | 2 |
| \((4, 2, 1, 3)\) | 4 | 2 | 1 | 3 |
| \((4, 2, 3, 1)\) | 4 | 2 | 1 | 1 |
| \((4, 3, 1, 2)\) | 4 | 3 | 1 | 2 |
| \((4, 3, 2, 1)\) | 4 | 3 | 2 | 1 |
| Total | 6 | 11 | 10 | 6 |
In the secretary experiment, set the number of candidates to \(n = 4\). Run the experiment 1000 times with each strategy \( k \in \{1, 2, 3, 4\} \)
For the case \(n = 5\), list the 120 permutations of \(\{1, 2, 3, 4, 5\}\) and verify the probabilities in the table below. Note that \(k = 3\) is optimal.
| \(k\) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| \(p_5(k)\) | \(\frac{24}{120}\) | \(\frac{50}{120}\) | \(\frac{52}{120}\) | \(\frac{42}{120}\) | \(\frac{24}{120}\) |
In the secretary experiment, set the number of candidates to \(n = 5\). Run the experiment 1000 times with each strategy \( k \in \{1, 2, 3, 4, 5\} \)
Well, clearly we don't want to keep doing this . Let's see if we can find a general analysis. With \(n\) candidates, let \(X_n\) denote the number (arrival order) of the best candidate, and let \(S_{n,k}\) denote the event of success for strategy \(k\) (we select the best candidate).
\(X_n\) is uniformly distributed on \(\{1, 2, \ldots, n\}\).
Proof
This follows since the candidates arrive in random order.
Next we will compute the conditional probability of success given the arrival order of the best candidate.
For \( n \in \N_+ \) and \( k \in \{2, 3, \ldots, n\} \), \[ \P(S_{n,k} \mid X_n = j) = \begin{cases} 0, & j \in \{1, 2, \ldots, k-1\} \\ \frac{k-1}{j-1}, & j \in \{k, k + 1, \ldots, n\} \end{cases} \]
Proof
For the first case, note that if the arrival number of the best candidate is \(j \lt k\), then strategy \(k\) will certainly fail. For the second cases, note that if the arrival order of the best candidate is \(j \ge k\), then strategy \(k\) will succeed if and only if one of the first \(k - 1\) candidates (the ones that are automatically rejected) is the best among the first \(j - 1\)
The two cases are illustrated below. The large dot indicates the best candidate. Red dots indicate candidates that are rejected out of hand, while blue dots indicate candidates that are considered.
Now we can compute the probability of success with strategy \(k\).
For \( n \in \N_+ \) \[ p_n(k) = \P(S_{n,k}) = \begin{cases} \frac{1}{n}, & k = 1 \\ \frac{k - 1}{n} \sum_{j=k}^n \frac{1}{j - 1}, & k \in \{2, 3, \ldots, n\} \end{cases} \]
Proof
When \( k = 1 \) we simply select the first candidate. This candidate will be the best one with probability \( 1 / n \). The result for \( k \in \{2, 3, \ldots, n\} \) follows from the previous two results, by conditioning on \(X_n\): \[ \P(S_{n,k}) = \sum_{j=1}^n \P(X_n = j) \P(S_{n,k} \mid X_n = j) = \sum_{j=k}^n \frac{1}{n} \frac{k - 1}{j - 1} \]
Values of the function \(p_n\) can be computed by hand for small \(n\) and by a computer algebra system for moderate \(n\). The graph of \(p_{100}\) is shown below. Note the concave downward shape of the graph and the optimal value of \(k\), which turns out to be 38. The optimal probability is about 0.37104.
The optimal strategy \(k_n\) that maximizes \(k \mapsto p_n(k)\), the ratio \(k_n / n\), and the optimal probability \(p_n(k_n)\) of finding the best candidate, as functions of \(n \in \{3, 4, \dots, 20\}\) are given in the following table:
| Candidates \(n\) | Optimal strategy \(k_n\) | Ratio \(k_n / n\) | Optimal probability \(p_n(k_n)\) |
|---|---|---|---|
| 3 | 2 | 0.6667 | 0.5000 |
| 4 | 2 | 0.5000 | 0.4583 |
| 5 | 3 | 0.6000 | 0.4333 |
| 6 | 3 | 0.5000 | 0.4278 |
| 7 | 3 | 0.4286 | 0.4143 |
| 8 | 4 | 0.5000 | 0.4098 |
| 9 | 4 | 0.4444 | 0.4060 |
| 10 | 4 | 0.4000 | 0.3987 |
| 11 | 5 | 0.4545 | 0.3984 |
| 12 | 5 | 0.4167 | 0.3955 |
| 13 | 6 | 0.4615 | 0.3923 |
| 14 | 6 | 0.4286 | 0.3917 |
| 15 | 6 | 0.4000 | 0.3894 |
| 16 | 7 | 0.4375 | 0.3881 |
| 17 | 7 | 0.4118 | 0.3873 |
| 18 | 7 | 0.3889 | 0.3854 |
| 19 | 8 | 0.4211 | 0.3850 |
| 20 | 8 | 0.4000 | 0.3842 |
Apparently, as we might expect, the optimal strategy \(k_n\) increases and the optimal probability \(p_n(k_n)\) decreases as \(n \to \infty\). On the other hand, it's encouraging, and a bit surprising, that the optimal probability does not appear to be decreasing to 0. It's perhaps least clear what's going on with the ratio. Graphical displays of some of the information in the table may help:
Could it be that the ratio \(k_n / n\) and the probability \(p_n(k_n)\) are both converging, and moreover, are converging to the same number? First let's try to establish rigorously some of the trends observed in the table.
The success probability \(p_n\) satisfies \[ p_n(k - 1) \lt p_n(k) \text{ if and only if } \sum_{j=k}^n \frac{1}{j-1} \gt 1 \]
It follows that for each \(n \in \N_+\), the function \(p_n\) at first increases and then decreases. The maximum value of \(p_n\) occurs at the largest \(k\) with \(\sum_{j=k}^n \frac{1}{j - 1} \gt 1\). This is the optimal strategy with \(n\) candidates, which we have denoted by \(k_n\).
As \(n\) increases, \(k_n\) increases and the optimal probability \(p_n(k_n)\) decreases.
Asymptotic Analysis
We are naturally interested in the asymptotic behavior of the function \(p_n\), and the optimal strategy as \(n \to \infty\). The key is recognizing \(p_n\) as a Riemann sum for a simple integral. (Riemann sums, of course, are named for Georg Riemann.)
If \(k(n)\) depends on \(n\) and \(k(n) / n \to x \in (0, 1)\) as \(n \to \infty\) then \(p_n[k(n)] \to -x \ln x\) as \(n \to \infty\).
Proof
First note that \[ p_n(k) = \frac{k-1}{n} \sum_{j=k}^n \frac{1}{n} \frac{n}{j-1} \] We recognize the sum above as the left Riemann sum for the the function \(f(t) = \frac{1}{t}\) corresponding to the partition of the interval \(\left[\frac{k-1}{n}, 1\right]\) into \((n - k) + 1\) subintervals of length \(\frac{1}{n}\) each: \(\left(\frac{k-1}{n}, \frac{k}{n}, \ldots, \frac{n-1}{n}, 1\right)\). It follows that \[ p_n[k(n)] \to x \int_x^1 \frac{1}{t} dt = -x \ln x \text{ as } n \to \infty\]
The optimal strategy \(k_n\) that maximizes \(k \mapsto p_n(k)\), the ratio \(k_n / n\), and the optimal probability \(p_n(k_n)\) of finding the best candidate, as functions of \(n \in \{10, 20, \dots, 100\}\) are given in the following table:
| Candidates \(n\) | Optimal strategy \(k_n\) | Ratio \(k_n / n\) | Optimal probability \(p_n(k_n)\) |
|---|---|---|---|
| 10 | 4 | 0.4000 | 0.3987 |
| 20 | 8 | 0.4000 | 0.3842 |
| 30 | 12 | 0.4000 | 0.3786 |
| 40 | 16 | 0.4000 | 0.3757 |
| 50 | 19 | 0.3800 | 0.3743 |
| 60 | 23 | 0.3833 | 0.3732 |
| 70 | 27 | 0.3857 | 0.3724 |
| 80 | 30 | 0.3750 | 0.3719 |
| 90 | 34 | 0.3778 | 0.3714 |
| 100 | 38 | 0.3800 | 0.3710 |
The graph below shows the true probabilities \(p_n(k)\) and the limiting values \(-\frac{k}{n} \, \ln\left(\frac{k}{n}\right)\) as a function of \(k\) with \(n = 100\).
For the optimal strategy \(k_n\), there exists \(x_0 \in (0, 1)\) such that \(k_n / n \to x_0\) as \(n \to \infty\). Thus, \(x_0 \in (0, 1)\) is the limiting proportion of the candidates that we reject out of hand. Moreover, \(x_0\) maximizes \(x \mapsto -x \ln x\) on \((0, 1)\).
The maximum value of \(-x \ln x\) occurs at \(x_0 = 1 / e\) and the maximum value is also \(1 / e\).
Proof
Thus, the magic number \(1 / e \approx 0.3679\) occurs twice in the problem. For large \(n\):
- Our approximate optimal strategy is to reject out of hand the first 37% of the candidates and then select the first candidate (if she appears) that is better than all of the previous candidates.
- Our probability of finding the best candidate is about 0.37.
The article
Who Solved the Secretary Problem?
by Tom Ferguson (1989) has an interesting historical discussion of the problem, including speculation that Johannes Kepler may have used the optimal strategy to choose his second wife. The article also discusses many interesting generalizations of the problem. A different version of the secretary problem, in which the candidates are assigned a score in \([0, 1]\), rather than a relative rank, is discussed in the section on Stopping Times in the chapter on Martingales


11: Relationships Between Variables
In the previous chapter, our discussion of variables and functions largely assumed that relationships were known or developed independent of any measurement or data. However, functional relationships between variables can also be derived from data. Here, we explore two concepts that help us understand the strength and nature of systematic relationships between variables.
11.1 Correlation
In common parlance, the word correlation suggests that two events or observations are linked with one another. In the analysis of data, the definition is much the same, but we can even be more specific about the manner in which events or observations are linked. The most straight-forward measure of correlation is the linear correlation coefficient , which is usually written r (and is, indeed, related to the r \(^{2}\) that we cite in assessing the fit of a regression equation). The value of r may range from -1 to 1, and the closer it is to the ends of this range (i.e., | r |→ 1), the stronger the correlation. We may say that two variables are positively correlated if r is close to +1, and negatively correlated if r is close to -1. Poorly correlated or uncorrelated variables will have r closer to 0.
In the margin are two plots comparing life-history and reproductive traits of various mammals. In the first one, Figure 11.1, the arrangement of points in a band from lower left to upper right on the graph is relatively strong, corresponding to a relatively high r of 0.73. In contrast, the correlation between litter number per year and litter size in Figure 11.2 is (surprisingly?) weak, producing more of a shotgun pattern and r a modest 0.36.
In the abstract, the mathematical formula for the correlation be- tween two variables, x and y , can be written:
\(r=\frac{1}{n} \sum_{i=1}^{n}\left(\frac{x_{i}-\bar{x}}{\sigma_{x}}\right)\left(\frac{y_{i}-\bar{y}}{\sigma_{y}}\right)\) (11.1)
where the subscript i corresponds to the i th observation, the overbar indicates mean values, and σ x and σ y are standard deviations. The specifics of this formula are not of great interest to us. The important thing to understand is that when positive changes in one variable are clearly linked with positive changes in a second variable, this indicates a good, positive correlation, r > 0. The same is true if negative changes in one variable correspond to negative changes in the other. However, positive changes in one variable corresponding to negative changes in another indicate negative correlation, r < 0.
It is also important to note that this is a good measure of correlation only for linear relationships, and even if two variables are closely interdependent, if their functional dependence is not linear, the r value will not be particularly helpful. Nevertheless, correlation can still help us identify key relation- ships when we first encounter a dataset. Consider the changes in weather variables measured at a meteorological station as a function of time. Weather data can be very overwhelming due to the number of variables and the sheer volume of data. One handy way to isolate some of the strongest interdependencies among variables of interest is to look for correlations. A correlation matrix plot is essentially a grid of plots where each variable is plotted against all the other variables in a square array of panels. Relationships with strong positive or negative correlations immediately jump out, suggesting which relatinships we might wish to investigate further. For example, let’s look at a month-long weather dataset downloaded from www.wunderground.com .
There is alot of information in these plots, so let’s look at them piece by piece. Notice that the panels on the diagonal from upper left to lower right would be a variable plotted against itself (
r
= 1), and they are therefore replaced by a density distribution for each variable. Also notice that since the upper right half would be a mirror image of the lower left, there is just a number in each of those panels rather than a plot. In any case, here we have just selected four variables of potential interest, and you can immediately see that there is a strong positive correlation between mean temperature and mean dew point, with
r
= 0.962. The strength of the correlations from these plots
(the six in the lower left) is indicated by the correlation coefficient in the (mirrored) corresponding panel in the upper right. There are also relatively strong negative correlations between temperature and pressure, and dew point and pressure. In contrast, we see weak correlations between humidity and temperature and humidity and pressure, as indicated by the low
r
values.
The important thing to remember from all of this is what the correlation coefficient can tell us: a high, positive correlation between two variables indicates that when one goes up, so does the other. A high negative correlation indicates that as one goes up, the other goes down. Low correlation coefficients indicate that a consistent linear relationship cannot be established. If correlation is established, how- ever, this analysis doesn’t yet provide details about the functional relationships present
11.2 Regression
Regression is the process of fitting a mathematical function to a set of data points using some criterion for judging “goodness-of-fit”. The resulting “best-fit” function may then be used to predict unknown values, to forecast future values, or to evaluate the dependence of one variable upon another. Goodness-of-fit can be determined by one of many statistical techniques that determine how well a function describes the variations in the data used to generate it. The most common criterion for goodness-of-fit is called “least-squares”, so you might sometimes see the whole process called least-squares regression. Least squares means what it sounds like, sort of. When a function (let’s write it y = f ( x )) is tested for goodness-of-fit, the difference between the y -values predicted by the test function, each of which we can call y ̄ , at a given x , and the y -values in the data iii set are found, squared, and added together for the entire dataset. The best-fit line is then the one for which the sum of the squares of the residuals are minimized (least). This is very commonly done for linear equations, but we can use the same techniques for nonlinear equations as well.
Some data sets that we may encounter just don’t appear to have linear trends though. In these cases, we can try transforming one or both variables\(^{1}\) or we can attempt to perform nonlinear regression. As with many of the statistical and spatial methods discussed in this book, the heavy lifting for most of these options can – and probably should – be done with computer software. However, we should still be aware of what is happening
Some common data transformations include logarithmic, exponential, and reciprocal. In these transformations, a modified variable is created by performing the selected operation on the original variable values.
| Cumulative catch | catch/effort |
| 86 | 2.46 |
| 137 | 1.76 |
| 169 | 1.14 |
| 178 | 0.29 |
11.2.1 Example: brook trout electrofishing (Problem 3.7)
Having isolated the age-0 brook trout from each electrofishing traverse and computed the catch per unit effort c ue of that subset, we may now employ the Leslie method to estimate the total population of age-0 brook trout in the study reach.
In this method, we create a dependent variable c \(_{ccumul.}\) corresponding to the cumulative number of fish removed in each pass, the “cumulative catch”. We then plot and perform a linear regression of the catch per unit effort as a function of cumulative catch, as illustrated in Figure 11.5. By the Leslie method if we extrapolate the best-fit line to a vertical-axis value c \(_{ue}\) = 0, the cumulative catch value where that occurs is the estimated total population. This value can be estimated from the graph itself, but the result is better if we solve the for the value directly from the best-fit line. The equation of the best-fit line for this regression is:
c \(_{ue}\) = −0.0208\(_{ccumul.}\) + 4.38. (11.2)
Note that the slope of this line (−0.0208), consistent with intuition, is a negative number. The y -intercept 4.38 corresponds to the hypothetical initial catch per unit effort at the very start of the first traverse. Rearranging and solving for c \(_{ccumul.}\) gives
c \(_{ccumul.}\) = 210.6 − 48.1\(_{cue}\) (11.3)
and we find that the estimated total population is 210.
Discuss in a paragraph the benefits and drawbacks of deciding, prior to any data analysis, what type of function to seek best-fit parameters for.
In Section 11.2.1, we skipped several steps in the algebraic manipulation that allowed solution for c \(_{ccumul.}\) Carry out all the intermediate steps, showing your work completely, and determine whether the solution cited above is acceptable.
Find a dataset that interests you within a public ecological or natural resource data repository\(^{2}\), identify variables within a dataset that may be related, and perform a regression to see the nature of that relationship.


9.5: Loving v. Virginia 388 US 1 (1967)
U.S. Supreme Court
Loving v. Virginia, 388 U.S. 1 (1967)
Loving v. Virginia
No. 395
Argued April 10, 1967
Decided June 12, 1967
388 U.S. 1
APPEAL FROM THE SUPREME COURT OF APPEALS OF VIRGINIA
Syllabus
Virginia’s statutory scheme to prevent marriages between persons solely on the basis of racial classifications held to violate the Equal Protection and Due Process Clauses of the Fourteenth Amendment. Pp. 388 U. S. 4 -12.
206 Va. 924, 147 S.E.2d 78, reversed.
Page 388 U. S. 2
MR. CHIEF JUSTICE WARREN delivered the opinion of the Court.
This case presents a constitutional question never addressed by this Court: whether a statutory scheme adopted by the State of Virginia to prevent marriages between persons solely on the basis of racial classifications violates the Equal Protection and Due Process Clauses of the Fourteenth Amendment. [ Footnote 1 ] For reasons which seem to us to reflect the central meaning of those constitutional commands, we conclude that these statutes cannot stand consistently with the Fourteenth Amendment.
In June, 1958, two residents of Virginia, Mildred Jeter, a Negro woman, and Richard Loving, a white man, were married in the District of Columbia pursuant to its laws. Shortly after their marriage, the Lovings returned to Virginia and established their marital abode in Caroline County. At the October Term, 1958, of the Circuit Court
Page 388 U. S. 3
of Caroline County, a grand jury issued an indictment charging the Lovings with violating Virginia’s ban on interracial marriages. On January 6, 199, the Lovings pleaded guilty to the charge, and were sentenced to one year in jail; however, the trial judge suspended the sentence for a period of 25 years on the condition that the Lovings leave the State and not return to Virginia together for 25 years. He stated in an opinion that:
“Almighty God created the races white, black, yellow, malay and red, and he placed them on separate continents. And, but for the interference with his arrangement, there would be no cause for such marriage. The fact that he separated the races shows that he did not intend for the races to mix.”
After their convictions, the Lovings took up residence in the District of Columbia. On November 6, 1963, they filed a motion in the state trial court to vacate the judgment and set aside the sentence on the ground that the statutes which they had violated were repugnant to the Fourteenth Amendment. The motion not having been decided by October 28, 1964, the Lovings instituted a class action in the United States District Court for the Eastern District of Virginia requesting that a three-judge court be convened to declare the Virginia anti-miscegenation statutes unconstitutional and to enjoin state officials from enforcing their convictions. On January 22, 1965, the state trial judge denied the motion to vacate the sentences, and the Lovings perfected an appeal to the Supreme Court of Appeals of Virginia. On February 11, 1965, the three-judge District Court continued the case to allow the Lovings to present their constitutional claims to the highest state court.
The Supreme Court of Appeals upheld the constitutionality of the anti-miscegenation statutes and, after
Page 388 U. S. 4
modifying the sentence, affirmed the convictions. [ Footnote 2 ] The Lovings appealed this decision, and we noted probable jurisdiction on December 12, 1966, 385 U.S. 986.
The two statutes under which appellants were convicted and sentenced are part of a comprehensive statutory scheme aimed at prohibiting and punishing interracial marriages. The Lovings were convicted of violating § 258 of the Virginia Code:
“ Leaving State to evade law. — If any white person and colored person shall go out of this State, for the purpose of being married, and with the intention of returning, and be married out of it, and afterwards return to and reside in it, cohabiting as man and wife, they shall be punished as provided in § 20-59, and the marriage shall be governed by the same law as if it had been solemnized in this State. The fact of their cohabitation here as man and wife shall be evidence of their marriage.”
Section 259, which defines the penalty for miscegenation, provides:
“ Punishment for marriage. — If any white person intermarry with a colored person, or any colored person intermarry with a white person, he shall be guilty of a felony and shall be punished by confinement in the penitentiary for not less than one nor more than five years.”
Other central provisions in the Virginia statutory scheme are § 20-57, which automatically voids all marriages between “a white person and a colored person” without any judicial proceeding, [ Footnote 3 ] and §§ 20-54 and 1-14 which,
Page 388 U. S. 5
respectively, define “white persons” and “colored persons and Indians” for purposes of the statutory prohibitions. [ Footnote 4 ] The Lovings have never disputed in the course of this litigation that Mrs. Loving is a “colored person” or that Mr. Loving is a “white person” within the meanings given those terms by the Virginia statutes.
Page 388 U. S. 6
Virginia is now one of 16 States which prohibit and punish marriages on the basis of racial classifications. [ Footnote 5 ] Penalties for miscegenation arose as an incident to slavery, and have been common in Virginia since the colonial period. [ Footnote 6 ] The present statutory scheme dates from the adoption of the Racial Integrity Act of 1924, passed during the period of extreme nativism which followed the end of the First World War. The central features of this Act, and current Virginia law, are the absolute prohibition of a “white person” marrying other than another “white person,” [ Footnote 7 ] a prohibition against issuing marriage licenses until the issuing official is satisfied that
Page 388 U. S. 7
the applicants’ statements as to their race are correct, [ Footnote 8 ] certificates of “racial composition” to be kept by both local and state registrars, [ Footnote 9 ] and the carrying forward of earlier prohibitions against racial intermarriage. [ Footnote 10 ]
I
In upholding the constitutionality of these provisions in the decision below, the Supreme Court of Appeals of Virginia referred to its 1965 decision in Naim v. Naim, 197 Va. 80, 87 S.E.2d 749, as stating the reasons supporting the validity of these laws. In Naim, the state court concluded that the State’s legitimate purposes were “to preserve the racial integrity of its citizens,” and to prevent “the corruption of blood,” “a mongrel breed of citizens,” and “the obliteration of racial pride,” obviously an endorsement of the doctrine of White Supremacy. Id. at 90, 87 S.E.2d at 756. The court also reasoned that marriage has traditionally been subject to state regulation without federal intervention, and, consequently, the regulation of marriage should be left to exclusive state control by the Tenth Amendment.
While the state court is no doubt correct in asserting that marriage is a social relation subject to the State’s police power, Maynard v. Hill, 125 U. S. 190 (1888), the State does not contend in its argument before this Court that its powers to regulate marriage are unlimited notwithstanding the commands of the Fourteenth Amendment. Nor could it do so in light of Meyer v. Nebraska, 262 U. S. 390 (1923), and Skinner v. Oklahoma, 316 U. S. 535 (1942). Instead, the State argues that the meaning of the Equal Protection Clause, as illuminated by the statements of the Framers, is only that state penal laws containing an interracial element
Page 388 U. S. 8
as part of the definition of the offense must apply equally to whites and Negroes in the sense that members of each race are punished to the same degree. Thus, the State contends that, because its miscegenation statutes punish equally both the white and the Negro participants in an interracial marriage, these statutes, despite their reliance on racial classifications, do not constitute an invidious discrimination based upon race. The second argument advanced by the State assumes the validity of its equal application theory. The argument is that, if the Equal Protection Clause does not outlaw miscegenation statutes because of their reliance on racial classifications, the question of constitutionality would thus become whether there was any rational basis for a State to treat interracial marriages differently from other marriages. On this question, the State argues, the scientific evidence is substantially in doubt and, consequently, this Court should defer to the wisdom of the state legislature in adopting its policy of discouraging interracial marriages.
Because we reject the notion that the mere “equal application” of a statute containing racial classifications is enough to remove the classifications from the Fourteenth Amendment’s proscription of all invidious racial discriminations, we do not accept the State’s contention that these statutes should be upheld if there is any possible basis for concluding that they serve a rational purpose. The mere fact of equal application does not mean that our analysis of these statutes should follow the approach we have taken in cases involving no racial discrimination where the Equal Protection Clause has been arrayed against a statute discriminating between the kinds of advertising which may be displayed on trucks in New York City, Railway Express Agency, Inc. v. New York, 336 U. S. 106 (1949), or an exemption in Ohio’s ad valorem tax for merchandise owned by a nonresident in a storage warehouse, Allied Stores of Ohio,
Page 388 U. S. 9
Inc. v. Bowers, 358 U. S. 522 (1959). In these cases, involving distinctions not drawn according to race, the Court has merely asked whether there is any rational foundation for the discriminations, and has deferred to the wisdom of the state legislatures. In the case at bar, however, we deal with statutes containing racial classifications, and the fact of equal application does not immunize the statute from the very heavy burden of justification which the Fourteenth Amendment has traditionally required of state statutes drawn according to race.
The State argues that statements in the Thirty-ninth Congress about the time of the passage of the Fourteenth Amendment indicate that the Framers did not intend the Amendment to make unconstitutional state miscegenation laws. Many of the statements alluded to by the State concern the debates over the Freedmen’s Bureau Bill, which President Johnson vetoed, and the Civil Rights Act of 1866, 14 Stat. 27, enacted over his veto. While these statements have some relevance to the intention of Congress in submitting the Fourteenth Amendment, it must be understood that they pertained to the passage of specific statutes, and not to the broader, organic purpose of a constitutional amendment. As for the various statements directly concerning the Fourteenth Amendment, we have said in connection with a related problem that, although these historical sources “cast some light” they are not sufficient to resolve the problem;
“[a]t best, they are inconclusive. The most avid proponents of the post-War Amendments undoubtedly intended them to remove all legal distinctions among ‘all persons born or naturalized in the United States.’ Their opponents, just as certainly, were antagonistic to both the letter and the spirit of the Amendments, and wished them to have the most limited effect.”
Brown v. Board of Education, 347 U. S. 483 , 347 U. S. 489 (1954). See also Strauder
Page 388 U. S. 10
v. West Virginia, 100 U. S. 303 , 100 U. S. 310 (1880). We have rejected the proposition that the debates in the Thirty-ninth Congress or in the state legislatures which ratified the Fourteenth Amendment supported the theory advanced by the State, that the requirement of equal protection of the laws is satisfied by penal laws defining offenses based on racial classifications so long as white and Negro participants in the offense were similarly punished. McLaughlin v. Florida, 379 U. S. 184 (1964).
The State finds support for its “equal application” theory in the decision of the Court in Pace v. Alabama, 106 U. S. 583 (1883). In that case, the Court upheld a conviction under an Alabama statute forbidding adultery or fornication between a white person and a Negro which imposed a greater penalty than that of a statute proscribing similar conduct by members of the same race. The Court reasoned that the statute could not be said to discriminate against Negroes because the punishment for each participant in the offense was the same. However, as recently as the 1964 Term, in rejecting the reasoning of that case, we stated “ Pace represents a limited view of the Equal Protection Clause which has not withstood analysis in the subsequent decisions of this Court.” McLaughlin v. Florida, supra, at 379 U. S. 188 . As we there demonstrated, the Equal Protection Clause requires the consideration of whether the classifications drawn by any statute constitute an arbitrary and invidious discrimination. The clear and central purpose of the Fourteenth Amendment was to eliminate all official state sources of invidious racial discrimination in the States. Slaughter-House Cases, 16 Wall. 36, 83 U. S. 71 (1873); Strauder v. West Virginia, 100 U. S. 303 , 100 U. S. 307 -308 (1880); Ex parte Virginia, 100 U. S. 339 , 100 U. S. 334 -335 (1880); Shelley v. Kraemer, 334 U. S. 1 (1948); Burton v. Wilmington Parking Authority, 365 U. S. 715 (1961).
Page 388 U. S. 11
There can be no question but that Virginia’s miscegenation statutes rest solely upon distinctions drawn according to race. The statutes proscribe generally accepted conduct if engaged in by members of different races. Over the years, this Court has consistently repudiated “[d]istinctions between citizens solely because of their ancestry” as being “odious to a free people whose institutions are founded upon the doctrine of equality.” Hirabayashi v. United States, 320 U. S. 81 , 320 U. S. 100 (1943). At the very least, the Equal Protection Clause demands that racial classifications, especially suspect in criminal statutes, be subjected to the “most rigid scrutiny,” Korematsu v. United States, 323 U. S. 214 , 323 U. S. 216 (1944), and, if they are ever to be upheld, they must be shown to be necessary to the accomplishment of some permissible state objective, independent of the racial discrimination which it was the object of the Fourteenth Amendment to eliminate. Indeed, two members of this Court have already stated that they
“cannot conceive of a valid legislative purpose . . . which makes the color of a person’s skin the test of whether his conduct is a criminal offense.”
McLaughlin v. Florida, supra, at 379 U. S. 198 (STEWART, J., joined by DOUGLAS, J., concurring).
There is patently no legitimate overriding purpose independent of invidious racial discrimination which justifies this classification. The fact that Virginia prohibits only interracial marriages involving white persons demonstrates that the racial classifications must stand on their own justification, as measures designed to maintain White Supremacy. [ Footnote 11 ] We have consistently denied
Page 388 U. S. 12
the constitutionality of measures which restrict the rights of citizens on account of race. There can be no doubt that restricting the freedom to marry solely because of racial classifications violates the central meaning of the Equal Protection Clause.
II
These statutes also deprive the Lovings of liberty without due process of law in violation of the Due Process Clause of the Fourteenth Amendment. The freedom to marry has long been recognized as one of the vital personal rights essential to the orderly pursuit of happiness by free men.
Marriage is one of the “basic civil rights of man,” fundamental to our very existence and survival. Skinner v. Oklahoma, 316 U. S. 535 , 316 U. S. 541 (1942). See also Maynard v. Hill, 125 U. S. 190 (1888). To deny this fundamental freedom on so unsupportable a basis as the racial classifications embodied in these statutes, classifications so directly subversive of the principle of equality at the heart of the Fourteenth Amendment, is surely to deprive all the State’s citizens of liberty without due process of law. The Fourteenth Amendment requires that the freedom of choice to marry not be restricted by invidious racial discriminations. Under our Constitution, the freedom to marry, or not marry, a person of another race resides with the individual, and cannot be infringed by the State.
These convictions must be reversed.
It is so ordered.
Page 388 U. S. 13
[ Footnote 1 ]
Section 1 of the Fourteenth Amendment provides:
“All persons born or naturalized in the United States and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.”
[ Footnote 2 ]
206 Va. 924, 147 S.E.2d 78 (1966).
[ Footnote 3 ]
Section 257 of the Virginia Code provides:
“ Marriages void without decree. — All marriages between a white person and a colored person shall be absolutely void without any decree of divorce or other legal process.”
Va.Code Ann. § 20-57 (1960 Repl. Vol.).
[ Footnote 4 ]
Section 20-54 of the Virginia Code provides:
“ Intermarriage prohibited; meaning of term ‘white persons.’   All laws heretofore passed and now in effect regarding the intermarriage of white and colored persons shall apply to marriages prohibited by this chapter.”
Va.Code Ann. § 20-54 (1960 Repl. Vol.).
The exception for persons with less than one-sixteenth “of the blood of the American Indian” is apparently accounted for, in the words of a tract issued by the Registrar of the State Bureau of Vital Statistics, by “the desire of all to recognize as an integral and honored part of the white race the descendants of John Rolfe and Pocathontas. . . .” Plecker, The New Family and Race Improvement, 17 Va.Health Bull., Extra No. 12, at 25-26 (New Family Series No. 5, 1925), cited in Wadlington, The Loving Case: Virginia’s Anti-Miscegenation Statute in Historical Perspective, 52 Va.L.Rev. 1189, 1202, n. 93 (1966).
Section 1-14 of the Virginia Code provides:
“ Colored persons and Indians defined. — Every person in whom there is ascertainable any Negro blood shall be deemed and taken to be a colored person, and every person not a colored person having one fourth or more of American Indian blood shall be deemed an American Indian; except that members of Indian tribes existing in this Commonwealth having one fourth or more of Indian blood and less than one sixteenth of Negro blood shall be deemed tribal Indians.”
Va.Code Ann. § 1-14 (1960 Repl. Vol.).
[ Footnote 5 ]
After the initiation of this litigation, Maryland repealed its prohibitions against interracial marriage, Md.Laws 1967, c. 6, leaving Virginia and 15 other States with statutes outlawing interracial marriage: Alabama, Ala.Const., Art. 4, § 102, Ala.Code, Tit. 14, § 360 (1958); Arkansas, Ark.Stat.Ann. § 55-104 (1947); Delaware, Del.Code Ann., Tit. 13, § 101 (1953); Florida, Fla.Const., Art. 16, § 24, Fla.Stat. § 741.11 (1965); Georgia, Ga.Code Ann. § 53-106 (1961); Kentucky, Ky.Rev.Stat.Ann. § 402.020 (Supp. 1966); Louisiana, La.Rev.Stat. § 14:79 (1950); Mississippi, Miss.Const., Art. 14, § 263, Miss.Code Ann. § 459 (1956); Missouri, Mo.Rev.Stat. § 451.020 (Supp. 1966); North Carolina, N.C.Const., Art. XIV, § 8, N.C.Gen.Stat. § 14-181 (1953); Oklahoma, Okla.Stat., Tit. 43, § 12 (Supp. 1965); South Carolina, S.C.Const., Art. 3, § 33, S.C.Code Ann. § 20-7 (1962); Tennessee, Tenn.Const., Art. 11, § 14, Tenn.Code Ann. § 36-402 (1955); Texas, Tex.Pen.Code, Art. 492 (1952); West Virginia, W.Va.Code Ann. § 4697 (1961).
Over the past 15 years, 14 States have repealed laws outlawing interracial marriages: Arizona, California, Colorado, Idaho, Indiana, Maryland, Montana, Nebraska, Nevada, North Dakota, Oregon, South Dakota, Utah, and Wyoming.
The first state court to recognize that miscegenation statutes violate the Equal Protection Clause was the Supreme Court of California. Perez v. Sharp, 32 Cal.2d 711, 198 P.2d 17 (1948).
[ Footnote 6 ]
For a historical discussion of Virginia’s miscegenation statutes, see Wadlington, supra, n 4.
[ Footnote 7 ]
Va.Code Ann. § 20-54 (1960 Repl. Vol.).
[ Footnote 8 ]
Va.Code Ann. § 20-53 (1960 Repl. Vol.).
[ Footnote 9 ]
Va.Code Ann. § 20-50 (1960 Repl. Vol.).
[ Footnote 10 ]
Va.Code Ann. § 254 (1960 Repl. Vol.).
[ Footnote 11 ]
Appellants point out that the State’s concern in these statutes, as expressed in the words of the 1924 Act’s title, “An Act to Preserve Racial Integrity,” extends only to the integrity of the white race. While Virginia prohibits whites from marrying any nonwhite (subject to the exception for the descendants of Pocahontas), Negroes, Orientals, and any other racial class may intermarry without statutory interference. Appellants contend that this distinction renders Virginia’s miscegenation statutes arbitrary and unreasonable even assuming the constitutional validity of an official purpose to preserve “racial integrity.” We need not reach this contention, because we find the racial classifications in these statutes repugnant to the Fourteenth Amendment, even assuming an even-handed state purpose to protect the “integrity” of all races.
MR. JUSTICE STEWART, concurring.
I have previously expressed the belief that “it is simply not possible for a state law to be valid under our Constitution which makes the criminality of an act depend upon the race of the actor.” McLaughlin v. Florida, 379 U. S. 184 , 379 U. S. 198 (concurring opinion). Because I adhere to that belief, I concur in the judgment of the Court.
Contributors and Attributions
- Authored by : US Governement. Located at : https://supreme.justia.com/cases/federal/us/388/1/case.html . License : Public Domain: No Known Copyright


7.1: Formulating a Thesis
You need a good thesis statement for your essay but are having trouble getting started. You may have heard that your thesis needs to be specific and arguable, but still wonder what this really means.
Let’s look at some examples. Imagine you’re writing about John Hughes’s film Sixteen Candles (1984).
You take a first pass at writing a thesis:
Sixteen Candles is a romantic comedy about high school cliques.
Is this a strong thesis statement? Not yet, but it’s a good start. You’ve focused on a topic–high school cliques–which is a smart move because you’ve settled on one of many possible angles. But the claim is weak because it’s not yet arguable. Intelligent people would generally agree with this statement—so there’s no real “news” for your reader. You want your thesis to say something surprising and debatable. If your thesis doesn’t go beyond summarizing your source, it’s descriptive and not yet argumentative.
The key words in the thesis statement are “romantic comedy” and “high school cliques.” One way to sharpen the claim is to start asking questions .
For example, how does the film represent high school cliques in a surprising or complex way? How does the film reinforce stereotypes about high school groups and how does it undermine them? Or why does the film challenge our expectations about romantic comedies by focusing on high school cliques? If you can answer one of those questions (or others of your own), you’ll have a strong thesis.
Tip : Asking “how” or “why” questions will help you refine your thesis, making it more arguable and interesting to your readers.
Take 2. You revise the thesis. Is it strong now?
Sixteen Candles is a romantic comedy criticizing the divisiveness created by high school cliques.
You’re getting closer. You’re starting to take a stance by arguing that the film identifies “divisiveness” as a problem and criticizes it, but your readers will want to know how this plays out and why it’s important. Right now, the thesis still sounds bland – not risky enough to be genuinely contentious.
Tip : Keep raising questions that test your ideas. And ask yourself the “so what” question. Why is your thesis interesting or important?
Take 3. Let’s try again. How about this version?
Although the film Sixteen Candles appears to reinforce stereotypes about high school cliques, it undermines them in important ways, questioning its viewers’ assumptions about what’s normal.
Bingo! This thesis statement is pretty strong. It challenges an obvious interpretation of the movie (that is just reinforces stereotypes), offering a new and more complex reading in its place. We also have a sense of why this argument is important. The film’s larger goal, we learn, is to question what we think we understand about normalcy.
What’s a Strong Thesis?
As we’ve just seen, a strong thesis statement crystallizes your paper’s argument and, most importantly, it’s arguable .
This means two things. It goes beyond merely summarizing or describing to stake out an interpretation or position that’s not obvious, and others could challenge for good reasons. It’s also arguable in the literal sense that it can be argued , or supported through a thoughtful analysis of your sources. If your argument lacks evidence, readers will think your thesis statement is an opinion or belief as opposed to an argument.
Exercises for Drafting an Arguable Thesis
A good thesis will be
focused
on your object of study (as opposed to making a big claim about the world) and will introduce the
key words
guiding your analysis.
To get started, you might experiment with some of these “mad libs.” They’re thinking exercises that will help propel you toward an arguable thesis.
By examining __________________ [topic/approach], we can see _____________________[thesis—the claim that’s surprising], which is important because ___________________________.[1]
Example:
“By examining Sixteen Candles through the lens of Georg Simmel’s writings on fashion, we can seethat the protagonist’s interest in fashion as an expression of her conflicted desire to be seen as both unique and accepted by the group. This is important because the film offers its viewers a glimpse into the ambivalent yearnings of middle class youth in the 1980s.
Although readers might assume _________________ [the commonplace idea you’re challenging], I argue that _________________________ [your surprising claim].
Example:
Although viewers might assume the romantic comedy Sixteen Candles is merely entertaining, I believe its message is political. The film uses the romance between Samantha, a middle class sophomore, and Jake, an affluent senior, to reinforce the fantasy that anyone can become wealthy and successful with enough cunning and persistence.
Still Having Trouble? Let’s Back Up…
It helps to understand why readers value the arguable thesis. What larger purpose does it serve?  The better you can anticipate the expectations of your readers, the better you’ll be able to persuade them to entertain seeing things your way.
Academic readers (and readers more generally) read to learn something new. They want to see the writer challenge commonplaces—either everyday assumptions about your object of study or truisms in the scholarly literature. In other words, academic readers want to be surprised so that their thinking shifts or at least becomes more complex by the time they finish reading your essay. Good essays problematize what we think we know and offer an alternative explanation in its place. They leave their reader with a fresh perspective on a problem.
We all bring important past experiences and beliefs to our interpretations of texts, objects, and problems. You can harness these observational powers to engage critically with what you are studying. The key is to be alert to what strikes you as strange, problematic, paradoxical, or puzzling about your object of study. If you can articulate this and a claim in response, you’re well on your way to formulating an arguable thesis in your introduction.
How do I set up a “problem” and an arguable thesis in response?
All good writing has a purpose or motive for existing. Your thesis is your surprising response to this problem or motive. This is why it seldom makes sense to start a writing project by articulating the thesis. The first step is to articulate the question or problem your paper addresses.
Here are some possible ways to introduce a conceptual problem in your paper’s introduction.
1. Challenge a commonplace interpretation (or your own first impressions).
How are readers likely to interpret this source or issue? What might intelligent readers think at first glance? (Or, if you’ve been given secondary sources or have been asked to conduct research to locate secondary sources, what do other writers or scholars assume is true or important about your primary source or issue?)
What does this commonplace interpretation leave out, overlook, or under-emphasize?
2. Help your reader see the complexity of your topic.
Identify and describe for your reader a paradox, puzzle, or contradiction in your primary source(s).
What larger questions does this paradox or contradiction raise for you and your readers?
3.
Summarize for your reader another scholar’s argument about your topic, primary source, or case study and tell your reader why this claim is interesting.
Now explain how you will extend this scholar’s argument to explore an issue or case study that the scholar doesn’t address fully.
4. If your assignment asks you to do research, identify a gap in another scholar’s or a group of scholars’ research.
Summarize for your reader another scholar’s argument about your topic, primary source, or case study and tell your reader why this claim is interesting. Or, summarize how scholars in the field tend to approach your topic.
Next, explain what important aspect this scholarly representation misses or distorts. Introduce your particular approach to your topic and its value
5. If your assignment asks you to do research, bring in a new lens for investigating your case study or problem.
Summarize for your reader how a scholar or group of scholars has approached your topic.
Introduce a theoretical source (possibly from another discipline) and explain how it helps you address this issue from a new and productive angle.
Tip : your introductory paragraph will probably look like this:
Testing Your Thesis
You can test your thesis statement’s arguability by asking the following questions:
Does my thesis only or mostly summarize my source?
If so, try some of the exercises above to articulate your paper’s conceptual problem or question.
Is my thesis arguable –can it be supported by evidence in my source, and is it surprising and contentious?
If not, return to your sources and practice the exercises above.
Is my thesis about my primary source or case study, or is it about the world?
If it’s about the world, revise it so that it focuses on your primary source or case study. Remember you need solid evidence to support your thesis.
“Formulating a Thesis” was written by Andrea Scott, Princeton University
Acknowledgements
I’d like to thank my current and former colleagues in the Princeton Writing Program for helping me think through and test ways of teaching the arguable thesis. Special thanks go to Kerry Walk, Amanda Irwin Wilkins, Judy Swan, and Keith Shaw. A shout-out to Mark Gaipa as well, whose cartoons on teaching source use remain a program favorite.
[1] Adapted from Erik Simpson’s “Five Ways of Looking at a Thesis” at http://www.math.grinnell.edu/~simpsone/Teaching/fiveways.html
Contributors and Attributions
- Formulating a Thesis. Authored by : Andrea Scott. Provided by : Princeton University. Located at : http://www.princeton.edu/main/ . Project : WritingCommons. License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives