10.3: R and Bootstrapping
All generalities like standard deviation and mean are normally taken from sample but meant to represent the whole statistical population. Therefore, it is possible that these estimations could be seriously wrong. Statistical techniques like bootstrapping were designed to minimize the risk of these errors. Bootstrap is based only on the given sample but try to estimate the whole population.
The idea of bootstrap was inspired by from Buerger and Raspe “Baron Munchausen’s miraculous adventures”, where the main character pulls himself (along with his horse) out of a swamp by his hair (Figure \(\PageIndex{1}\)). Statistical bootstrap was actively promoted by Bradley Efron since 1970s but was not used frequently until 2000s because it is computationally intensive. In essence, bootstrap is the re-sampling strategy which replaces part of sample with the subsample of its own. In R , we can simply sample() our data with the replacement .
First, we will bootstrap the mean (Figure \(\PageIndex{2}\)) using the advanced boot package:
Code \(\PageIndex{1}\) (R):
(Note that here and in many other places in this book number of replicates is 100. For the working purposes, however, we recommend it to be at least 1,000.)
Code \(\PageIndex{2}\) (R):
Package boot allows to calculate the 95% confidence interval:
Code \(\PageIndex{3}\) (R):
More basic bootstrap package bootstraps in a simpler way. To demonstrate, we will use the spur.txt data file. This data is a result of measurements of spur length on 1511 Dactylorhiza orchid flowers. The length of spur is important because only pollinators with mouth parts comparable to spur length can successfully pollinate these flowers.
Code \(\PageIndex{4}\) (R):
Jackknife is similar to the bootstrap but in that case observations will be taking out of the sample one by one without replacement:
Code \(\PageIndex{5}\) (R):
This is possible to bootstrap standard deviation and mean of this data even without any extra package, with for cycle and sample() :
Code \(\PageIndex{6}\) (R):
(Alternatively, tt could be an empty data frame, but this way takes more computer time which is important for bootstrap. What we did above, is the pre-allocation , useful way to save time and memory.)
Actually, spur length distribution does not follow the normal law ( check it yourself). It is better then to estimate median and median absolute deviation (instead of mean and standard deviation), or median and 95% range:
Code \(\PageIndex{7}\) (R):
(Note the use of replicate() function, this is another member of apply() family.)
This approach allows also to bootstrap almost any measures. Let us, for example, bootstrap 95% confidence interval for Lyubishchev’s K:
Code \(\PageIndex{8}\) (R):
Bootstrap and jackknife are related with numerous resampling techniques . There are multiple R packages (like coin ) providing resampling tests and related procedures:
Code \(\PageIndex{9}\) (R):
Bootstrap is also widely used in the machine learning. Above there was an example of Jclust() function from the asmisc.r set. There also are BootA() , BootRF() and BootKNN() to boorstrap non-supervised and supervised results.
) plants. Use bootstrap and resampling methods.


4: Behavior Near Trajectories - Linearization
Now we are going to discuss a method for analyzing stability that utilizes linearization about the object whose stability is of interest. For now, the "objects of interest" are specific solutions of a vector field.The structure of the solutions of linear, constant coefficient systems is covered in many ODE textbooks. My favorite is the book of Hirsch et al.1. It covers all of the linear algebra needed for analyzing linear ODEs that you probably did not cover in your linear algebra course. The book by Arnold is also very good, but the presentation is more compact, with fewer examples.
We begin by considering a general nonautonomous vector field:
\[\dot{x} = f(x, t), x \in \mathbb{R}^n, \label{4.1}\]
and we suppose that
\[\bar{x}(t, t_{0}, x_{0}), \label{4.2}\]
is the solution of Equation \ref{4.1} for which we wish to determine its stability properties. As when we introduced the definitions of stability, we proceed by localizing the vector field about the solution of interest. We do this by introducing the change of coordinates
\(x = y+\bar{x}\)
for Equation \ref{4.1} as follows:
\(\dot{x} = \dot{y}+\dot{\bar{x}} = f(y+\bar{x}, t)\),
or
\(\dot{y} = f(y+\bar{x}, t)\dot{\bar{x}}\),
\[= f(y+\bar{x}, t)f(\bar{x}, t), \label{4.3}\]
where we omit the arguments of \(\bar{x}(t, t_{0}, x_{0})\) for the sake of a less cumbersome notation. Next we Taylor expand \(f(y+\bar{x}, t)\) in \(y\) about the solution \(\bar{x}\), but we will only require the leading order terms explicitly
\[f(y+\bar{x}, t) = f(\bar{x}, t)+Df(\bar{x},t)y+\mathbb{O}(|y|^2), \label{4.4}\]
where \(Df\) denotes the derivative (i.e. Jacobian matrix) of the vector valued function f and \(\mathbb{O}(|y|^2)\) denotes higher order terms in the Taylor expansion that we will not need in explicit form. Substituting this into Equation \ref{4.4} gives:
\(\dot{y} = f(y+\bar{x}, t) - f(\bar{x}, t)\),
\(= f(\bar{x}, t)+Df(\bar{x}, t)y+\mathbb{O}(|y|^2)f(\bar{x}, t)\),
\[= Df(\bar{x}, t)y+\mathbb{O}(|y|^2). \label{4.5}\]
Keep in mind that we are interested in the behavior of solutions near \(\bar{x}(t, t_{0}, x_{0})\), i.e., for \(y\) small. Therefore, in that situation it seems reasonable that neglecting the \(\mathbb{O}(|y|^2)\) in Equation \ref{4.5} would be an approximation that would provide us with the particular information that we seek. For example, would it provide sufficient information for us to deter- mine stability? In particular,
\[\dot{y} = Df(\bar{x}, t)y, \label{4.6}\]
is referred to as the linearization of the vector field \(\dot{x} = f (x, t)\) about the solution \(\bar{x}(t, t_{0}, x_{0})\).
Before we answer the question as to whether or not Equation \ref{4.1} provides an adequate approximation to solutions of Equation \ref{4.5} for y "small", we will first study linear vector fields on their own.
Linear vector fields can also be classified as nonautonomous or autonomous . Nonautonomous linear vector fields are obtained by linearizing a nonautonomous vector field about a solution (and retaining only the linear terms). They have the general form:
\[\dot{y} = A(t)y, y(0) = y_{0}, \label{4.7}\]
where
\[A(t) \equiv Df(\bar{x}(t, t_{0}, x_{0}), t) \label{4.8}\]
is a \(n \times n\) matrix. They can also be obtained by linearizing an autonomous vector field about a time-dependent solution.
An autonomous linear vector field is obtained by linearizing an autonomous vector field about an equilibrium point. More precisely, let \(\dot{x} = f(x)\) denote an autonomous vector field and let \(x = x_{0}\) denote an equilibrium point, i.e. \(f(x_{0}) = 0\). The linearized autonomous vector field about this equilibrium point has the form:
\[\dot{y} = Df(x_{0})y, y(0) = y_{0}, \label{4.9}\]
or
\[\dot{y} = Ay, y(0) = y_{0}, \label{4.10}\]
where \(A \equiv Df(x_{0})\) is a \(n \times n\) matrix of real numbers. This is significant because (4.10) can be solved using techniques of linear algebra, but Equation \ref{4.7}, generally, cannot be solved in this manner. Hence, we will now describe the general solution of (4.10).
The general solution of Equation \ref{4.10} is given by:
\[y(t) = e^{At}y_{0}. \label{4.11}\]
In order to verify that this is the solution, we merely need to substitute into the right hand side and the left hand side of (4.10) and show that equality holds. However, first we need to explain what \(e^{At}\) is, i.e. the exponential of the \(n \times n\) matrix A (by examining Equation \ref{4.11} it should be clear that if Equation \ref{4.11} is to make sense mathematically, then \(e^{At}\) must be a \(n times n\) matrix).
Just like the exponential of a scalar, the exponential of a matrix is defined through the exponential series as follows:
\(e^{At} \equiv \mathbb{I}+At+\frac{1}{2!}A^{2}t^{2}+···+\frac{1}{n!}A^{n}t^{n}+\cdots\),
\[= \sum_{i=0}^{n}\frac{1}{i!}A^{i}t^{i}, \label{4.12}\]
where \(\mathbb{I}\) denotes the \(n \times n\) identity matrix. But we still must answer the question, "does this exponential series involving products of matrices make mathematical sense"? Certainly we can compute products of matrices and multiply them by scalars. But we have to give meaning to an infinite sum of such mathematical objects. We do this by defining the norm of a matrix and then considering the convergence of the series in norm. When this is done the "convergence problem" is exactly the same as that of the exponential of a scalar. Therefore the exponential series for a matrix converges absolutely for all t, and therefore it can be differentiated with respect to t term-by-term, and the resulted series of derivatives also converges absolutely.
Next we need to argue that Equation \ref{4.11} is a solution of Equation \ref{4.10}. If we differentiate the series (4.12) term by term, we obtain that:
\[\frac{d}{dt}e^{At} = Ae^{At} = e^{At}A, \label{4.13}\]
where we have used the fact that the matrices A and \(e^{At}\) commute (this is easy to deduce from the fact that A commutes with any power of A) It then follows from this calculation that:
\[\dot{y} = \frac{d}{dt}e^{At}y_{0} = Ae^{At}y_{0} = Ay. \label{4.14}\]
Therefore the general problem of solving (4.10) is equivalent to computing \(e^{At}\), and we now turn our attention to this task.
First, suppose that \(A\) is a diagonal matrix, say
\[A = \begin{pmatrix} {\lambda_{1}}&{0}&{\cdots}&{0}\\ {0}&{\lambda_{2}}&{\cdots}&{0}\\ {0}&{0}&{\cdots}&{0}\\ {0}&{0}&{\cdots}&{\lambda_{n}} \end{pmatrix} \label{4.15}\]
Then it is easy to see by substituting A into the exponential series (4.12) that:
\[e^{At} = \begin{pmatrix} {e^{\lambda_{1}t}}&{0}&{\cdots}&{0}\\ {0}&{e^{\lambda_{2}t}}&{\cdots}&{0}\\ {0}&{0}&{\cdots}&{0}\\ {0}&{0}&{\cdots}&{e^{\lambda}t} \end{pmatrix} \label{4.16}\]
Therefore our strategy will be to transform coordinates so that in the new coordinates A becomes diagonal (or as "close as possible" to diagonal, which we will explain shortly). Then \(e^{At}\) will be easily computable in these coordinates. Once that is accomplished, then we use the inverse of the transformation to transform the solution back to the original coordinate system.
Now we make these ideas precise. We let
\[y = Tu, u \in \mathbb{R}^n, y \in \mathbb{R}^n, \label{4.17}\]
where T is a \(n \times n\) matrix whose precise properties will be developed in the following.
This is a typical approach in ODEs. We propose a general coordinate transformation of the ODE, and then we construct it in a way that gives the properties of the ODE that we desire. Substituting (4.17) into (4.10) gives:
\[\dot{y} = T\dot{u} = Ay = ATu, \label{4.18}\]
T will be constructed in a way that makes it invertible, so that we have:
\[\dot{u} = T^{-1}ATu, u(0) = T^{-1}y(0). \label{4.19}\]
To simplify the notation we let:
\[T = T^{-1}AT, \label{4.20}\]
or
\[A = T^{-1}\Lambda T. \label{4.21}\]
Substituting (4.21) into the series for the matrix exponential (4.12) gives:
\(e^{At} = e^{T\Lambda T^{-1}t}\),
\[= \mathbb{1}+ T\Lambda T^{-1}t+\frac{1}{2!}(T\Lambda T^{-1})^{2}t^2+\cdots+\frac{1}{n!}(T\Lambda T^{-1})^{n}t^n+\cdots \label{4.22}\]
Now note that for any positive integer n we have:
\((T\Lambda T^{-1})^{n} = \underbrace{(T\Lambda T^{-1})(T\Lambda T^{-1})\cdots (T\Lambda T^{-1})(T\Lambda T^{-1})}_{n factors}\)
\[= T\Lambda^{n}T^{-1} \label{4.23}\]
Substituting this into (4.22) gives:
\(e^{At} = \sum_{n=0}^{\infty} \frac{1}{n!}(T\Lambda T^{-1})^{n}t^n\),
\(= T(\sum_{n=0}^{\infty} \frac{1}{n!}\Lambda^{n}t^n)T^{-1}\),
\[= Te^{\Lambda t}T^{-1} \label{4.24}\]
or
\[e^{At} = Te^{\Lambda t}T^{-1} \label{4.25}\]
Now we arrive at our main result. If T is constructed so that
\[\Lambda = T^{-1}AT \label{4.26}\]
is diagonal, then it follows from (4.16) and (4.25) that \(e^{At}\) can always be computed. So the ODE problem of solving (4.10) becomes a problem in linear algebra. But can a general \(n \times n\) matrix A always be diagonalized? If you have had a course in linear algebra, you know that the answer to this question is "no". There is a theory of the (real) that will apply here. However, that would take us into too great a diversion for this course. Instead, we will consider the three standard cases for \(2 \times 2\) matrices. That will suffice for introducing the the main ideas without getting bogged down in linear algebra. Nevertheless, it cannot be avoided entirely. You will need to be able to compute eigenvalues and eigenvectors of \(2 \times 2\) matrices, and understand their meaning.
The three cases of \(2 \times 2\) matrices that we will consider are characterized by their eigenvalues:
- two real eigenvalues, diagonalizable A,
- two identical eigenvalues, nondiagonalizable A,
- a complex conjugate pair of eigenvalues.
In the table below we summarize the form that these matrices can be transformed in to (referred to as the of A) and the resulting exponential of this canonical form.
Once the transformation to \(\Lambda\) has been carried out, we will use these results to deduce \(e^{\Lambda}\).


2.4: Division of Labor, Book 1
-
- Last updated
- Save as PDF
“History shows that as one type progresses, the other type fades away.”
NOTE ON SOURCE: This passage comes from Durkheim’s dissertation, completed in 1893, and first published in 1902 as De la Division du Travaile Sociale. The first English translation was done by George Simpson in 1933, but this version was found to have several shortcomings. A more approved translation was made in 1984 by W. D. Halls, edited by Lewis A. Coser. This translation was republished with some improvement by Steven Lukes in 1997. This is the recommended version if you would like to read more of the text than what is included here.
Introduction – Why this is important and what to look for
The Division of Labor in Society was divided into three books, as the previous selection demonstrated. The first book examines the function of the division of labor and introduces the distinction between societies held together by Mechanical Solidarity and those held together by Organic Solidarity. The passages below describe those differences. Pay close attention to all the ways in which these two types of society differ.
Chapter 3. Organic Solidarity due to the Division of Labor
Part 4. Conclusion
We recognize two kinds of positive solidarity, solidarity which produces integration. The first kind directly binds the individual to society while the second binds the individual indirectly , through reliance on the other people who collectively make up society.
Society is not seen in the same aspect in the two cases. In the first, what we call society is more or less composed of the beliefs and values held in common by all people. In contrast, the second case is composed of a system of different and unique functions which are united through interdependence.
In the first case, society is strong if the ideas and common tendencies are greater quantitatively and qualitatively than those ideas and habits held by individuals. This kind of solidarity can grow only at the expense of individual personality. In each of us there are two consciences, one common to our group and the other which is personal to us and distinct and that makes us an individual. Solidarity through likeness is at its greatest when the collective conscience takes over our entire consciousness, [ when what we think is what society thinks ]. But at that moment we have no personality. Our personalities can only emerge if the community to which we belong has less control of us. These are two opposing forces. If we want to think and act for ourselves, we cannot also be strongly inclined to think and act as everyone else. So, when this form of solidarity acts on us forcefully, our personality vanishes; we are no longer ourselves but the collective life.
The social molecules which cohere together in this way can act together only if they have no actions of their own; they are molecules of inorganic bodies. For that reason we propose to call this type of solidarity mechanical . This doesn’t mean it is produced by mechanical or artificial means, but only as an analogy to the cohesion uniting an inanimate body, as opposed to elements of a living body. The individual conscience is dependent upon the collective conscience and follows all its movements, without a life of its own, so to speak. In societies where this type of solidarity is strong, the individual does not really appear. Personal rights are generally not recognized.
It is utterly different when we consider the solidarity which is produced by the division of labor. While the first type implied that individuals resemble each other, this type presumes they are differentiated. While the first type is possible only so much as the individual personality is subsumed by the collective personality, the second type is possible only if each individual person has a sphere of action unique to him or her, and so an individual personality. It is necessary for the collective conscience to recede to allow the individual conscience to operate freely. The more it does so, the stronger the cohesion which results [as each becomes reliant on every other member fulfilling his or her unique sphere of action]. Each one depends more on society as labor is divided, and each person’s activity becomes more specialized.
Chapter 4. Further Proof the Preceding
Part 1. [Likeness in Mechanical Solidarity Societies]
The more societies are primitive, the more alike are its members.
Chapter 5. Progressive Preponderance of Organic Solidarity; Its Consequences
Part 2. [Three Conditions of Mechanical Solidarity]
Not only does mechanical solidarity bind people together less strongly over time, but we find it slackens as we progress socially.
Indeed, the strength of social links through likeness vary with respect to the following three conditions:
First, the relative proportion of collective and individual conscience. The links are stronger the more the first overshadows the second.
Second, the average intensity of the states of the collective conscience.
Third, The distinctiveness of these states. That is, the more specifically defined are the collective beliefs and practices, the less room there is for individualization. The more general and abstract are the rules, however, the more individual reflection plays a role. Centrifugal tendencies multiply at the cost of social cohesion.
Strong and defined states of the common conscious are the roots of penal law. However, the number of such laws is less today than before, and diminish progressively as societies approach our modern type.
[Durkheim discusses the relative proportion of what he calls “repressive” laws, violation of which offends the collective conscience, and “restitutive” laws, violation of which is less shocking and more personally related to the parties at hand (think broken contracts). He argues that, as societies advance, the proportional amount of repressive law diminishes as that of restitutive law increases. This is, indeed, a key part of Book 1, and is used to support his argument about changes from mechanical solidarity to organic solidarity over time. As anthropologists of law have criticized the empirical facts Durkheim used, limited as they were to what was available to him in the 19th century, you will not find more of this discussion in this reader, but you should know that Durkheim is using evidence, albeit imperfect, to build this theory, rather than engaging in “armchair philosophizing.” ]
Chapter 6. Progressive Preponderance of Organic Solidarity, continued
Part 1. Segmental Type
So we can say it is a historical law that mechanical solidarity, which at first stood alone, progressively loses ground and that, over time, little by little, organic solidarity predominates.
If we were to try to imagine an ideal type of society held together exclusively by likeness, we would have to conceive it as one wholly homogeneous, one in which none of its human members are distinguishable from one another; there would be no real organization to speak of. It would be a social protoplasm, a blob, a horde, if you will.
It is true we have yet to find any society that operates completely in this way. [ We do find some glimmers of it, among some Native American tribes, for example. We can designate hordes which form elements in more extensive groups as clans.]
We can call these societies segmented as they are formed by the repetition of like aggregations in them, like the rings of an earthworm. The term clan expresses the mixed nature of these segmented groups. The clan is a family because its members are kin to one another. These familial affinities are for the most part what keeps the group united. But these are not families the way we understand families, because kinship need not be by blood. The clan in fact contains a great many strangers. It can comprise several thousand persons. And it is the basic political unity as well, with the clan-heads the only social authorities.
The main point, however, is that the clan, just as the horde, of which it is but an extension, has no other solidarity than that derived from likeness. For segmented organization to be possible, the segments must resemble one another; otherwise, they would not be united.
In these societies, religion pervades all of social life. This is so because social life itself is almost exclusively composed of common beliefs and practices. Where the collective personality is the only one in existence, property also must be collective, so we find an early form of communism operating in these societies.
There is, then, a social structure of a specific kind which corresponds with mechanical solidarity. What characterizes it is a system of segments homogeneous and similar to each other.
Part 2. Organized Type
Quite another thing is the structure of societies where organic solidarity is preponderant.
They are constituted by a system of different organs, each of which has a special role, and which are themselves formed of differentiated parts. Social elements are not heaped together linearly as the rings of an earthworm, nor are they entwined with one another, but rather they are coordinated and subordinated to one another around a central organ which regulates the rest of the organism. Others may depend on this central organ, but the central organ depends on the others as well.  There is nothing superhuman or timeless about this central organ. There are only differences in degree between this organ and the others.
This social type rests on such different principles as that of the segmented type that it can develop only so much as it erases the segmented type. In organized societies, individuals are not grouped based on lineage or bloodline, but according to the particular nature of the social activity they engage in. Their natural context is not that of birth [ blood, race, etc. ] but of occupation. It is no longer real or fictitious kinship which marks the place of each, but the function which he or she fulfills.
No doubt, when this new organization began to appear, it tried to use the existing organization and to assimilate it. So, functions were often allocated based on original divisions of birth. In a way, classes (and castes in particular) probably have their origin thusly. But this mixed arrangement cannot last for long, because there is a fundamental contradiction between the two. It is only a very basic division of labor which can adapt to preexisting social divisions in this way. The division of labor can only grow by freeing itself from this confining framework. As soon as it passes a certain stage of development, there is no longer any relation between the hereditarily fixed properties of segments and the new skills and aptitudes called forth by the growth of functions needed in society. The social material must combine in new ways to organize itself upon these different foundations. The old structure, so far as it persists, is opposed to these new combinations. Which is why it must disappear.
Thus the history shows that as one type progresses, the other type fades away.
Just as we could not say there was any known wholly segmented society, we also observe that there is as yet no wholly organized society. We do see, however, that organic solidarity is progressing, and becoming more preponderant.
Our future investigations will show that our current occupational organization is not everything it should be, as abnormal causes have prevented it from attaining the degree of development which our social order now demands. [ More on that in Book 3 ]
Questions
- If thinking for ourselves and thinking for the community are mutually opposed, as Durkheim suggests, where do YOU lie on this continuum? If you had been born in, say, 1300CE, do you think your answer would have been different? What about 1300BCE? Why?
- Durkheim has been claimed as an early anthropologist, and much of his theory developed in The Division of Labor is based on observational and historical data about “primitive” peoples, including Native Americans, Aboriginal Australians, and early Jewish peoples. Much of this is inaccurate and/or biased by Eurocentric thinking. In Chapter 6, Durkheim draws copiously from early anthropological thinking to describe how “segmented” societies (the horde, the clan) operate. Given the problems with the data used, is his theory still valid? Explain your answer. For those of you who are familiar with the world of Star Trek , it might be helpful for you to think of the “horde” as The Borg.
- When Durkheim talks about “the central organ” that “regulates” the other members, to what is he referring? If you are asked to describe Durkheim’s theory of the state in modern society, would this passage help?
- It may be hard for us, who develop in what Durkheim would call organized societies, to recognize the pull of “birth” to which he refers in part 2 of Chapter 6. You may want to consider what it might be like to live in a society in which all that mattered was who your ancestors were. Can you think of historical examples when this might have been the case? Compare Durkheim’s “birth vs. occupation” to Weber’s “status vs. class.”
Concepts
Segmented Society
Collective Conscience
Organized Society
Mechanical Solidarity (and MS Societies)
Organic Solidarity (and OS Societies)


4.18: Putting It Together- Diversity in the Workplace
Although decades of advocacy and diversity training have had an impact, the results have been mixed. The upside: Deloitte’s research into the current state of inclusion found that 86% of respondents felt they could be themselves most or all of the time at work. [1] That statistic represents a significant improvement over a relatively short period of time. A survey conducted six years prior found that 61% of “respondents felt they had to hide at least one aspect of who they are.” [2] At that time, the conclusion was that “most inclusion programs require people to assimilate into the overall corporate culture” and that this need to “cover” directly impacts not only an individual’s sense of self but their commitment to the organization. [3]
On the downside, bias remains a constant, with over 60% of respondents reporting bias in their workplace. [4] Deloitte reports that 64% of employees surveyed “felt they had experienced bias in their workplace during the last year.” [5] Even more disturbing, 61% of those respondents “felt they experienced bias in the workplace at least once a month.” The percentages of respondents who indicated that they have observed bias in their workplaces during the last year and observe it on a monthly basis are are roughly the same at 64% and 63%, respectively.
Given the slow pace of progress on D&I , it may be worth looking at the situation as Melinda Briana Epler proposes in one of her ally recommendations. Specifically, if it’s not working, it may be you. Although we are generally aware that our perceptions are subjective, we are largely unaware that there can be a disconnect between our conscious thoughts and our unconscious beliefs or biases, primarily a product of socio-cultural conditioning. To see how this might play out, watch Yassmin Abdel-Magied’s “What does my headscarf mean to you?” TED Talk.
To take D&I to the next level, raise your awareness of unconscious bias and use design hacks and technology to circumvent automatic reactions. For more perspective on how to cultivate diversity, refer to GGSC’s How Do I Cultivate Diversity page. To quote Abdel-Magied: “If we want to live in a world where the circumstances of your birth do not dictate your future and where equal opportunity is ubiquitous, then each and every one of us has a role to play in making sure unconscious bias does not determine our lives.” Perhaps, with practice, egalitarianism will become our default.
- Jacobson, Adam. “ Inclusion Does Not Stop Workplace Bias, Deloitte Survey Shows .” Risk Management Monitor, July 26, 2019. ↵
- "The Bias Barrier: Allyship, Inclusion, and Everyday Behaviors." Deloitte. 2019. Accessed September 14, 2019. ↵
- Matuson, Roberta. “ Uncovering Talent: A New Model For Inclusion And Diversity .” Fast Company, September 12, 2013. ↵
- "The Bias Barrier: Allyship, Inclusion, and Everyday Behaviors." Deloitte. 2019. ↵
- Ibid. ↵
- Ibid. ↵
- “ Clean Up Corrosive Interpersonal Dynamics on Your Team with This System .” First Round Review, March 27, 2018. ↵
Contributors and Attributions
- Woman. Authored by : Zackary Drucker. Provided by : Broadly. Located at : https://broadlygenderphotos.vice.com/ . License : CC BY-NC-ND: Attribution-NonCommercial-NoDerivatives
- Untitled. Authored by : NESA by Makers. Provided by : Unsplash. Located at : https://unsplash.com/photos/IgUR1iX0mqM . License : CC0: No Rights Reserved . License Terms : Unsplash License
- What does my headscarf mean to you?. Authored by : Yassmin Abdel-Magied. Provided by : TED. Located at : https://youtu.be/18zvlz5CxPE . License : All Rights Reserved . License Terms : Standard YouTube License


25.1: Introduction
Chapter Objectives
After studying this chapter, you will be able to:
- Describe the composition of urine
- Label structures of the urinary system
- Characterize the roles of each of the parts of the urinary system
- Illustrate the macroscopic and microscopic structures of the kidney
- Trace the flow of blood through the kidney
- Outline how blood is filtered in the kidney nephron
- Provide symptoms of kidney failure
- List some of the solutes filtered, secreted, and reabsorbed in different parts of the nephron
- Describe the role of a portal system in the kidney
- Explain how urine osmolarity is hormonally regulated
- Describe the regulation of major ions by the kidney
- Summarize the role of the kidneys in maintaining acid–base balance
The urinary system has roles you may be well aware of: cleansing the blood and ridding the body of wastes probably come to mind. However, there are additional, equally important functions played by the system. Take for example, regulation of pH, a function shared with the lungs and the buffers in the blood. Additionally, the regulation of blood pressure is a role shared with the heart and blood vessels. What about regulating the concentration of solutes in the blood? Did you know that the kidney is important in determining the concentration of red blood cells? Eighty-five percent of the erythropoietin (EPO) produced to stimulate red blood cell production is produced in the kidneys. The kidneys also perform the final synthesis step of vitamin D production, converting calcidiol to calcitriol, the active form of vitamin D.
If the kidneys fail, these functions are compromised or lost altogether, with devastating effects on homeostasis. The affected individual might experience weakness, lethargy, shortness of breath, anemia, widespread edema (swelling), metabolic acidosis, rising potassium levels, heart arrhythmias, and more. Each of these functions is vital to your well-being and survival. The urinary system, controlled by the nervous system, also stores urine until a convenient time for disposal and then provides the anatomical structures to transport this waste liquid to the outside of the body. Failure of nervous control or the anatomical structures leading to a loss of control of urination results in a condition called incontinence.
This chapter will help you to understand the anatomy of the urinary system and how it enables the physiologic functions critical to homeostasis. It is best to think of the kidney as a regulator of plasma makeup rather than simply a urine producer. As you read each section, ask yourself this question: “What happens if this does not work?” This question will help you to understand how the urinary system maintains homeostasis and affects all the other systems of the body and the quality of one’s life.
Interactive Link
Watch this video from the Howard Hughes Medical Institute for an introduction to the urinary system.


7.1: Prelude to the Central Limit Theorem
-
- Last updated
- Save as PDF
Learning Objectives
By the end of this chapter, the student should be able to:
- Recognize central limit theorem problems.
- Classify continuous word problems by their distributions.
- Apply and interpret the central limit theorem for means.
- Apply and interpret the central limit theorem for sums.
Why are we so concerned with means? Two reasons are: they give us a middle ground for comparison, and they are easy to calculate. In this chapter, you will study means and the central limit theorem . The central limit theorem (clt for short) is one of the most powerful and useful ideas in all of statistics. There are two alternative forms of the theorem, and both alternatives are concerned with drawing finite samples size n from a population with a known mean, \(\mu\), and a known standard deviation, \(\sigma\). The first alternative says that if we collect samples of size \(n\) with a "large enough \(n\)," calculate each sample's mean, and create a histogram of those means, then the resulting histogram will tend to have an approximate normal bell shape. The second alternative says that if we again collect samples of size \(n\) that are "large enough," calculate the sum of each sample and create a histogram, then the resulting histogram will again tend to have a normal bell-shape.
In either case, it does not matter what the distribution of the original population is, or whether you even need to know it. The important fact is that the distribution of sample means and the sums tend to follow the normal distribution.
The size of the sample, \(n\), that is required in order to be "large enough" depends on the original population from which the samples are drawn (the sample size should be at least 30 or the data should come from a normal distribution). If the original population is far from normal, then more observations are needed for the sample means or sums to be normal. Sampling is done with replacement.

Each time a person rolls more than one die, he or she calculates the sample mean of the faces showing. For example, one person might roll five fair dice and get 2, 2, 3, 4, 6 on one roll.
The mean is \(\frac{2+2+3+4+6}{5} = 3.4\). The 3.4 is one mean when five fair dice are rolled. This same person would roll the five dice nine more times and calculate nine more means for a total of ten means.
Your instructor will pass out the dice to several people. Roll your dice ten times. For each roll, record the faces, and find the mean. Round to the nearest 0.5.
Your instructor (and possibly you) will produce one graph (it might be a histogram) for one die, one graph for two dice, one graph for five dice, and one graph for ten dice. Since the "mean" when you roll one die is just the face on the die, what distribution do these means appear to be representing?
- Draw the graph for the means using two dice. Do the sample means show any kind of pattern?
- Draw the graph for the means using five dice. Do you see any pattern emerging?
- Finally, draw the graph for the means using ten dice. Do you see any pattern to the graph? What can you conclude as you increase the number of dice?
As the number of dice rolled increases from one to two to five to ten, the following is happening:
- The mean of the sample means remains approximately the same.
- The spread of the sample means (the standard deviation of the sample means) gets smaller.
- The graph appears steeper and thinner.
You have just demonstrated the central limit theorem (clt). The central limit theorem tells you that as you increase the number of dice, the sample means tend toward a normal distribution (the sampling distribution).
Glossary
- Sampling Distribution
- Given simple random samples of size \(n\) from a given population with a measured characteristic such as mean, proportion, or standard deviation for each sample, the probability distribution of all the measured characteristics is called a sampling distribution.


<IP_ADDRESS>: Enabling and Using Interactive Matplotlib Features
Matplotlib
is a Python library which supports a broad range of plotting functions and features. If you have ever created and displayed a plot using
matplotlib
yourself, then you may have noticed that the plot has some interactive features allowing you to pan, reset view, save the image, and more. These features are not available by default on Libretexts, but by using the
ipympl widget
you can enable all of those same functions for your plots. We will follow a
short and simple example
to demonstrate this.
To compare, run the following cell and check the output. Without the
ipympl
widget,
matplotlib
plots will simply appear like an embedded image.
Next, we will enable the
ipympl
backend which controls the interactive features of
matplotlib
and then we will draw a 3D plot, because we can. Run the code, and you should see that there is a vertical toolbar on the left-hand side of the plot which allows us to interact with the figure. At the moment, this toolbar does not work perfectly with 3D plots on Libretexts, so you may find that the 'rest original view' and 'forward/previous view' buttons do not work. You may still freely rotate the plot around.
Once the widget is enabled, that change will persist throughout the whole page until the kernel is restarted (either by explicitly pressing the 'restart' button or by refreshing the page). If you go back and rerun the first cell after the one directly above, you will notice that it also has interactive features enabled. Whichever widget you have most recently enabled within a kernel session is the one that will be activated for the entire webpage. To return to the static plotting images, use
%matplotlib inline
. This allows you to include both static and interactive plots on your Libretexts page; just call the relevant
%matplotlib
magic command
and you can switch between the two styles.
You can also adjust the interactive features of your plot with
fig.canvas
booleans. The next cell demonstrates this by disabling the header, toolbox, and footer. You can find more options like this on this ipympl
example page
. Current limitations of the Libretexts code cell plugin prevent later code cells from interacting with earlier ones, so you will have to set all your
fig.canvas
options within the same cell that you display the plot.
The
github page
for
ipympl
and
usage guide
for
matplotlib
are two external resources that may help you debug your code. If you run into any bugs or difficulties and those websites still do not resolve your issue, feel free to email us at
<EMAIL_ADDRESS>and we can provide you with more assistance.
The ipywidgets also work with ipympl. For example:


1.2: Individual Decision Making
-
- Last updated
- Save as PDF
- Boundless
- Boundless
Scarcity Leads to Tradeoffs and Choice
When scarce resources are used, actors are forced to make choices that have an opportunity cost.
Learning objectives
Give examples of economic trade-offs.
A fundamental concept in economics is that of scarcity. In contrast to its colloquial usage, scarcity in economics connotes not that something is nearly impossible to find, but simply that it is not unlimited. For example, the number of available hours in a day is a scarce resource: there is a finite amount of time available to you to do work, hang out with friends, and relax. Most resources are scarce in most situations.
Since resources tend to be scarce, anyone that uses the resource has to make a decision about how to use it. Suppose, for example, that you are a drink manufacturer. To produce a beverage, you have to use some scarce resources: the plastic for the bottle, the workers’ time, a machine to fill the bottles, etc. If you choose to make one bottle of water, you have chosen to not make a bottle of soda. Your scarce resources force you to make a choice and a trade-off producing one product or another.
Tradeoffs : Since resources are scarce for a drink manufacturer, it must make a tradeoff between producing bottles of water and bottles of soda.
Like producers, consumers also have to make choices. Often, consumers must choose between current consumption (“I want to buy an ice cream”) and future consumption (“I should rather save my money so I can buy an ice cream tomorrow”). Since consumers’ resources such as time, attention, and money are limited, they must choose how to best allocate them by making tradeoffs.
The concept of trade-offs due to scarcity is formalized by the concept of opportunity cost. The opportunity cost of a choice is the value of the best alternative forgone. In other words, if you can only produce bottles of soda and water, the opportunity cost of producing a bottle of water is the value of producing a bottle of soda. Similarly, there is an opportunity cost in everything: the opportunity cost of you reading this is what you could be doing with your time instead (say, watching a movie). When scarce resources are used (and just about everything is a scarce resource), people and firms are forced to make choices that have an opportunity cost.
Individuals Face Opportunity Costs
Individuals face opportunity costs when they choose one course of action over another.
Learning objectives
Distinguish between explicit costs and opportunity costs
When individuals make decisions, they are necessarily deciding between taking one course of action over another. In doing so, they are choosing both what to do and, by extension, what not to do. The value of the next best choice forgone is called the opportunity cost. In other words, the opportunity cost of a course of action is the value the of the option that the individual chose not to take.
Individuals face opportunity costs in both economic and non-economic decisions. One of the easiest way to imagine an individual’s opportunity costs is to imagine a student who decides to study. By choosing to study, the student is implicitly choosing to not go to a party, hang out with friends, or catch up on some much-needed sleep. In this example, the opportunity cost is not easily expressed in dollars and cents, but is just as real.
Opportunity Cost : By choosing to go to spend time and money on things like classes and computers, you are necessarily choosing not to spend it on something else, like going on vacation. This is an opportunity cost.
Rational individuals will try to minimize their opportunity costs. By doing so, individuals are maximizing the amount that they can get out of their resources (time, money, effort, etc.). This makes sense: individuals should seek to get the most and give up the least.
As economic actors, individuals face opportunity costs as well.  You could have chosen to spend your money on books or rent or a spring break trip; whichever one of those options is most valuable to you (beside purchasing a new computer) is the opportunity cost.
Such logic applies for every economic decision: purchasing one good means that an individual has chosen to spend resources one way instead of another. Opportunity costs are an important consideration for economists and business people, but are faced by individuals even when they are not making classically economic decisions.
Individuals Make Decisions at the Margins
Individuals will choose the option that yields the greatest net marginal benefit.
Learning objectives
Apply the concepts of marginal analysis and utility to decision-making
When individuals make decisions, they do so by looking at the additional cost and benefit of the decision. The cost or benefit of the single decision is called the marginal cost or the marginal benefit. This is different from the total or average: net marginal benefit (marginal benefit minus marginal cost) is the amount that total benefit will change due to the single decision. For example, if the cost of making 9 pieces of pizza is $90 and the cost of making 10 pieces is $110, the marginal cost of producing the tenth piece of pizza is $20. In theory, individuals will only choose an option if marginal benefit exceeds marginal cost.
Marginal and Total Utility : Marginal utility is the amount that a certain action will change total utility. Individuals use net marginal utility to make decisions.
Let’s take an example. Suppose you are buying a car and have three choices:
- Car A, which costs $10,000
- Car B, which costs $12,000
- Car C, which costs $15,000
The prices represent the marginal costs of each car; purchasing the car will add the cost of the car to your total costs. Also suppose Car A provides you $15,000 worth of utility, Car B provides $15,000, and Car C provides $25,000. Those utilities, in dollar terms, are the marginal benefit of each car.
In order to make the decision, you look at the marginal cost and marginal benefit of each car. By subtracting the cost from the benefit, Car A offers $5,000 of marginal benefit, Car B offers $3,000, and Car C offers $10,000. Obviously, Car C is the best choice because, at the margins, it offers the most benefit to you.
Note that you are concerned not with your total or average cost and benefit (assuming no resource or other external restrictions), but with the marginal cost and benefit. As a decision maker, you want to know how much the decision will change your current state, so you look at the margins, not the overall picture. That is not to say that things like the total cost are unimportant, but that, assuming there are enough resources, individuals will look at the marginal change each option will provide to his/her life or to the firm and chose the one with the greatest net marginal benefit.
Marginal Benefits and Costs for Pollution
The tools of marginal analysis can illustrate the marginal costs and the marginal benefits of reducing pollution. When the quantity of environmental protection is low (quantity QaQa) and pollution is extensive, there are cheap and easy ways to reduce pollution, and the marginal benefits of doing so are quite high. At QaQa, it makes sense to allocate more resources to fight pollution.
However, as environmental protection increases, the cheap and easy ways of reducing pollution decrease, and pollution can only be reduced with costly methods. In other words, the largest marginal benefits are achieved first, followed by decreasing marginal benefits. As the quantity of environmental protection increases to QbQb, the gap between marginal benefits and marginal costs decreases. At point QcQc , the marginal costs will exceed the marginal benefits. At this level of environmental protection, society is not allocating resources efficiently, because too many resources are being given up to reduce pollution.
Marginal Costs and Marginal Benefits of Environmental Protection : Reducing pollution is costly—resources must be sacrificed. The marginal costs of reducing pollution are generally increasing, because the least expensive and easiest reductions can be made first, leaving the more expensive methods for later. The marginal benefits of reducing pollution are generally declining, because the steps that provide the greatest benefit can be taken first, and steps that provide less benefit can wait until later.
Individuals Respond to Incentives
Incentives are ways to encourage or discourage certain behaviors or choices.
Learning objectives
Predict how pay incentives will influence a person’s work performance
An incentive is something that motivates an individual to perform an action. The study of incentive structures is central to the study of all economic activities (both in terms of individual decision-making and in terms of cooperation and competition within a larger institutional structure).
Perhaps the most notable incentive in economics is price. Price acts as a signal to suppliers to produce and to consumers to buy. For example, a sale is nothing more than a store providing an incentive to potential customers to buy. The lowering of the price makes the purchase a better idea for some customers; the sale seeks to persuade individuals to change their actions (namely, to buy the product).
Sales are Incentives : Sales are incentives for consumers to buy, because firms know consumers generally respond to lower prices by purchasing more.
Similarly, the increase in price acts as an incentive to suppliers to produce more of a good. If suppliers think they can sell their products for more, they will be inclined to produce more. The price acts, therefore, as an incentive to customers to buy and suppliers to produce.
Types of Incentives
Incentives come in many other forms, however. Broadly, most incentives can be grouped into one of four categories:
- Remunerative incentives: The incentive comes in the form of some sort of material reward – especially money – in exchange for acting in a particular way. Wages, prices, and bribery are all examples of remunerative incentives. This is the type of incentive that is typically associated with economics.
- Moral incentives: This occurs when a certain choice is widely regarded as the right thing to do, or as particularly admirable, or where the failure to act in a certain way is condemned as indecent. Societies and cultures are two main sources of moral incentives.
- Coercive incentives: The incentive is a promise of some sort of punishment if the wrong decision is made. For example, the promise of imprisonment is a coercive incentive for people to not steal.
- Natural Incentives: Things such as curiosity, mental or physical exercise, admiration, fear, anger, pain, joy, the pursuit of truth, and a sense of control of people or oneself can cause individuals to make certain decisions.
Economics is mainly concerned with remunerative incentives, though when discussing government regulations, coercive incentives often come into play. By manipulating incentives, individuals (as well as businesses and governments) hope to encourage some behaviors and discourage others.
Incentives and Performance
Companies leverage incentives-based strategies to drive performance and optimize employee decision-making and behaviors through meaningful reward systems. While there are both advantages and drawbacks to this type of approach, remunerative (financial) incentives are highly attractive options for employers in a variety of industries and businesses. Providing incentives such as variable income, where an individual can obtain more personal rewards for successfully creating a product or making a sale, often drives up production for highly motivated employees.
An example of this would be a manufacturing facility making widgets. The floor manager shifts the wage system from an hourly wage perspective to a straight piece rate system. The more widgets a worker creates, the higher his or her prospective income will be. Under this incentive system less productive workers may stay the same, but highly productive workers will respond by increasing their production.
Key Points
- Scarce resources diminish as they are used and almost all resources are scarce.
- In order to use a scarce resource, you are inherently using the resource for one purpose and not an alternative.
- The cost of using a resource is called the opportunity cost: the value of the next best alternative that you could be using the resource for instead.
- The opportunity cost is the value of the next best alternative foregone.
- Every decision necessarily means giving up other options, which all have a value.
- The opportunity cost is the value one could have derived from using the same resources another way, though this is not always easily quantifiable.
- The marginal cost or benefit is the amount that a decision will change the total cost or benefit from where it is currently.
- Individuals will make choice that maximizes the net marginal benefit (marginal benefit – marginal cost).
- While total or average cost and benefit are important, provided enough resources, individuals will look only at the net marginal benefit.
- Price is one of the main incentives studied in economics. Price incentivizes producers to supply a certain amount, and consumers to purchase a certain amount.
- Economics is mainly concerned with studying remunerative incentives (those that concern material reward).
- Individuals, firms, and governments all change incentives in hopes of encouraging desired outcomes.
Key Terms
- Scarce : Insufficient to meet demand.
- Opportunity cost : The value of the best alternative forgone.
- Opportunity Costs : The value of the best alternative forgone, in a situation in which a choice needs to be made between several mutually exclusive alternatives given limited resources.
- marginal benefit : The additional benefit from taking a course of action.
- marginal cost : The additional cost from taking a course of action.
- incentive : Something that motivates an individual to perform an action.
- Incentive Structure : The cumulative set of promised rewards and/or punishments that encourage actors to make a set of decisions.
LICENSES AND ATTRIBUTIONS
CC LICENSED CONTENT, SPECIFIC ATTRIBUTION
- Opportunity cost. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Opportunity_cost . License : CC BY-SA: Attribution-ShareAlike
- Steven Hinson, The Invisible Hand and Allocative Efficiency. October 18, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m42970/latest/ . License : CC BY: Attribution
- Microeconomics/Goods and Scarcity. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Microec...s_and_Scarcity . License : CC BY-SA: Attribution-ShareAlike
- Scarce. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/Scarce . License : CC BY-SA: Attribution-ShareAlike
- Opportunity cost. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/Opportunity+cost . License : CC BY-SA: Attribution-ShareAlike
- Steven Hinson, The Invisible Hand and Allocative Efficiency. October 18, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m42970/latest/ . License : CC BY: Attribution
- Principles of Economics/Opportunity Costs. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Princip...ortunity_Costs . License : CC BY-SA: Attribution-ShareAlike
- Microeconomics/Opportunity Cost. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Microec...portunity_Cost . License : CC BY-SA: Attribution-ShareAlike
- Amy Ando, Evaluating Projects and Policies. October 31, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m38611/latest/ . License : CC BY: Attribution
- Opportunity cost. Provided by : WIKIPEDIA. Located at : en.Wikipedia.org/wiki/Opportunity_cost . License : CC BY-SA: Attribution-ShareAlike
- Opportunity Costs. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Opportunity%20Costs . License : CC BY-SA: Attribution-ShareAlike
- Steven Hinson, The Invisible Hand and Allocative Efficiency. October 18, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m42970/latest/ . License : CC BY: Attribution
- Caneel Bay Turtle Bay Beach 4. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...ay_Beach_4.jpg . License : CC BY-SA: Attribution-ShareAlike
- The Benefits and Costs of U.S. Environmental Laws. Provided by : Principles of Economics by Phil Schatz. Located at : http://philschatz.com/economics-book...ts/m48672.html . License : CC BY: Attribution
- Marginal analysis - Marginal Benefit including marginal revenue and Marginal cost. Provided by : MicroEcon201. Located at : microecon201.wikispaces.com/M...Marginal++cost . License : CC BY-SA: Attribution-ShareAlike
- Macroeconomics/Economics. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Macroec...ginal_analysis . License : CC BY-SA: Attribution-ShareAlike
- Principles of Economics/Marginal Utility. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Princip...rginal_Utility . License : CC BY-SA: Attribution-ShareAlike
- marginal benefit. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/marginal_benefit . License : CC BY-SA: Attribution-ShareAlike
- marginal cost. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/marginal_cost . License : CC BY-SA: Attribution-ShareAlike
- Steven Hinson, The Invisible Hand and Allocative Efficiency. October 18, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m42970/latest/ . License : CC BY: Attribution
- Caneel Bay Turtle Bay Beach 4. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...ay_Beach_4.jpg . License : CC BY-SA: Attribution-ShareAlike
- UtilityQuantified. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...Quantified.svg . License : CC BY-SA: Attribution-ShareAlike
- Marginal Costs and Marginal Benefits of Environmental Protection. Provided by : Principles of Economics. Located at : http://philschatz.com/economics-book...ts/m48672.html . License : CC BY-SA: Attribution-ShareAlike
- Managing Groups and Teams/Which attributes are fundamental to team cohesion?. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Managin...n?%23Incentive . License : CC BY-SA: Attribution-ShareAlike
- Incentive. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Incentive . License : CC BY-SA: Attribution-ShareAlike
- Amy Ando, Solutions: Property Rights, Regulations, and Incentive Policies. November 1, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m38956/latest/ . License : CC BY: Attribution
- incentive. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/incentive . License : CC BY-SA: Attribution-ShareAlike
- Boundless. Provided by : Boundless Learning. Located at : www.boundless.com//economics/...tive-structure . License : CC BY-SA: Attribution-ShareAlike
- Steven Hinson, The Invisible Hand and Allocative Efficiency. October 18, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m42970/latest/ . License : CC BY: Attribution
- Caneel Bay Turtle Bay Beach 4. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...ay_Beach_4.jpg . License : CC BY-SA: Attribution-ShareAlike
- UtilityQuantified. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/Fi...Quantified.svg . License : CC BY-SA: Attribution-ShareAlike
- Marginal Costs and Marginal Benefits of Environmental Protection. Provided by : Principles of Economics. Located at : http://philschatz.com/economics-book...ts/m48672.html . License : CC BY-SA: Attribution-ShareAlike
- Sale sign. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:Sale_sign.jpg . License : CC BY-SA: Attribution-ShareAlike


5.2: Sketching Figures with Non-isometric Lines
Figure \(\PageIndex{1}\) shows an object that is basically rectangular but has one face machined at an angle. You can easily construct an isometric sketch of the basic rectangular block. To show the machined face, it is necessary to plot the appropriate points of intersection and join those points to produce the correct angle.
Sketch a light outline of the basic rectangular block using the size measurements given in Figure \(\PageIndex{1}\). Mark the number of units indicated along the length (L) and the depth (D), as shown in Figure \(\PageIndex{2}\).
Lightly sketch lines parallel to the original block outlines from the marked points on the front and side surfaces, as shown in Figure \(\PageIndex{3}\).
Join the two points on the front face and the two ends of the lines you have just sketched across the object (Figure \(\PageIndex{4}\)). Once you are sure your sketch is correct, erase the light lines that originally outlined the block and darken the outline of the completed block as shown in Figure \(\PageIndex{5}\).