15.25: Assignment- Incorporating and Documenting Sources
STEP 1 : Using the passage below, construct a paraphrase of a portion of the text , naming the source and incorporating the paraphrase within your own logic. Follow the “quote sandwich” method explained in this section: begin with an introduction, then a signal phrase (like “according to…” or “[the author] states that…” or “In the passage from . . .”), then the paraphrased sentences, and an explanation of the paraphrase in your own words.
Passage for Step 1
Here is a passage from a world history textbook. Use this for Step 1.
Like so many things desired by Europeans and supplied by Asians—at first luxury items for the elite such as silk or porcelain, but increasingly products like tea from China for the mass market—cotton textiles were produced well and cheaply in India. The British textile manufacturers focused on the “cheap” part and complained that with relatively higher wages, British manufacturers could not compete. India had a competitive advantage in the eighteenth century, being able to undersell in the world market virtually any other producer of textiles. Some thought the reason for cheap Indian textiles was because of a low living standard, or a large population earning depressed wages, but all of those have been shown to not be true: Indian textile workers in the eighteenth century had just as high a standard of living as British workers. So, if it was not a low standard of living that gave India its competitive advance, what did?In a word: agriculture. Indian agriculture was so productive that the amount of food produced, and hence its cost, was significantly lower than in Europe. In the preindustrial age, when working families spent 60-80 percent of their earnings on food, the cost of food was the primary determinant of their real wages (i.e. how much a pound, dollar, a real, or a pagoda could buy). In India (and China and Japan as well), the amount of grain harvested from a given amount of seed was in the ration of 20:1 (e.g., twenty bushels of rice harvested for every one planted), whereas in England it was at best 8:1. Asian agriculture thus was more than twice as efficient as British (and by extension European) agriculture, and food—the major component in the cost of living—cost less in Asia.
STEP 2 : Next, repeat the process in step one, but this time construct a successful paraphrase sandwich of a passage from an article you are using in your research . If you are not currently working on a research project, then choose any topic that interests you and find an article in your library database that you can use for this activity.
This time, include correct MLA or APA in-text citations (also known as parenthetical citations) to the end of the paraphrased passage.
Create a MLA Works Cited or APA References page that includes an entry for this source.
STEP 3 : Next, repeat the process in above, but this time construct a successful summary sandwich of a passage from an article you are using in your research . If you are not currently working on a research project, then choose any topic that interests you and find an article in your library database that you can use for this activity.
Include correct MLA or APA in-text citations (also known as parenthetical citations) to the end of the summarized passage.
Create a MLA Works Cited or APA References page that includes an entry for this source.
| Criteria | Proficient | Developing | Not Evident | Points |
|---|---|---|---|---|
| Paraphrased passage | Uses the quote sandwich to paraphrase some of the sample text. The sandwich includes an introduction, a signal phrase, paraphrased sentences, and an explanation of the paraphrase in your own words. | Paraphrases the passage but does not fully introduce or explain the paraphrase within a quote sandwich. | Incomplete paraphrase | __/8 |
| Paraphrased passage from research | Uses the quote sandwich to paraphrase a portion from another text. The sandwich includes an introduction, a signal phrase, paraphrased sentences, and an explanation of the paraphrase in your own words. This includes an in-text and end-of-text citation. | Paraphrases the passage but does not fully introduce or explain the paraphrase within a quote sandwich. Citation may not be formatted correctly or is missing. | Incomplete paraphrase | __/8 |
| Summarized passage from research | Uses the quote sandwich to summarize some of the sample text. The sandwich includes an introduction, a signal phrase, paraphrased sentences, and an explanation of the paraphrase in your own words. This includes an in-text and end-of-text citation. | Summarizes the passage but does not fully introduce or explain the paraphrase within a quote sandwich. Citation may not be formatted correctly or is missing. | Incomplete summary | __/4 |
| Total: | __/20 |
Contributors and Attributions
- Assignment: MLA Documentation. Provided by : Lumen Learning. License : CC BY: Attribution
- Activity from Listening to Sources, Talking to Sources. Authored by : Amy Guptill. Located at : pressbooks.opensuny.org/writing-in-college-from-competence-to-excellence/chapter/5/. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
- Quote Sandwich Activity. Authored by : Alexis McMillan-Clifton. Provided by : Tacoma Community College. License : CC BY: Attribution


3.1: Maximization Applications
-
- Last updated
- Save as PDF
Learning Objectives
In this section, you will learn to:
- Recognize the typical form of a linear programing problem
- Formulate maximization linear programming problems
- Graph feasibility regions for maximization linear programming problems
- Determine optimal solutions for maximization linear programming problems.
Application problems in business, economics, and social and life sciences often ask us to make decisions on the basis of certain conditions. The conditions or constraints often take the form of inequalities. In this section, we will begin to formulate, analyze, and solve such problems, at a simple level, to understand the many components of such a problem.
A typical linear programming problem consists of finding an extreme value of a linear function subject to certain constraints. We are either trying to maximize or minimize the value of this linear function, such as to maximize profit or revenue, or to minimize cost. That is why these linear programming problems are classified as maximization or minimization problems , or just optimization problems . The function we are trying to optimize is called an objective function , and the conditions that must be satisfied are called constraints .
A typical example is to maximize profit from producing several products, subject to limitations on materials or resources needed for producing these items; the problem requires us to determine the amount of each item produced. Another type of problem involves scheduling; we need to determine how much time to devote to each of several activities in order to maximize income from (or minimize cost of) these activities, subject to limitations on time and other resources available for each activity.
In this chapter, we will work with problems that involve only two variables, and therefore, can be solved by graphing. In the next chapter, we’ll learn an algorithm to find a solution numerically. That will provide us with a tool to solve problems with more than two variables. At that time, with a little more knowledge about linear programming, we’ll also explore the many ways these techniques are used in business and wide variety of other fields.
We begin by solving a maximization problem.
Example \(\PageIndex{1}\)
Niki holds two part-time jobs, Job I and Job II. She never wants to work more than a total of 12 hours a week. She has determined that for every hour she works at Job I, she needs 2 hours of preparation time, and for every hour she works at Job II, she needs one hour of preparation time, and she cannot spend more than 16 hours for preparation.
If Nikki makes $40 an hour at Job I, and $30 an hour at Job II, how many hours should she work per week at each job to maximize her income?
Solution
We start by choosing our variables.
- Let \(x\) = The number of hours per week Niki will work at Job I.
- Let \(y\) = The number of hours per week Niki will work at Job II.
Now we write the objective function. Since Niki gets paid $40 an hour at Job I, and $30 an hour at Job II, her total income I is given by the following equation.
\[I = 40x + 30y \nonumber \]
Our next task is to find the constraints. The second sentence in the problem states, "She never wants to work more than a total of 12 hours a week." This translates into the following constraint:
\[x + y \leq 12 \nonumber \]
The third sentence states, "For every hour she works at Job I, she needs 2 hours of preparation time, and for every hour she works at Job II, she needs one hour of preparation time, and she cannot spend more than 16 hours for preparation." The translation follows.
\[2x + y \leq 16 \nonumber \]
The fact that \(x\) and \(y\) can never be negative is represented by the following two constraints:
\[x \geq 0 \text{, and } y \geq 0 \nonumber. \nonumber \]
Well, good news! We have formulated the problem. We restate it as
\begin{array}{ll}
\textbf { Maximize } & \mathrm{I}=40 \mathrm{x}+30 \mathrm{y} \\
\textbf { Subject to: } & \mathrm{x}+\mathrm{y} \leq 12 \\
& 2 \mathrm{x}+\mathrm{y} \leq 16 \\
& \mathrm{x} \geq 0 ; \mathrm{y} \geq 0
\end{array}
In order to solve the problem, we graph the constraints and shade the region that satisfies all the inequality constraints.
Any appropriate method can be used to graph the lines for the constraints. However often the easiest method is to graph the line by plotting the x-intercept and y-intercept.
The line for a constraint will divide the plane into two region, one of which satisfies the inequality part of the constraint. A test point is used to determine which portion of the plane to shade to satisfy the inequality. Any point on the plane that is not on the line can be used as a test point.
- If the test point satisfies the inequality, then the region of the plane that satisfies the inequality is the region that contains the test point.
- If the test point does not satisfy the inequality, then the region that satisfies the inequality lies on the opposite side of the line from the test point.
In the graph below, after the lines representing the constraints were graphed using an appropriate method from Chapter 1, the point (0,0) was used as a test point to determine that
- (0,0) satisfies the constraint \(x + y \leq 12\) because \(0 + 0 < 12\)
- (0,0) satisfies the constraint \(2x + y \leq 16\) because \(2(0) + 0 < 16\)
Therefore, in this example, we shade the region that is below and to the left of both constraint lines, but also above the x axis and to the right of the y axis, in order to further satisfy the constraints \(x \geq 0\) and \(y \geq 0\).
The shaded region where all conditions are satisfied is called the feasibility region or the feasibility polygon.
The Fundamental Theorem of Linear Programming states that the maximum (or minimum) value of the objective function always takes place at the vertices of the feasibility region.
Therefore, we will identify all the vertices (corner points) of the feasibility region. We call these points critical points. They are listed as (0, 0), (0, 12), (4, 8), (8, 0). To maximize Niki's income, we will substitute these points in the objective function to see which point gives us the highest income per week. We list the results below.
| Critical Points | Income |
|---|---|
| (0, 0) | 40(0) + 30(0) = $0 |
| (0, 12) | 40(0) + 30(12) = $360 |
| (4, 8) | 40(4) + 30(8) = $400 |
| (8, 0) | 40(8) + 30(0) = $320 |
Clearly, the point (4, 8) gives the most profit: $400.
Therefore, we conclude that Niki should work 4 hours at Job I, and 8 hours at Job II.
Example \(\PageIndex{2}\)
A factory manufactures two types of gadgets, regular and premium. Each gadget requires the use of two operations, assembly and finishing, and there are at most 12 hours available for each operation. A regular gadget requires 1 hour of assembly and 2 hours of finishing, while a premium gadget needs 2 hours of assembly and 1 hour of finishing. Due to other restrictions, the company can make at most 7 gadgets a day. If a profit of $20 is realized for each regular gadget and $30 for a premium gadget, how many of each should be manufactured to maximize profit?
Solution
We choose our variables.
- Let \(x\) = The number of regular gadgets manufactured each day.
- and \(y\) = The number of premium gadgets manufactured each day.
The objective function is
\[P = 20x + 30y \nonumber \]
We now write the constraints. The fourth sentence states that the company can make at most 7 gadgets a day. This translates as
\[x + y \leq 7 \nonumber \]
Since the regular gadget requires one hour of assembly and the premium gadget requires two hours of assembly, and there are at most 12 hours available for this operation, we get
\[x + 2y \leq 12 \nonumber \]
Similarly, the regular gadget requires two hours of finishing and the premium gadget one hour. Again, there are at most 12 hours available for finishing. This gives us the following constraint.
\[2x + y \leq 12 \nonumber \]
The fact that \(x\) and \(y\) can never be negative is represented by the following two constraints:
\[x \geq 0 \text{, and } y \geq 0 \nonumber. \nonumber \]
We have formulated the problem as follows:
\[\begin{array}{ll}
\textbf { Maximize } & \mathrm{P}=20 \mathrm{x}+30 \mathrm{y} \\
\textbf { Subject to: } & \mathrm{x}+\mathrm{y} \leq 7 \\
& \mathrm{x}+2\mathrm{y} \leq 12 \\
& 2\mathrm{x} +\mathrm{y} \leq 12 \\
& \mathrm{x} \geq 0 ; \mathrm{y} \geq 0
\end{array} \nonumber \]
In order to solve the problem, we next graph the constraints and feasibility region.
Again, we have shaded the feasibility region, where all constraints are satisfied.
Since the extreme value of the objective function always takes place at the vertices of the feasibility region, we identify all the critical points. They are listed as (0, 0), (0, 6), (2, 5), (5, 2), and (6, 0). To maximize profit, we will substitute these points in the objective function to see which point gives us the maximum profit each day. The results are listed below.
| Critical Point | Income |
|---|---|
| (0, 0) | 20(0) + 30(0) = $0 |
|
(0, 6) |
20(0) + 30(6) = $180 |
|
(2, 5) |
20(2) + 30(5) = $190 |
|
(5, 2) |
20(5) + 30(2) = $160 |
| (6,0) |
20(6) + 30(0) = $120 |
The point (2, 5) gives the most profit, and that profit is $190.
Therefore, we conclude that we should manufacture 2 regular gadgets and 5 premium gadgets daily to obtain the maximum profit of $190.
So far we have focused on “ standard maximization problems ” in which
- The objective function is to be maximized
- All constraints are of the form \(ax + by \leq c\)
- All variables are constrained to by non-negative (\(x ≥ 0\), \(y ≥ 0\))
We will next consider an example where that is not the case. Our next problem is said to have “ mixed constraints ”, since some of the inequality constraints are of the form \(ax + by ≤ c\) and some are of the form \(ax + by ≥ c\). The non-negativity constraints are still an important requirement in any linear program.
Example \(\PageIndex{3}\)
Solve the following maximization problem graphically.
\[\begin{array}{ll}
\textbf { Maximize } & \mathrm{P}=10 \mathrm{x}+15 \mathrm{y} \\
\textbf { Subject to: } & \mathrm{x}+\mathrm{y} \geq 1 \\
& \mathrm{x}+2\mathrm{y} \leq 6 \\
& 2\mathrm{x} +\mathrm{y} \leq 6 \\
& \mathrm{x} \geq 0 ; \mathrm{y} \geq 0
\end{array} \nonumber \]
Solution
The graph is shown below.
The five critical points are listed in the above figure. The reader should observe that the first constraint \(x + y ≥ 1\) requires that the feasibility region must be bounded below by the line \(x + y =1\); the test point (0,0) does not satisfy \(x + y ≥ 1\), so we shade the region on the opposite side of the line from test point (0,0).
| Critical point | Income |
|---|---|
| (1, 0) | 10(1) + 15(0) = $10 |
| (3, 0) | 10(3) + 15(0) = $30 |
| (2, 2) | 10(2) + 15(2) = $50 |
| (0, 3) | 10(0) + 15(3) = $45 |
| (0,1) | 10(0) + 15(1) = $15 |
Clearly, the point (2, 2) maximizes the objective function to a maximum value of 50.
It is important to observe that that if the point (0,0) lies on the line for a constraint, then (0,0) could not be used as a test point. We would need to select any other point we want that does not lie on the line to use as a test point in that situation.
Finally, we address an important question. Is it possible to determine the point that gives the maximum value without calculating the value at each critical point?
The answer is yes.
For example \(\PageIndex{2}\), we substituted the points (0, 0), (0, 6), (2, 5), (5, 2), and (6, 0), in the objective function \(P = 20x + 30y\), and we got the values $0, $180, $190, $160, $120, respectively.
Sometimes that is not the most efficient way of finding the optimum solution. Instead we could find the optimal value by also graphing the objective function.
To determine the largest \(P\), we graph \(P = 20x + 30y\) for any value \(P\) of our choice. Let us say, we choose \(P = 60\). We graph \(20x + 30y = 60\).
Now we move the line parallel to itself, that is, keeping the same slope at all times. Since we are moving the line parallel to itself, the slope is kept the same, and the only thing that is changing is the P. As we move away from the origin, the value of P increases. The largest possible value of P is realized when the line touches the last corner point of the feasibility region.
The figure below shows the movements of the line, and the optimum solution is achieved at the point (2, 5). In maximization problems, as the line is being moved away from the origin, this optimum point is the farthest critical point.
We summarize:
The Maximization Linear Programming Problems
- Write the objective function.
-
Write the constraints.
- For the standard maximization linear programming problems, constraints are of the form: \(ax + by ≤ c\)
- Since the variables are non-negative, we include the constraints: \(x ≥ 0\); \(y ≥ 0\).
- Graph the constraints.
- Shade the feasibility region.
- Find the corner points.
-
Determine the corner point that gives the maximum value.
- This is done by finding the value of the objective function at each corner point.
- This can also be done by moving the line associated with the objective function.


3.4: Introduction to Challenges of Multichannel Retailers
What you’ll learn to do: Recognize the challenges faced by multichannel retailers and how retailers must adapt
Multi-channel retailers work to engage shoppers at the point of purchase in a variety of channels, in order to be present where and when these potential customers are ready to buy. The complexity of doing this effectively continues to increase as outlets, especially those online, multiply: branded websites, e-commerce sites, apps, social media and marketplaces. Thus, a multi-channel marketing strategy requires organizational discipline and infrastructure to ensure that the customer experience is consistent across channels.
Contributors and Attributions
CC licensed content, Original
- Introduction to Challenges of Multichannel Retailers. Authored by : Patrick Williams. Provided by : Lumen Learning. License : CC BY: Attribution


21.4.4: Carbon Dating and Estimating Fossil Age
-
- Last updated
- Save as PDF
- Boundless
- Boundless
Learning Objectives
- Summarize the available methods for dating fossils
Determining Fossil Ages
Paleontology seeks to map out how life evolved across geologic time. A substantial hurdle is the difficulty of working out fossil ages. There are several different methods for estimating the ages of fossils, including:
- stratigraphy
- biostratigraphy
- carbon dating
Stratigraphy
Paleontologists rely on stratigraphy to date fossils. Stratigraphy is the science of understanding the strata, or layers, that form the sedimentary record. Strata are differentiated from each other by their different colors or compositions and are exposed in cliffs, quarries, and river banks. These rocks normally form relatively horizontal, parallel layers, with younger layers forming on top.
If a fossil is found between two layers of rock whose ages are known, the fossil’s age is thought to be between those two known ages. Because rock sequences are not continuous, but may be broken up by faults or periods of erosion, it is difficult to match up rock beds that are not directly adjacent.
Biostratigraphy
Fossils of species that survived for a relatively short time can be used to match isolated rocks: this technique is called biostratigraphy. For instance, the extinct chordate Eoplacognathus pseudoplanus is thought to have existed during a short range in the Middle Ordovician period. If rocks of unknown age have traces of E. pseudoplanus , they have a mid-Ordovician age. Such index fossils must be distinctive, globally distributed, and occupy a short time range to be useful. Misleading results can occur if the index fossils are incorrectly dated.
Relative Dating
Stratigraphy and biostratigraphy can in general provide only relative dating (A was before B), which is often sufficient for studying evolution. This is difficult for some time periods, however, because of the barriers involved in matching rocks of the same age across continents. Family-tree relationships can help to narrow down the date when lineages first appeared. For example, if fossils of B date to X million years ago and the calculated “family tree” says A was an ancestor of B, then A must have evolved earlier.
It is also possible to estimate how long ago two living branches of a family tree diverged by assuming that DNA mutations accumulate at a constant rate. However, these “molecular clocks” are sometimes inaccurate and provide only approximate timing. For example, they are not sufficiently precise and reliable for estimating when the groups that feature in the Cambrian explosion first evolved, and estimates produced by different approaches to this method may vary as well.
Carbon Dating
Together with stratigraphic principles, radiometric dating methods are used in geochronology to establish the geological time scale. Beds that preserve fossils typically lack the radioactive elements needed for radiometric dating (” radiocarbon dating ” or simply “carbon dating”). The principle of radiocarbon dating is simple: the rates at which various radioactive elements decay are known, and the ratio of the radioactive element to its decay products shows how long the radioactive element has existed in the rock. This rate is represented by the half-life, which is the time it takes for half of a sample to decay.
The half-life of carbon-14 is 5,730 years, so carbon dating is only relevant for dating fossils less than 60,000 years old. Radioactive elements are common only in rocks with a volcanic origin, so the only fossil-bearing rocks that can be dated radiometrically are volcanic ash layers. Carbon dating uses the decay of carbon-14 to estimate the age of organic materials, such as wood and leather.
Key Points
- Determining the ages of fossils is an important step in mapping out how life evolved across geologic time.
- The study of stratigraphy enables scientists to determine the age of a fossil if they know the age of layers of rock that surround it.
- Biostratigraphy enables scientists to match rocks with particular fossils to other rocks with those fossils to determine age.
- Paleontology seeks to map out how life evolved across geologic time. A substantial hurdle is the difficulty of working out fossil ages.
- Scientists use carbon dating when determining the age of fossils that are less than 60,000 years old, and that are composed of organic materials such as wood or leather.
Key Terms
- half-life : The time required for half of the nuclei in a sample of a specific isotope to undergo radioactive decay.
- stratigraphy : The study of rock layers and the layering process.
- radiocarbon dating : A method of estimating the age of an artifact or biological vestige based on the relative amounts of various isotopes of carbon present in a sample.


1.E: Vectors in Euclidian Space (Exercises)
1.1: Introduction
A
1.1.1. Calculate the magnitudes of the following vectors:\(\text{(a) } \textbf{v} = (2,-1) \quad \text{(b) } \textbf{v} = (2,-1,0) \quad \text{(c) } \textbf{v} = (3,2,-2) \quad \text{(d) }\textbf{v} = (0,0,1) \quad \text{ (e) } \textbf{v} = (6,4,-4)\)
1.1.2. For the points \(P =(1,-1,1)\), \(Q=(2,-2,2)\), \(R=(2,0,1)\), \(S=(3,-1,2)\), does \(\overrightarrow{PQ} = \overrightarrow{RS}\)?
1.1.3 For the points \(P =(0,0,0)\), \(Q=(1,3,2)\), \(R=(1,0,1)\), \(S=(2,3,4)\), does \(\overrightarrow{PQ} = \overrightarrow{RS}\)?
B
1.1.4 Let \(\textbf{v} = (1,0,0)\) and \(\textbf{w} = (a,0,0)\) be vectors in \(\mathbb{R}^{3}\). Show that \(\norm{\textbf{w}} = |a| \,\norm{\textbf{v}}\).
1.1.5 Let \(\textbf{v} = (a,b,c)\) and \(\textbf{w} = (3a,3b,3c)\) be vectors in \(\mathbb{R}^{3}\). Show that \(\norm{\textbf{w}} = 3 \,\norm{\textbf{v}}\).
C
Figure 1.1.9
1.1.6
Though we will see a simple proof of Theorem 1.1 in the next section, it is possible to prove it using methods similar to those in the proof of Theorem 1.2. Prove the special case of Theorem 1.1 where the points \(P = (x_{1}, y_{1}, z_{1})\) and \(Q = x_{2}, y_{2}, z_{2})\) satisfy the following conditions:
\(x_{2} > x_{1} > 0\), \(y_{2} > y_{1} > 0\), and \(z_{2} > 1 > 0\).
\(\textit{Hint: Think of Case 4 in the proof of Theorem (1.2), and consider Figure 1.1.9.}\)
1.2: Vector Algebra
A
1.2.1 Let \(\textbf{v} = (-1,5,-2)\) and \(\textbf{w} = (3,1,1)\).
- Find \(\textbf{v} - \textbf{w}\).
- Find \(\textbf{v} + \textbf{w}\).
- Find \(\frac{\textbf{v}}{\norm{\textbf{v}}}\).
- Find \(\norm{\frac{1}{2}(\textbf{v} - \textbf{w})}\).
- Find \(\norm{\frac{1}{2}(\textbf{v} + \textbf{w})}\).
- Find \(-2\,\textbf{v} + 4\,\textbf{w}\).
- Find \(\textbf{v} - 2\,\textbf{w}\).
- Find the vector \(\textbf{u}\) such that \(\textbf{u} + \textbf{v} + \textbf{w} = \textbf{i}\).
- Find the vector \(\textbf{u}\) such that \(\textbf{u} + \textbf{v} + \textbf{w} = 2\,\textbf{j} + \textbf{k}\).
- Is there a scalar \(m\) such that \(m(\textbf{v} + 2\,\textbf{w}) = \textbf{k}\)? If so, find it.
1.2.2. For the vectors \(\textbf{v}\) and \(\textbf{w}\) from Exercise 1, is \(\norm{\textbf{v} - \textbf{w}} = \norm{\textbf{v}} - \norm{\textbf{w}}\)? If not, which quantity is larger?
1.2.3.
For the vectors \(\textbf{v}\) and \(\textbf{w}\) from Exercise 1, is \(\norm{\textbf{v} + \textbf{w}} = \norm{\textbf{v}} + \norm{\textbf{w}}\)? If not, which quantity is larger?
B
1.2.4.
Prove Theorem 1.5 (f) for \(\mathbb{R}^{3}\).
1.2.5. Prove Theorem 1.5 (g) for \(\mathbb{R}^{3}\).
C
1.2.6. We know that every vector in \(\mathbb{R}^{3}\) can be written as a scalar combination of the vectors \(\textbf{i}\), \(\textbf{j}\), and \(\textbf{k}\). Can every vector in \(\mathbb{R}^{3}\) be written as a scalar combination of just \(\textbf{i}\) and \(\textbf{j}\), i.e. for any vector \(\textbf{v}\) in \(\mathbb{R}^{3}\), are there scalars \(m\), \(n\) such that \(\textbf{v} = m\,\textbf{i} + n\,\textbf{j}\)? Justify your answer.
1.3: Dot Product
A
1.3.1. Let \(\textbf{v} = (5,1,-2)\) and \(\textbf{w} = (4,-4,3)\). Calculate \(\textbf{v} \cdot \textbf{w}\).
1.3.2. Let \(\textbf{v} = -3\,\textbf{i} - 2\,\textbf{j} - \textbf{k}\) and \(\textbf{w} = 6\,\textbf{i} + 4\,\textbf{j} + 2\,\textbf{k}\). Calculate \(\textbf{v} \cdot \textbf{w}\).
For Exercises 3-8, find the angle \(\theta\) between the vectors \(\textbf{v}\) and \(\textbf{w}\).
1.3.3. \(\textbf{v} = (5,1,-2)\), \(\textbf{w} = (4,-4,3)\)
1.3.4. \(\textbf{v} = (7,2,-10)\), \(\textbf{w} = (2,6,4)\)
1.3.5. \(\textbf{v} = (2,1,4)\), \(\textbf{w} = (1,-2,0)\)
1.3.6. \(\textbf{v} = (4,2,-1)\), \(\textbf{w} = (8,4,-2)\)
1.3.7. \(\textbf{v} = -\,\textbf{i} + 2\,\textbf{j} + \textbf{k}\), \(\textbf{w} = -3\,\textbf{i} + 6\,\textbf{j} + 3\,\textbf{k}\)
1.3.8. \(\textbf{v} = \textbf{i}\), \(\textbf{w} = 3\,\textbf{i} + 2\,\textbf{j} + 4\textbf{k}\)
1.3.9. Let \(\textbf{v} = (8,4,3)\) and \(\textbf{w} = (-2,1,4)\). Is \(\textbf{v} \perp \textbf{w}\)? Justify your answer.
1.3.10. Let \(\textbf{v} = (6,0,4)\) and \(\textbf{w} = (0,2,-1)\). Is \(\textbf{v} \perp \textbf{w}\)? Justify your answer.
1.3.11. For \(\textbf{v}\), \(\textbf{w}\) from Exercise 5, verify the Cauchy-Schwarz Inequality \(|\textbf{v} \cdot \textbf{w}| \le \norm{\textbf{v}}\,\norm{\textbf{w}}\).
1.3.12. For \(\textbf{v}\), \(\textbf{w}\) from Exercise 6, verify the Cauchy-Schwarz Inequality \(|\textbf{v} \cdot \textbf{w}| \le \norm{\textbf{v}}\,\norm{\textbf{w}}\).
1.3.13. For \(\textbf{v}\), \(\textbf{w}\) from Exercise 5, verify the Triangle Inequality \(\norm{\textbf{v} + \textbf{w}} \le \norm{\textbf{v}} + \norm{\textbf{w}}\).
1.3.14. For \(\textbf{v}\), \(\textbf{w}\) from Exercise 6, verify the Triangle Inequality \(\norm{\textbf{v} + \textbf{w}} \le \norm{\textbf{v}} + \norm{\textbf{w}}\).
B
Note: Consider only vectors in \(\mathbb{R}^{3}\) for Exercises 15-25.
1.3.15. Prove Theorem 1.9 (a).
1.3.16. Prove Theorem 1.9 (b).
1.3.17. Prove Theorem 1.9 (c).
1.3.18. Prove Theorem 1.9 (d).
1.3.19. Prove Theorem 1.9 (e).
1.3.20. Prove Theorem 1.10 (a).
1.3.21. Prove or give a counterexample: If \(\textbf{u} \cdot \textbf{v} = \textbf{u} \cdot \textbf{w}\), then \(\textbf{v} =\textbf{w}\).
C
1.3.22. Prove or give a counterexample: If \(\textbf{v} \cdot \textbf{w} = 0\) for all \(\textbf{v}\), then \(\textbf{w} =\textbf{0}\).
1.3.23. Prove or give a counterexample: If \(\textbf{u} \cdot \textbf{v} = \textbf{u} \cdot \textbf{w}\) for all \(\textbf{u}\), then \(\textbf{v} = \textbf{w}\).
1.3.24. Prove that \(|\norm{\textbf{v}} - \norm{\textbf{w}}| \le \norm{\textbf{v} - \textbf{w}}\) for all \(\textbf{v}, \textbf{w}\).
Figure 1.3.5
1.3.25. For nonzero vectors \(\textbf{v}\) and \(\textbf{w}\), the \(\textit{projection}\) of \(\textbf{v}\) onto \(\textbf{w}\) (sometimes written as \(proj_{\textbf{w}}\textbf{v}\)) is the vector \(\textbf{u}\) along the same line \(L\) as \(\textbf{w}\) whose terminal point is obtained by dropping a perpendicular line from the terminal point of \(\textbf{v}\) to \(L\) (see Figure 1.3.5). Show that
\[\norm{\textbf{u}} = \frac{|\textbf{v} \cdot \textbf{w}|}{\norm{\textbf{w}}}.\]
\(\textit{(Hint: Consider the angle between}\) \(\textbf{v}\) \(\textit{and}\) \(\textbf{w}\).\(\textit{)}\)
1.3.26. Let \(\alpha\), \(\beta\), and \(\gamma\) be the angles between a nonzero vector \(\textbf{v}\) in \(\mathbb{R}^{3}\) and the vectors \(\textbf{i}\), \(\textbf{j}\), and \(\textbf{k}\), respectively. Show that \(\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1\).
(Note: \(\alpha\), \(\beta\), \(\gamma\) are often called the \(\textit{direction angles}\) of \(\textbf{v}\), and \(\cos \alpha\), \(\cos \beta\), \(\cos \gamma\) are called the \(\textit{direction cosines}\).)
1.4: Cross Product
A
For Exercises 1-6, calculate \(\textbf{v} \times \textbf{w}\).
1.4.1.
\(\textbf{v} = (5,1,-2)\), \(\textbf{w} = (4,-4,3)\)
1.4.2.
\(\textbf{v} = (7,2,-10)\), \(\textbf{w} = (2,6,4)\)
1.4.3.
\(\textbf{v} = (2,1,4)\), \(\textbf{w} = (1,-2,0)\)
1.4.4.
\(\textbf{v} = (1,3,2)\), \(\textbf{w} = (7,2,-10)\)
1.4.5.
\(\textbf{v} = -\,\textbf{i} + 2\,\textbf{j} + \textbf{k}\), \(\textbf{w} = -3\,\textbf{i} + 6\,\textbf{j} + 3\,\textbf{k}\)
1.4.6.
\(\textbf{v} = \textbf{i}\), \(\textbf{w} = 3\,\textbf{i} + 2\,\textbf{j} + 4\textbf{k}\)
For Exercises 7-8, calculate the area of the triangle \(\triangle PQR\).
1.4.7
. \(P = (5,1,-2)\), \(Q = (4,-4,3)\), \(R =(2,4,0)\)
1.4.8.
\(P = (4,0,2)\), \(Q = (2,1,5)\), \(R =(-1,0,-1)\)
For Exercises 9-10, calculate the area of the parallelogram \(PQRS\).
1.4.9.
\(P = (2,1,3)\), \(Q = (1,4,5)\), \(R = (2,5,3)\), \(S = (3,2,1)\)
1.4.10.
\(P = (-2,-2)\), \(Q = (1,4)\), \(R = (6,6)\), \(S = (3,0)\)
For Exercises 11-12, find the volume of the parallelepiped with adjacent sides \(\textbf{u}, \textbf{v}, \textbf{w}\).
1.4.11.
\(\textbf{u} = (1,1,3)\), \(\textbf{v} = (2,1,4)\), \(\textbf{w} = (5,1,-2)\)
1.4.12.
\(\textbf{u} = (1,3,2)\), \(\textbf{v} = (7,2,-10)\), \(\textbf{w} = (1,0,1)\)
For Exercises 13-14, calculate \(\textbf{u} \cdot (\textbf{v} \times \textbf{w})\) and \(\textbf{u} \times (\textbf{v} \times \textbf{w})\).
1.4.13.
\(\textbf{u} = (1,1,1)\), \(\textbf{v} = (3,0,2)\), \(\textbf{w} = (2,2,2)\)
1.4.14.
\(\textbf{u} = (1,0,2)\), \(\textbf{v} = (-1,0,3)\), \(\textbf{w} = (2,0,-2)\)
1.4.15. Calculate \((\textbf{u} \times \textbf{v}) \cdot (\textbf{w} \times \textbf{z})\) for \(\textbf{u} = (1,1,1)\), \(\textbf{v} = (3,0,2)\), \(\textbf{w} = (2,2,2)\), \(\textbf{z} = (2,1,4)\).
B
1.4.16. If \(\textbf{v}\) and \(\textbf{w}\) are unit vectors in \(\mathbb{R}^{3}\), under what condition(s) would \(\textbf{v} \times \textbf{w}\) also be a unit vector in \(\mathbb{R}^{3}\;\)? Justify your answer.
1.4.17. Show that if \(\textbf{v} \times \textbf{w} = \textbf{0}\) for all \(\textbf{w}\) in \(\mathbb{R}^{3}\), then \(\textbf{v} = \textbf{0}\).
1.4.18. Prove Theorem 1.14(b).
1.4.19. Prove Theorem 1.14(c).
1.4.20. Prove Theorem 1.14(d).
1.4.21. Prove Theorem 1.14(e).
1.4.22. Prove Theorem 1.14(f).
1.4.23. Prove Theorem 1.16.
1.4.24. Prove Theorem 1.17. ( Hint: Expand both sides of the equation .)
1.4.25. Prove the following for all vectors \(\textbf{v}, \,\textbf{w}\text{ in }\mathbb{R}^ 3\):
(a) \(\norm{\textbf{v} \times \textbf{w}}^{2} + | \textbf{v} \cdot \textbf{w}|^{2} = \norm{\textbf{v}}^2 \, \norm{\textbf{w}}^2 \)
(b) If \(\textbf{v} \cdot \textbf{w} = 0\) and \(\textbf{v} \times \textbf{w} = \textbf{0}\), then \(\textbf{v} = \textbf{0}\) or \(\textbf{w} = \textbf{0}\).
C
1.4.26. Prove that in Example 1.8 the formula for the area of the triangle \(\triangle PQR\) yields the same value no matter which two adjacent sides are chosen. To do this, show that \(\frac{1}{2}\,\norm{\textbf{u} \times (-\textbf{w})} = \frac{1}{2}\,\norm{\textbf{v} \times \textbf{w}}\), where \(\textbf{u} = PR\), \(-\textbf{w} = PQ\), and \(\textbf{v} = QR\), \(\textbf{w} = QP\) as before. Similarly, show that \(\frac{1}{2}\,\norm{(-\textbf{u}) \times (-\textbf{v})} = \frac{1}{2}\,\norm{\textbf{v} \times \textbf{w}}\), where \(-\textbf{u} = RP\) and \(-\textbf{v} = RQ\).
1.4.27. Consider the vector equation \(\textbf{a} \times \textbf{x} = \textbf{b}\) in \(\mathbb{R}^{3}\), where \(\textbf{a} \ne \textbf{0}\). Show that:
- \(\textbf{a} \cdot \textbf{b} = 0\)
- \(\textbf{x} = \dfrac{\textbf{b} \times \textbf{a}}{\norm{\textbf{a}}^{2}} + k \textbf{a}\) is a solution to the equation, for any scalar \(k\).
1.4.28.
Prove the \(\textit{Jacobi identity}\):
$$\textbf{u} \times (\textbf{v} \times \textbf{w}) + \textbf{v} \times (\textbf{w} \times \textbf{u}) + \textbf{w} \times (\textbf{u} \times \textbf{v}) = \textbf{0}\]
1.4.29. Show that \(\textbf{u}, \textbf{v}, \textbf{w}\) lie in the same plane in \(\mathbb{R}^{3}\) if and only if \(\textbf{u} \cdot (\textbf{v} \times \textbf{w}) = 0\).
1.4.30.
For all vectors \(\textbf{u}, \textbf{v}, \textbf{w}, \textbf{z}\) in \(\mathbb{R}^{3}\), show that
\[(\textbf{u} \times \textbf{v}) \times (\textbf{w} \times \textbf{z}) =
(\textbf{z} \cdot (\textbf{u} \times \textbf{v}))\textbf{w} -
(\textbf{w} \cdot (\textbf{u} \times \textbf{v}))\textbf{z}\]
and that
\[(\textbf{u} \times \textbf{v}) \times (\textbf{w} \times \textbf{z}) =
(\textbf{u} \cdot (\textbf{w} \times \textbf{z}))\textbf{v} -
(\textbf{v} \cdot (\textbf{w} \times \textbf{z}))\textbf{u}\]
Why do both equations make sense geometrically?
Contributors and Attributions
-
Michael Corral (Schoolcraft College). The content of this page is distributed under the terms of the GNU Free Documentation License, Version 1.2 .


2.2: Statistical software
Graphical systems
There are two groups of statistical software. First, graphical systems which at a glance do not differ much from spreadsheets but supplied with much more statistical functions and have the powerful graphical and report modules. The typical examples are SPSS and MiniTab.
As all visual systems, they are flexible but only within the given range. If you need something new (new kind of plot, new type of calculation, unusual type of data input), the only possibility is to switch to non-visual side and use macros or sub-programs. But even more important is that visual ideology is not working well with more than one user, and does not help if the calculation should be repeated in different place with different people or several years after. That breaks reproducibility , one of the most important principle of science. Last but not least, in visual software statistical algorithms are hidden from end-user so if even you find the name of procedure you want, it is not exactly clear what program is going to do.
Statistical environments
This second group of programs uses the command-line interface (CLI). User enters commands, the system reacts. Sounds simple, but in practice, statistical environments belong to the most complicated systems of data analysis. Generally speaking, CLI has many disadvantages. It is impossible, for example, to choose available command from the menu. Instead, user must remember which commands are available. Also, this method is so similar to programming that users of statistical environments need to have some programming skills.
As a reward, the user has the full control over the system: combine all types of analysis, write command sequences into scripts which could be run later at any time, modify graphic output, easily extend the system and if the system is open source, modify the core statistical environment. The difference between statistical environment and graphical system is like the difference between supermarket and vending machine!
SAS is the one of the most advanced and powerful statistical environments. This commercial system has extensive help and the long history of development. Unfortunately, SAS is frequently overcomplicated even for the experienced programmer, has many “vestiges” of 1970s (when it was written), closed-source and extremely expensive...


Skip to main content
6: Social and Emotional Development in Early Childhood (Ages 3 to 6)
-
-
Last updated
-
-
Save as PDF
-
-
-
6.0: What Does Psychology Say?
-
Seyoun notices changes in her son Hwan's behavior since he started preschool, observing that he has become more hesitant around peers. She considers factors like their Korean-speaking household and strict grandparents, which might affect his self-perception and interaction. Though Hwan is less aggressive and receptive in social settings, Seyoun questions his emotional development, racial identity understanding, and whether to encourage more social initiative or family time.
-
-
6.1: Social and Emotional Development in Early Childhood
-
This section discusses the objectives of understanding Erik Erikson's psychosocial development theory, specifically the stage of initiative versus guilt in early childhood. It highlights how young children, like Meghan, develop self-confidence, emotional regulation, social skills, and personality traits.
-
-
6.2: Identity in Context- Gender Development and Racial Identity in Early Childhood
-
The section discusses the development of gender, racial, and ethnic identity in early childhood. Key concepts include gender identity development through stages of labeling, stability, and constancy, as well as the influence of family, peers, and media on gender socialization. The text also addresses racial and ethnic identity milestones such as ethnic labeling, knowledge acquisition, and constancy.
-
-
6.3: Families as Context in Early Childhood
-
This page explores various aspects of parenting and their impact on child development. It covers different parenting styles identified by Diana Baumrind, including authoritarian, authoritative, permissive, and uninvolved, and discusses their effects on children. The page also examines the role of siblings in development, the psychological impact of divorce on children, and the use of rewards and punishments in parenting.
-
-
6.4: Social Contexts- Peers, Play, and Friendship in Early Childhood
-
This page explores aspects of social development in early childhood, focusing on friendships, play, aggression, social withdrawal, and prosocial behavior. Children establish peer interactions that shape social, emotional, and cognitive growth. Play, whether structured or free, fosters these skills. While some children exhibit aggression due to environmental factors, others may withdraw due to shyness or anxiety.
-
-
6.5: Media Exposure and Literacy in Early Childhood
-
The page discusses the impact of media diets on children in early childhood, highlighting both positive and negative effects.


5.6: Predicting Products from Chemical Reactions
Part of the lure of chemistry is that things don’t always work out the way you expect. You plan a reaction, anticipate the products and, quite often, the results astound you! The exercise, then, is trying to figure out what was formed, why, and whether your observation leads to other useful generalizations. The first step in this process of discovery is anticipating or predicting the products which are likely to be formed in a given chemical reaction. The guidelines we describe here will accurately predict the products of most classes of simple chemical reactions. As your experience in chemistry grows, however, you will begin to appreciate the unexpected!
In simple synthesis reactions involving reaction of elements, such as aluminum metal reacting with chlorine gas, the product will be a simple compound containing both elements. In this case, it is easiest to consider the common charges that the elements adopt as ions and build your product accordingly. Aluminum is a Group III element and will typically form a +3 ion. Chlorine, being Group VII , will accept one electron and form a monoanion. Putting these predictions together, the product is likely to be AlCl 3 . In fact, if aluminum metal and chlorine gas are allowed to react, solid AlCl 3 is the predominant product.
2 Al (s) + 3 Cl 2 (g) → 2 AlCl 3 (s)
The synthesis reaction involving the non-metals hydrogen gas and bromine can be approached similarly. The product will contain both elements. Hydrogen, Group I, has one valence electron and will form one covalent bond. Bromine, Group VII, has seven valence electrons and will form one covalent bond. The likely product is therefore HBr , with one covalent bond between the hydrogen and the bromine.
H 2 (g) + Br 2 (g) < → 2 HBr (g)
For a single-replacement reaction, recall that (in general) metals will replace metals and non-metals will replace non-metals. For the reaction between lead(IV) chloride and fluorine gas, the fluorine will replace the chlorine, leading to a compound between lead and fluorine and the production of elemental chlorine . The lead can be viewed as a “spectator” in the reaction and the product is likely to be lead(IV) fluoride . The complete equation is shown below.
PbCl 4 (s) + 2 F 2 (g) → PbF 4 (s) + 2 Cl 2 (g)
In single-replacement reactions in which metals (or carbon or hydrogen) are expected to replace metals, first you should check the activity series to see if any reaction is anticipated. Remember that metals can only replace metals that are less active than themselves (to the right in the Table). If the reaction is predicted to occur, use the same general guidelines that we used above. For example, solid iron reacting with aqueous sulfuric acid (H 2 SO 4 ). In this reaction the question is whether iron will displace hydrogen and form hydrogen gas. Consulting the activity series, we see that hydrogen is to the right of iron, meaning that the reaction is expected to occur. Next, we reason that iron will replace hydrogen, leading to the formation of iron sulfate, where the sulfate is the “spectator” ion. The formation of hydrogen gas requires a change in oxidation number in the hydrogen of +1 to zero. Two hydrogen atoms must therefore be reduced (a decrease in oxidation number) and the two electrons required for the reduction must come from the iron. The charge on the iron is therefore most likely to be +2 (it starts off at zero and donates two electrons to the hydrogens). The final product is therefore most likely iron(II) sulfate . The complete equation is shown below.
Fe (s) + H 2 SO 4 (aq) → FeSO 4 (aq) + H 2 (g)
Decomposition reactions are the most difficult to predict, but there are some general trends that are useful. For example, most metal carbonates will decompose on heating to yield the metal oxide and carbon dioxide .
NiCO 3 (s) → NiO (s) + CO 2 (g)
Metal hydrogen carbonates also decompose on heating to give the metal carbonate , carbon dioxide and water .
2 NaHCO 3 (s) → Na 2 CO 3 (s) + H 2 O (g) + CO 2 (g)
Finally, many oxygen-containing compounds will decompose on heating to yield oxygen gas and “other compounds”. Identifying these compounds and building an understanding of why and how they are formed is one of the challenges of chemistry. Some examples:
H 2 O 2 (aq) → O 2 (g) + H 2 O (l)
2 HgO (s) → O 2 (g) + 2 Hg (l)
2 KClO 3 (s) → 3 O 2 (g) + 2 KCl (s)
The potential products in double-replacement reactions are simple to predict; the anions and cations simply exchange. Remember, however, that one of the products must precipitate , otherwise no chemical reaction has occurred. For the reaction between lead(II) nitrate and potassium iodide, the products are predicted to be lead(II) iodide and potassium nitrate. No redox occurs, and the product, lead iodide, precipitates from the solution as a bright yellow solid. The question of how do you predict this type of solubility trend is addressed in the next section.
Pb(NO 3 ) 2 (aq) + 2 KI (aq) < → PbI 2 (s) + 2 KNO 3 (aq)


8.5: The Electoral Process
8.5.1 Gerrymandering
In the United States, boundaries play an important role in the electoral process, but in this case, district, and precinct boundaries are significant in contrast to the country boundaries that have been previously discussed. Political parties in power will sometimes rearrange the boundaries of voting districts in order to ensure victory in elections, and this practice is called gerrymandering . Gerrymandering strategies can involve drawing the boundaries so that the majority of voters in a district favor the party in power. Another method is to segregate the opposition voters into several different districts ( Figure 8.6 ). While gerrymandering is not generally illegal in the United States, it can be challenged in court when it appears to clearly discriminate against minority populations. For example, when legislative districts where redrawn in 2015 in Gwinnett and Henry counties in Georgia, the NAACP filed a federal lawsuit because of the perception that the adjustment violated the rights of minority black voters. A 2013 Supreme Court decision, however, declared that a requirement in the 1965 Voting Rights Act for federal oversight of redistricting is not constitutional, and this decision may have an impact on such lawsuits.
8.5.2 Presidential Elections
One of the more intriguing aspects of political geography in the United States is presidential elections and how the Electoral College process has a decidedly spatial component. An examination of the presidential results from the 2016 election reveals interesting regional affiliations ( Figure 8.7 ). Northeastern and Mid-Atlantic states, as well as far Western states, all tended to vote Democrat, while the rest of the electoral map is Republican. Although it appears that the Republican victory was decisive from a spatial and Electoral College perspective, from the standpoint of the popular vote, Democrats actually had more votes than Republicans. The dichotomy is explained by the very large populations in both California and New York, predominantly Democrat states. California’s large population is reflected in a high number (55) of Electoral College votes, more than any other state.
Another marked trend in the voting map is evident when the election results are examined by county ( Figure 8.8 ). Ethnic groups tend to vote in particular voting blocks, and the impact of ethnic groups can be seen more clearly by looking at individual counties. For example, a crescent of blue (Democrat) can be seen running through south-central Alabama and into eastern Mississippi. African Americans tend to vote Democrat, and these blue counties contain predominantly African American voters.
Similarly, the blue counties in western Mississippi have a majority African American population. Another example of the influence of ethnic groups is evident in extreme southern Texas. Hispanics generally vote Democrat, and most counties in southern Texas are dominated by Hispanics.
Related to ethnic affiliations by county is the trend for rural voters to lean toward Republican and urbanites to vote Democrat. Rural areas tend to have white majorities, and whites tend to vote Republican. Rural voters also have lower numbers of college graduates compared to urban areas. In the 2016 presidential election in both Georgia and Texas, urban counties tended to vote Democrat, and rural counties went to the Republicans. With increasing urbanization and diversity in the United States, the electoral map will continue to undergo significant changes in the future.


8.2.2: Obedience, Power, and Leadership
-
- Last updated
- Save as PDF
- Anonymous
- LibreTexts
Learning Objectives
- Describe and interpret the results of Stanley Milgram’s research on obedience.
- Compare the different types of power proposed by John French and Bertram Raven and explain how they produce conformity.
- Define leadership and explain how effective leaders are determined by the person, the situation, and the person-situation interaction.
One of the fundamental aspects of social interaction is that some individuals have more influence than others. Social power can be defined as the ability of a person to create conformity even when the people being influenced may attempt to resist those changes (Fiske, 1993; Keltner, Gruenfeld, & Anderson, 2003). Bosses have power over their workers, parents have power over their children, and, more generally, we can say that those in authority have power over their subordinates. In short, power refers to the process of social influence itself—those who have power are those who are most able to influence others.
Milgram’s Studies on Obedience to Authority
The powerful ability of those in authority to control others was demonstrated in a remarkable set of studies performed by Stanley Milgram (1974). Milgram was interested in understanding the factors that lead people to obey the orders given by people in authority. He designed a study in which he could observe the extent to which a person who presented himself as an authority would be able to produce obedience, even to the extent of leading people to cause harm to others.
Like many other social psychologists, Milgram’s interest in conformity stemmed in part from his desire to understand how the presence of a powerful person—particularly the German dictator Adolph Hitler who ordered the killing of millions of people during World War II—could produce obedience. Under Hitler’s direction the German SS troops oversaw the execution of 6 million Jews as well as other “undesirables,” including political and religious dissidents, homosexuals, the mentally and physically disabled, and prisoners of war.
Milgram used newspaper ads to recruit men (and in one study, women) from a wide variety of backgrounds to participate in his research. When the research participant arrived at the lab, he or she was introduced to a man whom the participant believed was another research participant but who was actually an experimental confederate. The experimenter explained that the goal of the research was to study the effects of punishment on learning. After the participant and the confederate both consented to be in the study, the researcher explained that one of them would be the teacher and the other the learner. They were each given a slip of paper and asked to open it and to indicate what it said. In fact both papers read teacher , which allowed the confederate to pretend that he had been assigned to be the learner and thus to assure that the actual participant was always the teacher.
While the research participant (now the teacher) looked on, the learner was taken into the adjoining shock room and strapped to an electrode that was to deliver the punishment. The experimenter explained that the teacher’s job would be to sit in the control room and to read a list of word pairs to the learner. After the teacher read the list once, it would be the learner’s job to remember which words went together. For instance, if the word pair was blue-sofa , the teacher would say the word blue on the testing trials and the learner would have to indicate which of four possible words ( house , sofa , cat , or carpet ) was the correct answer by pressing one of four buttons in front of him.
After the experimenter gave the “teacher” a sample shock (which was said to be at 45 volts) to demonstrate that the shocks really were painful, the experiment began. The research participant first read the list of words to the learner and then began testing him on his learning. The shock panel, as shown in Figure 7.7, was presented in front of the teacher, and the learner was not visible in the shock room. The experimenter sat behind the teacher and explained to him that each time the learner made a mistake the teacher was to press one of the shock switches to administer the shock. Moreover, which switch was to be pressed increased by one level with each mistake, such that each mistake required a stronger shock.
Once the learner (who was, of course, actually an experimental confederate) was alone in the shock room, he unstrapped himself from the shock machine and brought out a tape recorder that he used to play a prerecorded series of responses that the teacher could hear through the wall of the room.
As you can see in Table 7.1, the teacher heard the learner say “ugh!” after the first few shocks. After the next few mistakes, when the shock level reached 150 volts, the learner was heard to exclaim “Let me out of here. I have a heart trouble!” As the shock reached about 270 volts the learner’s protests became more vehement, and after 300 volts the learner proclaimed that he was not going to answer any more questions. From 330 volts and up the learner was silent. The experimenter responded to participants’ questions at this point, if they asked any, with a scripted response indicating that they should continue reading the questions and applying increasing shock when the learner did not respond.
| 75 volts | Ugh! |
| 90 volts | Ugh! |
| 105 voles | Ugh! ( louder ) |
| 120 volts | Ugh! Hey, this really hurts. |
| 135 volts | Ugh!! |
| 150 volts | Uh!! Experimenter! That’s all. Get me out of here. I told you I had heart trouble. My heart’s starting to bother me now. Get me out of here, please. My heart’s starting to bother me. I refuse to go on. Let me out! |
| 165 volts | Ugh! Let me out! ( shouting ) |
| 180 volts | Ugh! I can’t stand the pain. Let me out of here! ( shouting ) |
| 195 volts | Ugh! Let me out of here! Let me out of here! My heart’s bothering me. Let me out of here! You have no right to keep me here! Let me out! Let me out of here! Let me out! Let me out of here! My heart’s bothering me. Let me out! Let me out! |
| 210 volts | Ugh!! Experimenter! Get me out of here. I’ve had enough. I won’t be in the experiment any more. |
| 225 volts | Ugh! |
| 240 volts | Ugh! |
| 255 volts | Ugh! Get me out of here. |
| 270 volts | ( agonized scream ) Let me out of here. Let me out of here. Let me out of here. Let me out. Do you hear? Let me out of here. |
| 285 volts | ( agonized scream ) |
| 300 volts | ( agonized scream ) I absolutely refuse to answer any more. Get me out of here. You can’t hold me here. Get me out. Get me out of here. |
| 315 volts | ( intensely agonized scream ) Let me out of here. Let me out of here. My heart’s bothering me. Let me out, I tell you. ( hysterically ) Let me out of here. Let me out of here. You have no right to hold me here. Let me out! Let me out! Let me out! Let me out of here! Let me out! Let me out! |
The results of Milgram’s research were themselves quite shocking. Although all of the participants gave the initial mild levels of shock, responses varied after that. Some refused to continue after about 150 volts, despite the insistence of the experimenter to continue to increase the shock level. Still others, however, continued to present the questions, and to administer the shocks, under the pressure of the experimenter, who demanded that they continue. In the end, 65% of the participants continued giving the shock to the learner all the way up to the 450 volts maximum, even though that shock was marked as “danger: severe shock” and there had been no response heard from the participant for several trials. In sum, well over half of the men who participated had, as far as they knew, shocked another person to death, all as part of a supposed experiment on learning.
In case you are thinking that such high levels of obedience would not be observed in today’s modern culture, there is fact evidence that they would be. Studies similar to Milgram’s findings have been conducted all over the world (Blass, 1999) and have found similar levels of conformity, with obedience rates ranging from a high of 90% in Spain and the Netherlands (Meeus & Raaijmakers, 1986) to a low of 16% among Australian women (Kilham & Mann, 1974).
Recently, Milgram’s results were almost exactly replicated, using men and women from a wide variety of ethnic groups, in a study conducted by Jerry Burger at Santa Clara University. In this replication of the Milgram experiment, 65% of the men and 73% of the women agreed to administer increasingly painful electric shocks when they were ordered to by an authority figure (Borge, 2010). In the replication, however, the participants were not allowed to go beyond the 150 volt shock switch.
Although it might be tempting to conclude that Milgram’s experiments demonstrate that people are innately evil creatures who are ready to shock others to death, Milgram did not believe that this was the case. Rather, he felt that it was the social situation, and not the people themselves, that was responsible for the behavior. To demonstrate this, Milgram conducted research that explored a number of variations on his original procedure, each of which demonstrated that changes in the situation could dramatically influence the amount of conformity. These variations are summarized in Figure 7.8.
This figure presents the percentage of participants in Stanley Milgram’s (1974) studies on obedience who were maximally obedient (that is, who gave all 450 volts of shock) in some of the variations that he conducted.
In the initial study the authority’s status and power was maximized—the experimenter had been introduced as a respected scientist at a respected university. However, in replications of the study in which the experimenter’s authority was decreased, obedience also declined. In one replication the status of the experimenter was reduced by having the experiment take place in a building located in Bridgeport, Connecticut, rather than at the labs on the Yale University campus, and the research was ostensibly sponsored by a private commercial research firm instead of by the university. In this study less obedience was observed (only 48% of the participants delivered the maximum shock). Full obedience was also reduced (to 20%) when the experimenter’s ability to express his authority was limited by having him sit in an adjoining room and communicate to the teacher by telephone. And when the experimenter left the room and had another student (actually a confederate) give the instructions for him, conformity was also reduced to 20%.
In addition to the role of authority, Milgram’s studies also confirmed the role of unanimity in producing conformity. When another research participant (again an experimental confederate) began by giving the shocks but then later refused to continue and the participant was asked to take over, only 10% were obedient. And if two experimenters were present but only one proposed shocking while the other argued for stopping the shocks, all the research participants took the more benevolent advice and did not shock. Finally, in another condition the teacher sat next to the learner and was forced to wear an insulated glove and to hold the learner’s hand on the shock pad as he administered the shock. Again, obedience was reduced, this time to only 30%.
Perhaps most telling regarding the important role of the situation were the studies in which Milgram allowed the participants to choose their own shock levels or in which one of the experimenters suggested that they should not actually use the shock machine. In these situations, there was virtually no shocking. These conditions show that people do not like to harm others, and when given a choice they will not. On the other hand, the social situation can create powerful, and potentially deadly, social influence.
Social Psychology in the Public Interest
The Zimbardo Prison Studies and Abu Ghraib
In Milgram’s research we can see a provocative demonstration of how people who have power can control the behavior of others. Can our understanding of the social psychological factors that produce conformity and obedience help us explain the events that occurred in 2004 at Abu Ghraib, the Iraqi prison in which U.S. soldiers physically and psychologically tortured their Iraqi prisoners? The social psychologist Philip Zimbardo thinks so. He notes the parallels between the events that occurred at Abu Ghraib and the events that occurred in the “prison study” that he conducted in 1971.
In that study, Zimbardo and his colleagues set up a mock prison. They selected 23 student volunteers and divided them into two groups. One group was chosen to be the “prisoners.” They were picked up at their homes by actual police officers, “arrested,” and brought to the prison to be guarded by the other group of students—the “guards.” The two groups were placed in a setting that was designed to look like a real prison, and the role-play began.
The Zimbardo Prison Study
The study was expected to run for two weeks. However, on the second day, the prisoners tried to rebel against the guards. The guards quickly moved to stop the rebellion by using both psychological punishment and physical abuse. In the ensuing days, the guards denied the prisoners food, water, and sleep; shot them with fire-extinguisher spray; threw their blankets into the dirt; and stripped them naked. On the fifth night the experimenters witnessed the guards putting bags over the prisoners’ heads, chaining their legs, and marching them around. At this point the researchers stopped the experiment early. The conclusions of Zimbardo’s research were clear: People may be so profoundly influenced by their social situation that they become coldhearted jail masters who torture their victims.
Zimbardo’s research can help us understand the events that occurred at Abu Ghraib. Zimbardo acted as an expert witness in the trial of Sergeant Chip Frederick, who was sentenced to eight years in prison for his role in the abuse at Abu Ghraib. Frederick was the Army reservist who was put in charge of the night shift at Tier 1A, where the detainees were abused. During this trial Frederick said, “What I did was wrong, and I don’t understand why I did it.” Zimbardo believes that Frederick acted exactly like the students in the prison study did. He worked in a prison that was overcrowded, filthy, and dangerous, and where he was expected to maintain control over the Iraqi prisoners—in short, the situation he found himself in was very similar to that of Zimbardo’s prison study.
In a recent interview, Zimbardo argued (you can tell that he is a social psychologist) that “human behavior is more influenced by things outside of us than inside.” He believes that, despite our moral and religious beliefs and despite the inherent goodness of people, there are times when external circumstances can overwhelm us and we do things we never thought we were capable of doing. He argued that “if you’re not aware that this can happen, you can be seduced by evil. We need inoculations against our own potential for evil. We have to acknowledge it. Then we can change it” (Driefus, 2007).
You may wonder whether the extreme behavior of the guards and prisoners in Zimbardo’s prison study was unique to the particular social context that he created. Recent research by Stephen Reicher and Alex Haslam (2006) suggests that this is indeed the case. In their research, they recreated Zimbardo’s prison experiment while making some small, but important, changes. For one, the prisoners were not “arrested” before the study began, and the setup of the jail was less realistic. Furthermore, the researchers in this experiment told the “guards” and the “prisoners” that the groups were arbitrary and could change over time (that is, that some prisoners might be able to be promoted to guards). The results of this study were entirely different than those found by Zimbardo. Although this study was also stopped early, this was more because the guards felt uncomfortable in their superior position than because the prisoners were being abused. This “prison” simply did not feel like a real prison to the participants, and as a result they did not take on the roles they were assigned. Again, the conclusions are clear—the specifics of the social situation, more than the people themselves, are often the most important determinants of behavior.
Types of Power
One of the most influential theories of power was developed by Bertram Raven and John French (French & Raven, 1959; Raven, 1992). Raven identified five different types of power— reward power , coercive power , legitimate power , referent power , and expert power (shown in Table 7.2), arguing that each type of power involves a different type of social influence and that the different types vary in terms of whether their use will create public conformity or private acceptance. Understanding the types of power is important because it allows us to see more clearly the many ways that people can influence others. Let’s consider these five types of power, beginning with those that are most likely to produce public conformity only and moving on to those that are more likely to produce private acceptance.
| Reward power | The ability to distribute positive or negative rewards |
| Coercive power | The ability to dispense punishments |
| Legitimate power | Authority that comes from a belief on the part of those being influenced that the person has a legitimate right to demand conformity. |
| Referent power | Influence based on identification with, attraction to, or respect for the power-holder |
| Expert power | Power that comes from other’s beliefs that the power-holder possesses superior skills and abilities |
| Note. French and Raven proposed five types of power, which differ in their likelihood of producing public conformity or private acceptance. |
|---|
Reward Power
Reward power occurs when one person is able to influence others by providing them with positive outcomes . Bosses have reward power over employees because they are able to increase employees’ salary and job benefits, and teachers have reward power over students because they can assign student high marks. The variety of rewards that can be used by the powerful is almost endless and includes verbal praise or approval, the awarding of status or prestige, and even direct financial payment.
The ability to wield reward power over those we want to influence is contingent on the needs of the person being influenced. Power is greater when the person being influenced has a strong desire to obtain the reward, and power is weaker when the individual does not need the reward. A boss will have more influence on an employee who has no other job prospects than on one who is being sought after by other corporations, and expensive presents will be more effective in persuading those who cannot buy the items with their own money. Because the change in behavior that results from reward power is driven by the reward itself, its use is usually more likely to produce public conformity than private acceptance.
Coercive Power
Coercive power is power that is based on the ability to create negative outcomes for others, for instance by bullying, intimidating, or otherwise punishing . Bosses have coercive power over employees if they are able (and willing) to punish employees by reducing their salary, demoting them to a lower position, embarrassing them, or firing them. And friends can coerce each other through teasing, humiliation, and ostracism. As with reward power, coercive power is effective only when the person being influenced is dependent upon the power-holder. Furthermore, coercion works only if the person being influenced does not leave the group entirely—people who are punished too much are likely to look for other situations that provide more positive outcomes.
In many cases power-holders use reward and coercive power at the same time—for instance, by both increasing salaries as a result of positive performance but also threatening to reduce them if the performance drops. Because the use of coercion has such negative consequences, authorities are generally more likely to use reward than coercive power (Molm, 1997). Coercion is usually more difficult to use, since it often requires energy to keep the person from avoiding the punishment by leaving the situation altogether. And coercive power is less desirable for both the power-holder and the person being influenced because it creates an environment of negative feelings and distrust that is likely to make interactions difficult, undermine satisfaction, and lead to retaliation against the power-holder (Tepper et al., 2009).
As with reward power, coercive power is more likely to produce public conformity than private acceptance. Furthermore, in both cases the effective use of the power requires that the power-holder continually monitor the behavior of the target to be sure that he or she is complying. This monitoring may itself lead to a sense of mistrust between the two individuals in the relationship. The power-holder feels (perhaps unjustly) that the target is only complying due to the monitoring, whereas the target feels (again perhaps unjustly) that the power-holder does not trust him or her.
Legitimate Power
Whereas reward and coercive power are likely to produce conformity, other types of power, which are not so highly focused around reward and punishment, are more likely to create changes in attitudes as well as behavior (private acceptance). In many ways, then, these sources of power are stronger because they produce real belief change. Legitimate power is power vested in those who are appointed or elected to positions of authority , such as teachers, politicians, policemen, and judges, and their power is successful because members of the group accept it as appropriate. We accept that governments can levy taxes and that judges can decide the outcomes of court cases because we see these groups and individuals as valid parts of our society. Individuals with legitimate power can exert substantial influence on their followers. Those with legitimate power may not only create changes in the behavior of others but also have the power to create and change the social norms of the group.
In some cases legitimate power is given to the authority figure as a result of laws or elections, or as part of the norms, traditions, and values of the society. The power that the experimenter had over the research participants in Milgram’s study on obedience seems to have been primarily the result of his legitimate power as a respected scientist at an important university. In other cases legitimate power comes more informally, as a result of being a respected group member. People who contribute to the group process and follow group norms gain status within the group and therefore earn legitimate power.
In some cases legitimate power can even be used successfully by those who do not seem to have much power. You may recall that after Hurricane Katrina hit the city of New Orleans in 2005, the people there demanded that the federal government help them rebuild the city. Although these people did not have much reward or coercive power, they were nevertheless perceived as good and respected citizens of the United States. Many U.S. citizens tend to believe that people who do not have as much as others (for instance, those who are very poor) should be treated fairly and that these people may legitimately demand resources from those who have more. This might not always work, but to the extent that it does it represents a type of legitimate power—power that comes from a belief in the appropriateness or obligation to respond to the requests of others with legitimate standing.
Referent Power
People with referent power have an ability to influence others because they can lead those others to identify with them . In this case, the person who provides the influence is (a) a member of an important reference group—someone we personally admire and attempt to emulate; (b) a charismatic, dynamic, and persuasive leader; or (c) a person who is particularly attractive or famous (Heath, McCarthy, & Mothersbaugh, 1994; Henrich & Gil-White, 2001; Kamins, 1989; Wilson & Sherrell, 1993). A young child who mimics the opinions or behaviors of an older sibling or a famous baseball player, or a churchgoer who follows the advice of a respected church leader, is influenced by referent power. Referent power generally produces private acceptance rather than public conformity (Kelman, 1961).
The influence brought on by referent power may occur in a passive sense because the person being emulated does not necessarily attempt to influence others, and the person who is being influenced may not even realize that the influence is occurring. In other cases, however, the person with referent power (such as the leader of a cult) may make full use of his or her status as the target of identification or respect to produce change. In either case, referent power is a particularly strong source of influence because it is likely to result in the acceptance of the opinions of the important other.
Expert Power
French and Raven’s final source of power is expert power . Experts have knowledge or information, and conforming to those whom we perceive to be experts is useful for making decisions about issues for which we have insufficient expertise. Expert power thus represents a type of informational influence based on the fundamental desire to obtain valid and accurate information, and where the outcome is likely to be private acceptance . Conformity to the beliefs or instructions of doctors, teachers, lawyers, and computer experts is an example of expert influence; we assume that these individuals have valid information about their areas of expertise, and we accept their opinions based on this perceived expertise (particularly if their advice seems to be successful in solving problems). Indeed, one method of increasing one’s power is to become an expert in a domain. Expert power is increased for those who possess more information about a relevant topic than others do because the others must turn to this individual to gain the information. You can see, then, that if you want to influence others, it can be useful to gain as much information about the topic as you can.
Research Focus
Does Power Corrupt?
Having power provides some benefits for those who have it. In comparison to those with less power, people who have more power over others are more confident and more attuned to potential opportunities in their environment (Anderson & Berdahl, 2002). They are also more likely than are people with less power to take action to meet their goals (Anderson & Galinsky, 2006; Galinsky, Gruenfeld, & Magee, 2003).
Despite these advantages of having power, a little power goes a long way and having too much can be dangerous, for both the targets of the power and the power-holder himself or herself. In an experiment by David Kipnis (1972), college students played the role of “supervisors” who were supposedly working on a task with other students (the “workers”). According to random assignment to experimental conditions, one half of the supervisors were able to influence the workers through legitimate power only, by sending them messages attempting to persuade them to work harder. The other half of the supervisors were given increased power. In addition to being able to persuade the workers to increase their output through the messages, they were also given both reward power (the ability to give small monetary rewards) and coercive power (the ability to take away earlier rewards).
Although the workers (who were actually preprogrammed) performed equally well in both conditions, the participants who were given more power took advantage of it by more frequently contacting the workers and more frequently threatening them. The students in this condition relied almost exclusively on coercive power rather than attempting to use their legitimate power to develop positive relations with the subordinates.
Although it did not increase the workers’ performance, having the extra power had a negative effect on the power-holders’ images of the workers. At the end of the study, the supervisors who had been given extra power rated the workers more negatively, were less interested in meeting them, and felt that the only reason the workers did well was to obtain the rewards.
The conclusion of these researchers is clear: Having power may lead people to use it, even though it may not be necessary, which may then lead them to believe that their subordinates are performing only because of the threats. Although using excess power may be successful in the short run, power that is based exclusively on reward and coercion is not likely to produce a positive environment for either the power-holder or the subordinate. People with power may also be more likely to stereotype people with less power than they have (Depret & Fiske, 1999) and may be less likely to help other people who are in need (van Kleef et al., 2008).
Although this research suggests that people may use power when it is available to them, other research has found that this is not equally true for all people—still another case of a person-situation interaction. Serena Chen and her colleagues (Chen, Lee-Chai, & Bargh, 2001) found that students who had been classified as more self-oriented (in the sense that they considered relationships in terms of what they could and should get out of them for themselves) were more likely to misuse their power, whereas students who were classified as other-oriented were more likely to use their power to help others
Leaders and Leadership
One type of person who has power over others, in the sense that they are able to influence them, are leaders. Leaders are in a position in which they can exert leadership , which is the ability to direct or inspire others to achieve goals (Chemers, 2001; Hogg, 2010). Leaders have many different influence techniques at their disposal: In some cases they may give commands and enforce them with reward or coercive power, resulting in public conformity with the commands. In other cases they may rely on well-reasoned technical arguments or inspirational appeals, making use of legitimate, referent, or expert power, with the goal of creating private acceptance and leading their followers to achieve.
Leadership is a classic example of the combined effects of the person and the social situation. Let’s consider first the person part of the equation and then turn to consider how the person and the social situation work together to create effective leadership.
Personality and Leadership
One approach to understanding leadership is to focus on person variables. Personality theories of leadership are explanations of leadership based on the idea that some people are simply “natural leaders” because they possess personality characteristics that make them effective (Zaccaro, 2007).
One personality variable that is associated with effective leadership is intelligence. Being intelligent improves leadership, as long as the leader is able to communicate in a way that is easily understood by his or her followers (Simonton, 1994, 1995). Other research has found that a leader’s social skills, such as the ability to accurately perceive the needs and goals of the group members, are also important to effective leadership. People who are more sociable, and therefore better able to communicate with others, tend to make good leaders (Kenny & Zaccaro, 1983; Sorrentino & Boutillier, 1975). Other variables that relate to leadership effectiveness include verbal skills, creativity, self-confidence, emotional stability, conscientiousness, and agreeableness (Cronshaw & Lord, 1987; Judge, Bono, Ilies, & Gerhardt, 2002; Yukl, 2002). And of course the individual’s skills at the task at hand are important. Leaders who have expertise in the area of their leadership will be more effective than those who do not.
Because so many characteristics seem to be related to leader skills, some researchers have attempted to account for leadership not in terms of individual traits but in terms of a package of traits that successful leaders seem to have. Some have considered this in terms of charisma (Beyer, 1999; Conger & Kanungo, 1998). Charismatic leaders are leaders who are enthusiastic, committed, and self-confident; who tend to talk about the importance of group goals at a broad level; and who make personal sacrifices for the group . Charismatic leaders express views that support and validate existing group norms but that also contain a vision of what the group could or should be. Charismatic leaders use their referent power to motivate, uplift, and inspire others. And research has found a positive relationship between a leader’s charisma and effective leadership performance (Simonton, 1988).
Another trait-based approach to leadership is based on the idea that leaders take either transactional or transformational leadership styles with their subordinates (Avolio &Yammarino, 2003; Podsakoff, MacKenzie, Moorman, & Fetter, 1990). Transactional leaders are the more regular leaders who work with their subordinates to help them understand what is required of them and to get the job done. Transformational leaders , on the other hand, are more like charismatic leaders—they have a vision of where the group is going and attempt to stimulate and inspire their workers to move beyond their present status and to create a new and better future. Transformational leaders are those who can reconfigure or transform the group’s norms (Reicher & Hopkins, 2003).
Leadership as an Interaction Between the Person and the Situation
Despite the fact that there appear to be at least some personality traits that relate to leadership ability, the most important approaches to understanding leadership take into consideration both the personality characteristics of the leader and the situation in which the leader is operating. In some cases the situation itself is important. For instance, you might remember that President George W. Bush’s ratings as a leader increased dramatically after the September 11, 2001, terrorist attacks on the World Trade Center. This is a classic example of how a situation can influence the perceptions of a leader’s skill. In other cases, however, both the situation and the person are critical.
One well-known person-situation approach to understanding leadership effectiveness was developed by Fred Fiedler and his colleagues (Ayman, Chemers, & Fiedler, 1995). The contingency model of leadership effectiveness is a model of leadership effectiveness that focuses on both person variables and situational variables . Fielder conceptualized the leadership style of the individual as a relatively stable personality variable and measured it by having people consider all the people they had ever worked with and describe the person that they least liked to work with (their least preferred coworker ). Those who indicated that they only somewhat disliked their least preferred coworker are relationship-oriented types of people, who are motivated to have close personal relationships with others. However, those who indicated that they did not like this coworker very much were classified as task-oriented types, who are motivated primarily by getting the job done.
In addition to classifying individuals according to their leadership styles, Fiedler also classified the situations in which groups had to perform their tasks, both on the basis of the task itself and on the basis of the leader’s relationship to the group members. Specifically, as shown in Figure 7.9, Fiedler thought that three aspects of the group situation were important:
- The degree to which the leader already has a good relationship with the group and the support of the group members ( leader-member relations )
- The extent to which the task is structured and unambiguous ( task structure )
- The leader’s level of power or support in the organization ( position power )
Furthermore, Fielder believed that these factors were ordered in terms of their importance, such that leader-member relationships were more important than task structure, which was in turn more important than position power. As a result, he was able to create eight levels of the “situational favorableness” of the group situation, which roughly range from most favorable to least favorable for the leader. The most favorable relationship involves good relationships, a structured task, and strong power for the leader, whereas the least favorable relationship involves poor relationships, an unstructured task, and weak leader power.
The contingency model is interactionist because it proposes that individuals with different leadership styles will differ in effectiveness in different group situations. Task-oriented leaders are expected to be most effective in situations in which the group situation is very favorable because this gives the leader the ability to move the group forward, or in situations in which the group situation is very unfavorable and in which the extreme problems of the situation require the leader to engage in decisive action. However, in the situations of moderate favorableness, which occur when there is a lack of support for the leader or when the problem to be solved is very difficult or unclear, the more relationship-oriented leader is expected to be more effective. In short, the contingency model predicts that task-oriented leaders will be most effective either when the group climate is very favorable and thus there is no need to be concerned about the group members’ feelings, or when the group climate is very unfavorable and the task-oriented leader needs to take firm control.
Still another approach to understanding leadership is based on the extent to which a group member embodies the norms of the group. The idea is that people who accept group norms and behave in accordance with them are likely to be seen as particularly good group members and therefore become leaders (Hogg, 2001; Hogg & Van Knippenberg, 2003). Group members who follow group norms are seen as more trustworthy (Dirks & Ferrin, 2002) and are likely to engage in group-oriented behaviors to strengthen their leadership credentials (Platow & van Knippenberg, 2001).
Key Takeaways
- Social power can be defined as the ability of a person to create conformity, even when the people being influenced may attempt to resist those changes.
- Milgram’s studies on obedience demonstrated the remarkable extent to which the social situation and people with authority have the power to create obedience.
- One of the most influential theories of power was developed by French and Raven, who identified five different types of power—reward power, coercive power, legitimate power, referent power, and expert power. The types vary in terms of whether their use is more likely to create public conformity or private acceptance.
- Although power can be abused by those who have it, having power also creates some positive outcomes for individuals.
- Leadership is determined by person variables, situational variables, and by the person-situation interaction. The contingency model of leadership effectiveness is an example of the latter.
Exercises and Critical Thinking
- Write a paragraph that expresses your opinions about the Holocaust or about another example of obedience to authority. Consider how social psychological research on obedience informs your interpretation of the event.
- Write a report on Dr. Philip Zimbardo’s book, The Lucifer Effect: Understanding How Good People Turn Evil , http://lucifereffect.com .
- Provide an example of someone who has each of the types of power discussed in this section.
- Consider a leader whom you have worked with in the past. What types of leadership did that person use? Were they effective?
- Choose a recent event that involved a very effective leader or one that involved a very poor one. Analyze the leadership in terms of the topics discussed in this chapter.
References
Anderson, C., & Berdahl, J. L. (2002). The experience of power: Examining the effects of power on approach and inhibition tendencies. Journal of Personality and Social Psychology, 83 , 1362–1377.
Anderson, C., & Galinsky, A. D. (2006). Power, optimism, and risk-taking. European Journal of Social Psychology, 36 , 511–536.
Avolio, B. J., & Yammarino, F. J. (2003). Transformational and charismatic leadership: The road ahead . Oxford, UK: Elsevier Press.
Ayman, R., Chemers, M. M., & Fiedler, F. (1995). The contingency model of leadership effectiveness: Its level of analysis. Leadership Quarterly, 6 (2), 147–167.
Beyer, J. M. (1999). Taming and promoting charisma to change organizations. Leadership Quarterly, 10 (2), 307–330.
Blass, T. (1999). The Milgram paradigm after 35 years: Some things we now know about obedience to authority. Journal of Applied Social Psychology, 29 , 955–978.
Borge, C. (2007). Basic instincts: The science of evil. Retrieved from http://a.abcnews.com/Primetime/Story?id=2765416&page=1 .
Chemers, M. M. (2001). Leadership effectiveness: An integrative review. In M. A. Hogg & R. S. Tindale (Eds.), Blackwell handbook of social psychology: Group processes (pp. 376–399). Oxford, UK: Blackwell.
Chen, S., Lee-Chai, A. Y., & Bargh, J. A. (2001). Relationship orientation as a moderator of the effects of social power. Journal of Personality and Social Psychology, 80 (2), 173–187.
Conger, J. A., & Kanungo, R. N. (1998). Charismatic leadership in organizations . Thousand Oaks, CA: Sage.
Cronshaw, S. F., & Lord, R. G. (1987). Effects of categorization, attribution, and encoding processes on leadership perceptions. Journal of Applied Psychology, 72 , 97–106.
Depret, E., & Fiske, S. T. (1999). Perceiving the powerful: Intriguing individuals versus threatening groups. Journal of Experimental Social Psychology, 35 (5), 461–480.
Dirks, K. T., & Ferrin, D. L. (2002). Trust in leadership: Meta-analytic findings and implications for research and practice. Journal of Applied Psychology, 87 , 611–628.
Driefus, C. (2007, April 3). Finding hope in knowing the universal capacity for evil. New York Times .
Fiske, S. T. (1993). Controlling other people: The impact of power on stereotyping. American Psychologist, 48 , 621–628.
French, J. R. P., & Raven, B. H. (1959). The bases of social power. In D. Cartwright (Ed.), Studies in social power (pp. 150–167). Ann Arbor, MI: Institute for Social Research.
Galinsky A. D., Gruenfeld, D. H, & Magee, J. C. (2003). From power to action. Journal of Personality and Social Psychology, 85 , 453–466.
Heath, T. B., McCarthy, M. S., and Mothersbaugh, D. L. (1994). Spokesperson fame and vividness effects in the context of issue-relevant thinking: The moderating role of competitive setting. Journal of Consumer Research, 20 , 520–534.
Henrich, J., & Gil-White, F. (2001). The evolution of prestige: Freely conferred status as a mechanism for enhancing the benefits of cultural transmission. Evolution and Human Behavior, 22 , 1–32.
Hogg, M. A. (2001). A social identity theory of leadership. Personality and Social Psychology Review, 5 ,184–200.
Hogg, M. A. (2010). Influence and leadership. In S. F. Fiske, D. T. Gilbert, & G. Lindzey (Eds.), Handbook of social psychology (Vol. 2, pp. 1166–1207). New York, NY: Wiley.
Hogg, M. A., & van Knippenberg, D. (2003). Social identity and leadership processes in groups. Advances in Experimental Social Psychology, 35 , 1–52.
Judge, T, Bono, J., Ilies, R., & Gerhardt, M. (2002). Personality and leadership: A qualitative and quantitative review, Journal of Applied Psychology, 87 , 765–780.
Kamins, A. M. (1989). Celebrity and non-celebrity in two-sided context. Journal of Advertising Research, 29 , 34–42.
Kelman, H. (1961). Processes of opinion change. Public Opinion Quarterly, 25 , 57–78.
Keltner, D., Gruenfeld, D. H., & Anderson, C. (2003). Power, approach, and inhibition. Psychological Review, 110 (2), 265–284.
Kenny, D. A., & Zaccaro, S. J. (1983). An estimate of variance due to traits in leadership. Journal of Applied Psychology, 68 , 678–685.
Kilham, W., & Mann, L. (1974). Level of destructive obedience as a function of transmitter and executant roles in the Milgram obedience paradigm. Journal of Personality and Social Psychology, 29 , 692–702.
Kipnis, D. (1972). Does power corrupt? Journal of Personality and Social Psychology, 24 , 33–41.
Meeus, W. H., & Raaijmakers, Q. A. (1986). Administrative obedience: Carrying out orders to use psychological-administrative violence. European Journal of Social Psychology, 16 , 311–324.
Milgram, S. (1974). Obedience to authority: An experimental view . New York, NY: Harper & Row.
Molm, L. D. (1997). Coercive power in social exchange . New York, NY: Cambridge University Press.
Platow, M. J., & van Knippenberg, D. (2001). A social identity analysis of leadership endorsement: The effects of leader in-group prototypicality and distributive intergroup fairness. Personality and Social Psychology Bulletin, 27 , 1508–1519.
Podsakoff, P. M., MacKenzie, S. B., Moorman, R. H., & Fetter, R. (1990). Transformational leader behaviors and their effects on followers’ trust in leader, satisfaction, and organizational citizenship behaviors. Leadership Quarterly, 1 , 107–142.
Raven, B. H. (1992). A power/interaction model of interpersonal influence: French and Raven thirty years later. Journal of Social Behavior and Personality, 7 (2), 217–244.
Reicher, S. D., & Hopkins, N. (2003). On the science of the art of leadership. In D. van Knippenberg and M. A. Hogg (Eds.), Leadership and power: Identity processes in groups and organizations (pp. 197–209). London, UK: Sage.
Reicher, S., & Haslam, S. A. (2006). Rethinking the psychology of tyranny: The BBC prison study. British Journal of Social Psychology, 45 (1), 1–40.
Simonton, D. K. (1988). Presidential style: Personality, biography and performance. Journal of Personality and Social Psychology, 55 , 928–936.
Simonton, D. K. (1994). Greatness: Who makes history and why . New York, NY: Guilford Press.
Simonton, D. K. (1995). Personality and intellectual predictors of leadership. In D. H. Saklofske et al. (Eds.), International handbook of personality and intelligence. Perspectives on individual differences (pp. 739–757). New York, NY: Plenum.
Sorrentino, R. M., & Boutillier, R. G. (1975). The effect of quantity and quality of verbal interaction on ratings of leadership ability. Journal of Experimental Social Psychology, 11 , 403–411.
Stanford Prison Experiment. Retrieved from http://www.prisonexp.org/links.htm .
Tepper, B. J., Carr, J. C., Breaux, D. M., Geider, S., Hu, C., & Hua, W. (2009). Abusive supervision, intentions to quit, and employees’ workplace deviance: A power/dependence analysis. Organizational Behavior and Human Decision Processes, 109 (2), 156–167.
van Kleef, G. A., Oveis, C., van der Löwe, I., LuoKogan, A., Goetz, J., & Keltner, D. (2008). Power, distress, and compassion: Turning a blind eye to the suffering of others. Psychological Science, 19 (12), 1315–1322.
Wilson, E. J., &. Sherrell, D. L. (1993). Source effects in communication and persuasion: A meta-analysis of effect size. Journal of the Academy of Marketing Science, 21 , 101–112.
Yukl, G. A. (2002). Leadership in organizations . Upper Saddle River, NJ: Prentice Hall.
Zaccaro, S. J. (2007). Trait-based perspectives of leadership. American Psychologist, 62 , 6–16.