# Crisp-Set Qualitative Comparative Analysis (csqca)

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page 33 3 Crisp-Set Qualitative Comparative Analysis (csqca) Benoît Rihoux Gisèle De Meur Goals of This Chapter After reading this chapter, you should be able to: Understand the key operations of Boolean algebra and use the correct conventions of that language Transform tabular data into Venn diagrams and vice versa; interpret Venn diagrams Replicate a standard csqca procedure, step by step, using the software In the course of this procedure, dichotomize your variables (conditions and outcome) in an informed way, and at a later stage, use an appropriate strategy for solving contradictory configurations Weigh the pros and cons of using logical remainders (non-observed cases) Read and interpret the minimal formulas obtained at the end of the csqca csqca was the first QCA technique developed, in the late 1980s, by Charles Ragin and programmer Kriss Drass. Ragin s research in the field of historical sociology led him to search for tools for the treatment of complex sets of binary data that did not exist in the mainstream statistics literature. He adapted for his own research, with the help of Drass, Boolean algorithms that had been developed in the 1950s by electrical engineers to simplify switching circuits, most notably Quine (1952) and McCluskey (1966). In these so-called minimization algorithms (see Box 3.2), he had found an instrument for identifying patterns of multiple conjunctural causation and a tool to simplify complex data structures in a logical and holistic manner (Ragin, 1987, p. viii). 33

2 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page CONFIGURATIONAL COMPARATIVE METHODS csqca is the most widely used QCA technique so far. In this chapter, a few basic operations of Boolean algebra will first be explicated, so the reader can grasp the nuts and bolts of csqca. Then, using a few variables from the interwar project (see Chapter 2), the successive steps, arbitrations, and good practices of a standard application of csqca will be presented. THE FOUNDATION OF CRISP- SET QCA: BOOLEAN ALGEBRA IN A NUTSHELL George Boole, a 19th-century British mathematician and logician, was the first to develop an algebra suitable for variables with only two possible values, such as propositions that are either true or false (Boole, 1847; 1958 [1854]). Following intuitions by Leibniz one century before him, Boole is the originator of mathematical logic that allows us to substitute for verbal reasoning a genuine symbolic calculation (Diagne, 1989, p. 8). This algebra has been studied further by many mathematicians and logicians over the last few decades. It has been central to the development of electronic circuits, computer science, and computer engineering, which are based on a binary language, and has led to many applications, mostly in experimental and applied scientific disciplines. Only a few basic principles and operations will be presented here (for more details, see, e.g., Caramani, 2008; De Meur & Rihoux, 2002; Ragin, 1987, pp ; Schneider & Wagemann, 2007, forthcoming). Boolean algebra, as any language, uses some conventions that need to be understood before engaging in csqca. Box 3.1 It is amain conception Conventions of causalityand according Operations to which: of Boolean Algebra 1. The main conventions of Boolean algebra are as follows: An uppercase letter represents the [1] value for a given binary variable. Thus [A] is read as: variable A is large, present, high,... A lowercase letter represents the [0] value for a given binary variable. Thus [a] is read as: variable A is small, absent, low,... A dash symbol [-] represents the don t care value for a given binary variable, meaning it can be either present (1) or absent (0). This also could be a value we don t know about (e.g., because it is irrelevant or the data is missing). It is not an intermediate value between [1] and [0].

3 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page 35 CRISP-SET QUALITATIVE COMPARATIVE ANALYSIS (CSQCA) Boolean algebra uses a few basic operators, the two chief ones being the following: Logical AND, represented by the [*] (multiplication) symbol. NB: It can also be represented with the absence of a space: [A*B] can also be written as: [AB]. Logical OR, represented by the [+] (addition) symbol. 3. The connection between conditions and the outcome: The arrow symbol [ ] 1 is used to express the (usually causal) link between a set of conditions on the one hand and the outcome we are trying to explain on the other. With this very basic language, it is possible to construct very long and elaborate expressions and also to conduct a complex set of operations. One key operation, which lies at the heart of csqca, is called Boolean minimization. Box 3.2 What Is Boolean Minimization? It is the reduction of a long, complex expression into a shorter, more parsimonious expression. It can be summarized verbally as follows: if two Boolean expressions differ in only one causal condition yet produce the same outcome, then the causal condition that distinguishes the two expressions can be considered irrelevant and can be removed to create a simpler, combined expression (Ragin, 1987, p. 93). Let us use a very simple example. Consider the following Boolean expression, with three condition variables (R, B, and I) and one outcome variable (O) (Formula 1): R * B * I + R * B * i O This expression can be read as follows: [The presence of R, combined with the presence of B and with the presence of I] OR [The presence of R, combined with the presence of B and with the absence of I] lead to the presence of outcome O. Notice that, no matter which value the [I] condition takes (0 or 1), the [I] outcome value is the same. This means, in verbal reasoning, that the [I] condition is superfluous; it can thus be removed from the initial expression. Indeed, if we remove the [I] condition, we are left with a much shorter, reduced expression (which is called a prime implicant) (Formula 2): R * B O (Continued)

4 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page CONFIGURATIONAL COMPARATIVE METHODS (Continued) It is read as follows: The presence of R, combined with the presence of B, leads to the presence of outcome O. This reduced expression meets the parsimony principle (see p. 10).We have been able to explain the O phenomenon in a more parsimonious way, but still leaving room for complexity, because for O to be present, a combination of the presence of R and the presence of B must occur. In other words, to relate this again to necessity and sufficiency (see p. 10): the presence of R is necessary (but not sufficient) for the outcome; likewise, the presence of B is necessary (but not sufficient) for the outcome. Because neither of the two conditions is sufficient for the outcome, they must be combined (or intersected, through Boolean multiplication, see Box 3.1) and, together, they could possibly 2 form a necessary and sufficient combination of conditions leading to the outcome. Boolean minimization can also be grasped in a more visual way. Let us consider again the same example and provide more explicit labels for the three conditions: respectively, R stands for RIGHT, B stands for BELOW, and I stands for INSIDE. As these are binary conditions, each divides the universe of cases into two parts: those that meet the condition (value 1) and those that do not (value 0). So we have three conditions: RIGHT (1 value) versus not-right (0 value) BELOW (1 value) versus not-below (0 value) INSIDE (1 value) versus not-inside (0 value) Because each condition divides the universe into two parts, the set of three conditions divides it in 2 * 2 * 2 = 8 zones, called elementary zones. Within each zone, the values for conditions of each case are the same. Let us also suppose that we know the value of the outcome variable for each one of these 8 zones. The data can first be presented in tabular format (see Table 3.1). The first column of Table 3.1 contains the case labels ( caseid ), from case1 to case8. The following three columns contain all logically possible combinations (there are 8 of them i.e., 2 3 ; see p. 27) of the binary RIGHT, BELOW, and INSIDE conditions. Finally, the fifth column contains the value of the OUTCOME for each one of the 8 combinations. The first six cases display a [0] outcome (Boolean notation: [outcome]), while the last two display a [1] outcome (Boolean notation: [OUTCOME]). The Boolean expression that was minimized above (see Formulas 1 and 2 in Box 3.2.) corresponds simply to a translation of the two bottom rows of this table.

5 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page 37 CRISP-SET QUALITATIVE COMPARATIVE ANALYSIS (CSQCA) 37 Table 3.1 Raw Data Table (3-Condition Example) caseid Right Below Inside Outcome case case case case case case case case This same data can be represented in a visual way, in a Venn diagram (see also p. 23), showing visually that each condition cuts the universe of cases in two (Figure 3.1) and making the 8 basic zones visible. In this Venn diagram: Condition [RIGHT] cuts the space vertically: All cases on the right-hand side of the vertical line have a (1) value for this condition, whereas all cases on the lefthand side of this line ( not right ) have a (0) value for this condition. Condition [BELOW] cuts the space horizontally: All cases below the horizontal line have a (1) value for this condition, whereas all cases above this line ( not below ) have a (0) value for this condition. Finally, condition [INSIDE] cuts the space between what s inside or outside the square in the middle: All cases inside the square have a (1) value for this condition, whereas all cases outside the square ( not inside ) have a (0) value for this condition. This Venn diagram also contains information about where each case is located. In this simple example, each basic zone is occupied by a single case. In addition, the diagram contains information about the value of the outcome variable. Let us consider only the [1] outcome. It corresponds to the darkshaded area, at the bottom right of the diagram, and corresponds indeed to

6 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page CONFIGURATIONAL COMPARATIVE METHODS case2 case case3. case4. case6 3Inside 011. case Below 010. case5. case8 1 Right Figure 3.1 Venn Diagram Corresponding to Table 3.1 (3 Condition Variables)* * Venn diagram produced by the visualizer tool, TOSMANA software case7 and case8 (as in the last two lines of Table 3.1). This light shaded area can be expressed, in Boolean notation, as follows (Formula 3): RIGHT * BELOW * INSIDE + RIGHT * BELOW * inside OUTCOME Note that this formula is exactly the same as Formula 1. It is read as follows: The outcome is present for case7 [a case that is on the right AND below AND inside] OR for case8 [a case that is on the right AND below AND not inside]. This is a long formulation, which requires the description of each basic zone: It describes first the basic zone where case7 stands, then the one where case8 stands. Technically speaking, this formula contains two terms (each term is a combination of conditions linked by the AND operator), and each term contains all three conditions. This is precisely where the Boolean minimization intervenes: It makes possible describing these two zones with one simpler and shorter expression, consisting of only one term. Indeed, we notice, visually, that the two basic zones where cases 7 and 8 stand, when combined, form

7 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page 39 CRISP-SET QUALITATIVE COMPARATIVE ANALYSIS (CSQCA) 39 a larger square: All the cases that are located below the horizontal line, AND all the cases that are located on the right-hand side of the vertical line. This larger zone can be expressed as (Formula 4): RIGHT * BELOW OUTCOME Note that this formula is exactly the same as Formula 2. It means that we simply need to know about two conditions [RIGHT] and [BELOW] to account for the outcome, which corresponds to cases 7 and 8. We need not know whether or not those cases are inside or not inside the middle square, because this information is superfluous: What is shared by these two cases with a [1] outcome is that they are on the right AND below. So we can simply remove the [INSIDE/not inside] condition. As we shall see later, this is exactly the operation that is performed by the software on more complex data sets, with more conditions, and also with empty basic zones (i.e., with no cases observed). Of course, this makes the operations somewhat more complex, so it is best to let the computer perform the algorithms (in csqca, the software uses the Quine minimization algorithm). Now that we have introduced the basics of Boolean language and operations, we can present key practical steps of csqca. We shall pursue the same example as in Chapter 2, from the inter-war project, where the cases are 18 European countries. STEP 1: BUILDING A DICHOTOMOUS DATA TABLE Of course, building a relevant data table requires previous work: A wellthought-out comparative research design, and in particular rigorous case and variable selection (see Chapter 2). Remember, also, that at this stage the researcher is supposed to have gained adequate substantive knowledge about each case and theoretical knowledge about the most relevant variables (conditions, in particular) included in the analysis. Among the large variety of approaches dealing with the more general conditions favoring the emergence and consolidation of democratic political systems in different parts of the world (see p. 26: list of main categories of conditions in the inter-war project), we have selected one that addresses overall socioeconomic and structural factors. Of course, for a more comprehensive account other factors such as specific historical and cultural conditions, intermediate organizations, institutional arrangements, actor-related aspects, and so on must also be considered (for an application of such a more comprehensive

8 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page CONFIGURATIONAL COMPARATIVE METHODS design, see also Berg-Schlosser, 1998). For purposes of illustration, however, using the selected approach will suffice in short, for the sake of clarity, we need a relatively simplified theory, which does not entail too many conditions. As a reminder, the specific outcome we are trying to explain here is the survival or breakdown of democratic systems in Europe during the inter-war period. Why is it that some democratic systems survived, while others collapsed? In QCA terminology, this variable we seek to explain is called the outcome. The most influential study dealing with the more general socioeconomic preconditions of democracy was S. M. Lipset s Political Man (1960), in particular his chapter Economic Development and Democracy. There, he (re)stated the general hypothesis that the more well-to-do a nation, the greater the chances that it will sustain democracy (p. 31). Indeed, among the stable European democracies analyzed by Lipset were cases like Belgium, the Netherlands, Sweden, and Great Britain, which all showed high levels of wealth, industrialization, education, and urbanization. Under his (very broad) category of unstable democracies and dictatorships figured countries like Greece, Hungary, Italy, Poland, Portugal, and Spain, with lower levels in this regard. However, he also noted that Germany is an example of a nation where growing industrialization, urbanization, wealth and education favoured the establishment of a democratic system, but in which a series of adverse historical events prevented democracy from securing legitimacy and thus weakened its ability to withstand crisis. (p. 20) This statement certainly applies to Austria as well, but the kind of adverse historical events and their specific roots were not investigated by Lipset. Similarly, the fact that countries like Czechoslovakia, Finland, and France (which also had higher levels of development and democratic institutions, and which, as far as internal factors were concerned, survived the economic crisis of the 1930s) were grouped by Lipset in the same unstable category, was not very helpful from an analytical viewpoint. In later years, Lipset s work was followed by a number of conceptually and statistically more refined studies and drew considerable criticism as well. However, when he later reviewed his original study, he still found its basic tenets confirmed (Lipset, 1994; see also Diamond, 1992). For each of the four main dimensions discussed by Lipset (wealth, industrialization, education, and urbanization), we have selected one major indicator as listed in Table 3.2. and have provided the data for each one of the 18 cases (countries) considered. In this example, we have 18 cases (each case being a row in Table 3.2). The outcome variable (SURVIVAL) is already of a dichotomous nature:

9 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page 41 CRISP-SET QUALITATIVE COMPARATIVE ANALYSIS (CSQCA) 41 Table 3.2 Lipset s Indicators, Raw Data (4 Conditions) CASEID GNPCAP URBANIZA LITERACY INDLAB SURVIVAL AUS BEL CZE EST FIN FRA GER GRE HUN IRE ITA NET POL POR ROM SPA SWE UK Labels for conditions: CASEID: Case identification (country name) abbreviations: AUS Austria; BEL Belgium; CZE Czechoslovakia; EST Estonia; FIN Finland; FRA France; GER Germany; GRE Greece; HUN Hungary; IRE Ireland; ITA Italy; NET Netherlands; POL Poland; POR Portugal; ROM Romania; SPA Spain; SWE Sweden; UK United Kingdom GNPCAP: Gross National Product/Capita (ca. 1930) URBANIZA: Urbanization (population in towns with 20,000 and more inhabitants) LITERACY: Literacy INDLAB: Industrial Labor Force (including mining) (Details of sources in Berg-Schlosser & Mitchell, 2000, 2003.)

10 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page CONFIGURATIONAL COMPARATIVE METHODS The [0] outcome value stands for breakdown of democracy (for 10 cases), and the [1] outcome value stands for survival of democracy (for the other 8 cases). However, the four variables that are supposed to explain the outcome, which are called conditions 3 in QCA terminology, are continuous (intervallevel) variables. To be used in csqca, those original conditions must be dichotomized according to relevant thresholds. To dichotomize conditions, it is best to use empirical (case-based) and theoretical knowledge (see good practices box, below). In this example, we have chosen to set the dichotomization thresholds as follows 4 : [GNPCAP]: Gross National Product/Capita (ca. 1930): 0 if below 600 USD; 1 if above. [URBANIZA]: Urbanization (population in towns with 20,000 and more inhabitants): 0 if below 50%; 1 if above. [LITERACY]: 0 if below 75%; 1 if above. [INDLAB]: Industrial Labor Force (incl. mining): 0 if below 30% of active population; 1 if above. Box 3.3 Good Practices (3): How to Dichotomize Conditions in a Meaningful Way Always be transparent when justifying thresholds. It is best to justify the threshold on substantive and/or theoretical grounds. If this is not possible, use technical criteria (e.g., considering the distribution of cases along a continuum; see also p. 79). As a last resort, some more mechanical cutoff points such as the mean or median can be used, but one should check whether this makes sense considering the distribution of the cases. 5 Avoid artificial cuts dividing cases with very similar values. More elaborate technical ways can also be used, such as clustering techniques (see p. 130), but then you should evaluate to what extent the clusters make theoretical or empirical sense. No matter which technique or reasoning you use to dichotomize the conditions, make sure to code the conditions in the correct direction, so that their presence ([1] value) is theoretically expected to be associated with a positive outcome ([1] outcome value).

11 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page 43 CRISP-SET QUALITATIVE COMPARATIVE ANALYSIS (CSQCA) 43 We thus obtain a dichotomized data table (Table 3.3). Already before engaging in csqca proper, exploring this table visually, in a non-formal way, we can easily see that some cases corroborate very neatly the Lipset theory, by looking at the most extreme cases. Table 3.3 Lipset s Indicators, Dichotomized Data (4 Conditions) CASEID GNPCAP URBANIZA LITERACY INDLAB SURVIVAL AUS BEL CZE EST FIN FRA GER GRE HUN IRE ITA NET POL POR ROM SPA SWE UK

12 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page CONFIGURATIONAL COMPARATIVE METHODS For instance, Belgium (row 2) is a perfect survivor case, corroborating Lipset s theory: A [1] value on all four conditions leads to the [1] outcome value [SURVIVAL]. Conversely, Portugal (row 14) is a perfect breakdown case, corroborating Lipset s theory: A [0] value on all four conditions leads to the [0] outcome value [survival]. However, for many other cases, the picture looks more complex. STEP 2: CONSTRUCTING A TRUTH TABLE The first step for which we need csqca proper (in terms of specific software treatment 6 ) corresponds to a first synthesis of the raw data table. The result thereof is called a truth table. It is a table of configurations remember that a configuration is, simply, a given combination of conditions associated with a given outcome. There are five types of configurations, each of which may correspond to none, one, or more than one case. Box 3.4 Five Types of Configurations Configurations with a [1] outcome (among the observed cases); also called 1 configurations. Configurations with a [0] outcome (among the observed cases); also called 0 configurations. Configurations with a ( don t care ) outcome (among the observed cases); also called don t care configurations. It means that the outcome is indeterminate.this is to be avoided, since the researcher is supposed to be interested in explaining a specific outcome across well-selected cases. 7 Configurations with a «C» («contradiction») outcome, called contradictory configurations. Such a configuration leads to a «0» outcome for some observed cases, but to a «1» outcome for other observed cases. This is a logical contradiction, which must be resolved (see p. 48) before engaging further in the csqca. Finally, configurations with an L or R ( logical remainder ) outcome.these are logically possible combinations of conditions that have not been observed among the empirical cases. Table 3.4 displays the truth table corresponding to the dichotomized data in Table 3.3.

13 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page 45 CRISP-SET QUALITATIVE COMPARATIVE ANALYSIS (CSQCA) 45 Table 3.4 Truth Table of the Boolean Configurations CASEID GNPCAP URBANIZA LITERACY INDLAB SURVIVAL SWE, C FRA, AUS FIN, HUN, C POL, EST BEL, NET, C UK, GER CZE ITA, ROM, POR, SPA, GRE IRE Box 3.5 Good Practices (4):Things to Check to Assess the Quality of a Truth Table Check again that there is a mix of cases with a positive outcome and cases with a negative outcome (see Box 2.2, good practices for the case selection). Check that there are no counterintuitive configurations. In this example, these would be configurations in which all [0] condition values lead to a [1] outcome, or all [1] condition values lead to a [0] outcome. Check for cross-condition diversity; in particular, make sure that some conditions do not display exactly the same values across all cases; if they do, ask yourself whether those conditions are too proximate to one another (if they are, they can be merged). Check that there is enough variation for each condition (a general rule: at least 1/3 of each value) (see also Box 2.3, good practices for condition selection: a variable must vary...). If one of these criteria is not met, reconsider your selection of cases and/or conditions or possibly the way you have defined and operationalized the outcome. It is also useful, at this stage, to check for the necessity and sufficiency of each condition with regard to the outcome.

14 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page CONFIGURATIONAL COMPARATIVE METHODS This truth table (Table 3.4) shows only the configurations corresponding to the 18 observed cases. It already allows us to synthesize the evidence substantially, by transforming the 18 cases into 6 configurations. We find out the following: There are two distinct configurations with a [1] outcome, corresponding respectively to Czechoslovakia and Ireland. There is one configuration with a [0] outcome, corresponding to five cases (Italy, Romania, Portugal, Spain, and Greece). It fits quite neatly with the Lipset theory, because a [0] value for all four conditions leads to a [0] outcome (breakdown of democracy). We also notice that there are 3 contradictory configurations (the first 3 rows in the truth table), corresponding to no less than 11 cases out of the 18. In other words: Lipset s theory in the way we have operationalized it, at least does not enable us to account for 11 out of 18 cases. The third contradictory configuration is particularly troubling: It contains [1] values on all of the conditions and yet produces the [0] outcome for one case (namely, Germany), whereas it produces the expected [1] outcome for the other 3 cases (Belgium, Great Britain, and the Netherlands). The data in this truth table can once again be visualized through a Venn diagram, a bit more complex than Figure 3.1 because it contains 4 conditions instead of 3 (Figure 3.2). This Venn diagram has 16 basic zones (configurations) that is, 2 4 zones. It is constructed using the same logic as Figure 3.1. In this empirically grounded example, we can observe four types of configurations: Two configurations with a [1] outcome, covering respectively the cases of Czechoslovakia and Ireland. One configuration with a [0] outcome, covering the five cases of Italy, Romania, Portugal, Spain, and Greece. Three contradictory configurations, covering in all 11 cases (the shaded zones corresponding to the C label). Finally, many non-observed, logical remainder ( R ) configurations 10 altogether. Thus, there is limited diversity (see p. 27) in the data: As the 18 observed cases correspond to only 6 configurations, the remaining Boolean property space is devoid of cases. As will be shown at a later stage, these logical remainder configurations will constitute a useful resource for further analyses. One way to look at this evidence, from a purely numerical perspective, would be to state that the model fits 7 out of 18 cases. This is, however, not a correct

15 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page 47 CRISP-SET QUALITATIVE COMPARATIVE ANALYSIS (CSQCA) GRE, ITA, POR, ROM, SPA INDLAB EST, FIN,. AUS, FRA, HUN, POL SWE. 3LITERACY CZE. BEL, GER, NET, UK IRE URBANIZA GNPCAP 0 1 R C 1100 Figure 3.2 Venn Diagram Corresponding to Table 3.4 (4 Conditions)* * Venn diagram produced by the visualizer tool, TOSMANA software. way to proceed; remember that QCA is a case-oriented method (see p. 6), and that each case matters. From this perspective, it is a problem that so many contradictions occur. Hence these contradictions first have to be resolved before proceeding to the core of csqca namely, Boolean minimization. At this stage of the analysis, it is also useful to check for the necessity and sufficiency of each condition with regard to the outcome. Let us assume a model that contains three conditions, A, B and C. For condition A, for instance, assessing its consistency as a necessary condition means answering the following question: To what extent is the statement condition A is necessary for the outcome consistent? Technically, this can be computed as follows: [the number of cases with a [1] value on the condition AND a [1] outcome value, divided by the total number of cases with a [1] outcome value]. For more details, see Goertz (2006a), Ragin (2006b), and Schneider and Wagemann (2007, forthcoming).

19 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page 51 CRISP-SET QUALITATIVE COMPARATIVE ANALYSIS (CSQCA) 51 We thus obtain a new raw data table (Table 3.5, with one additional column as compared to Table 3.2), which is dichotomized (Table 3.6). The software then produces a new truth table (Table 3.7). Table 3.5 Lipset s Indicators, Raw Data, Plus a Fifth Condition CASEID GNPCAP URBANIZA LITERACY INDLAB GOVSTAB SURVIVAL AUS BEL CZE EST FIN FRA GER GRE HUN IRE ITA NET POL POR ROM SPA SWE UK Labels for conditions: same as Table 3.2, plus a fifth condition: GOVSTAB: Governmental stability (number of cabinets in period) (Case abbreviations and sources: same as Table 3.2.)

20 03-Rihoux-45607:03-Rihoux /18/ :37 PM Page CONFIGURATIONAL COMPARATIVE METHODS Table 3.6 Lipset s Indicators, Dichotomized Data, Plus a Fifth Condition CASEID GNPCAP URBANIZA LITERACY INDLAB GOVSTAB SURVIVAL AUS BEL CZE EST FIN FRA GER GRE HUN IRE ITA NET POL POR ROM SPA SWE UK This truth table (Table 3.7) is richer than the previous one (Table 3.4): By adding a condition, we move from 6 to 10 configurations, so indeed we have added diversity across the cases. This has enabled us to resolve most contradictions. Consider, for instance, the three cases of Austria, Sweden, and France, which formed a contradictory configuration when we considered only the four Lipset conditions (see first row in Table 3.4). By adding the [GOV- STAB] condition, we can now differentiate Austria (first row in Table 3.7), which has a [0] value on [GOVSTAB], from Sweden and France (fifth row in

### What is Qualitative Comparative Analysis (QCA)?

What is Qualitative Comparative Analysis (QCA)? Charles C. Ragin Department of Sociology and Department of Political Science University of Arizona Tucson, AZ 85721 USA www.fsqca.com www.compasss.org www.u.arizona.edu/~cragin

### Lasse Cronqvist. Email: lasse@staff.uni-marburg.de. Tosmana. TOol for SMAll-N Analysis. version 1.2. User Manual

Lasse Cronqvist Email: lasse@staff.uni-marburg.de Tosmana TOol for SMAll-N Analysis version 1.2 User Manual Release: 24 th of January 2005 Content 1. Introduction... 3 2. Installing Tosmana... 4 Installing

### For example, estimate the population of the United States as 3 times 10⁸ and the

CCSS: Mathematics The Number System CCSS: Grade 8 8.NS.A. Know that there are numbers that are not rational, and approximate them by rational numbers. 8.NS.A.1. Understand informally that every number

### QUALITATIVE COMPARATIVE ANALYSIS. Charles C. Ragin Department of Sociology University of Arizona Tucson, AZ 85718. http://www.fsqca.

QUALITATIVE COMPARATIVE ANALYSIS Charles C. Ragin Department of Sociology University of Arizona Tucson, AZ 85718 http://www.fsqca.com http://www.compasss.org http://www.u.arizona.edu/~cragin Why the U-Shaped

### ENVIRONMENTAL EXPENSES AND INNOVATIONS IN CONTRAST WITH ECONOMIC DEVELOPMENT

Institute of Agricultural and Food Economics National Research Institute, Warsaw, Poland ENVIRONMENTAL EXPENSES AND INNOVATIONS IN CONTRAST WITH ECONOMIC DEVELOPMENT Szczepan Figiel INTERNATIONAL CONFERENCE

### International comparisons of road safety using Singular Value Decomposition

International comparisons of road safety using Singular Value Decomposition Siem Oppe D-2001-9 International comparisons of road safety using Singular Value Decomposition D-2001-9 Siem Oppe Leidschendam,

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Using fsqca. A Brief Guide and Workshop for Fuzzy-Set Qualitative Comparative Analysis

Using fsqca A Brief Guide and Workshop for Fuzzy-Set Qualitative Comparative Analysis Ray Kent Department of Marketing University of Stirling r.a.kent@stir.ac.uk 2008 Preface Ray Kent of the University

### South Carolina College- and Career-Ready (SCCCR) Algebra 1

South Carolina College- and Career-Ready (SCCCR) Algebra 1 South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR) Mathematical Process

### Common Core State Standards. Standards for Mathematical Practices Progression through Grade Levels

Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for

### Crisp and Fuzzy Set Qualitative Comparative Analysis (QCA)

Crisp and Fuzzy Set Qualitative Comparative Analysis (QCA) Peer C. Fiss University of Southern California (I thank Charles Ragin for the use of his slides; modified for this workshop) QCA Background QCA

### Pythagorean Theorem. Overview. Grade 8 Mathematics, Quarter 3, Unit 3.1. Number of instructional days: 15 (1 day = minutes) Essential questions

Grade 8 Mathematics, Quarter 3, Unit 3.1 Pythagorean Theorem Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Prove the Pythagorean Theorem. Given three side lengths,

### B Financial and Human Resources

Chapter B Financial and Human Resources Invested In Education Education at a Glance OECD 2011 203 chapter B Classification of al expenditure Educational expenditure in this chapter is classified through

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### PA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis*

Habits of Mind of a Productive Thinker Make sense of problems and persevere in solving them. Attend to precision. PA Common Core Standards The Pennsylvania Common Core Standards cannot be viewed and addressed

### 2. Propositional Equivalences

2. PROPOSITIONAL EQUIVALENCES 33 2. Propositional Equivalences 2.1. Tautology/Contradiction/Contingency. Definition 2.1.1. A tautology is a proposition that is always true. Example 2.1.1. p p Definition

### Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

### Lucerne, Switzerland 13th-15th July 2007 OVERALL STANDING

OVERALL STANDING Rk Nation Pts. LinzAmster. Lucern W1x M1x W2- M2- W2x M2x M4- LW2x LM2x LM4- W4x M4x W8+ M8+ 1 GBR 161 62 40 59 6 4 4 6 6 3 5 8 8 1 4 4 2 GER 112 47 32 33 6 5 6 4 6 6 3 CHN 101 38 58 5

### NERI Quarterly Economic Facts Summer 2012. 4 Distribution of Income and Wealth

4 Distribution of Income and Wealth 53 54 Indicator 4.1 Income per capita in the EU Indicator defined National income (GDP) in per capita (per head of population) terms expressed in Euro and adjusted for

### Government at a Glance 2015

Government at a Glance 2015 Size of public procurement Strategic public procurement E-procurement Central purchasing bodies 135 Size of public procurement Public procurement refers to the purchase by governments

### Introduction to. Hypothesis Testing CHAPTER LEARNING OBJECTIVES. 1 Identify the four steps of hypothesis testing.

Introduction to Hypothesis Testing CHAPTER 8 LEARNING OBJECTIVES After reading this chapter, you should be able to: 1 Identify the four steps of hypothesis testing. 2 Define null hypothesis, alternative

### Fairfield Public Schools

Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### Evaluation Tool for Assessment Instrument Quality

REPRODUCIBLE Figure 4.4: Evaluation Tool for Assessment Instrument Quality Assessment indicators Description of Level 1 of the Indicator Are Not Present Limited of This Indicator Are Present Substantially

### Dualization and crisis. David Rueda

Dualization and crisis David Rueda The economic crises of the 20 th Century (from the Great Depression to the recessions of the 1970s) were met with significant increases in compensation and protection

### This chapter is all about cardinality of sets. At first this looks like a

CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### Introduction to Logic Circuits

April 5, 999 4:05 g02-ch2 Sheet number Page number 7 black chapter 2 Introduction to Logic Circuits 2. d2 d4, d7 d5 7 April 5, 999 4:05 g02-ch2 Sheet number 2 Page number 8 black 8 CHAPTER 2 Introduction

### Writing the Empirical Social Science Research Paper: A Guide for the Perplexed. Josh Pasek. University of Michigan.

Writing the Empirical Social Science Research Paper: A Guide for the Perplexed Josh Pasek University of Michigan January 24, 2012 Correspondence about this manuscript should be addressed to Josh Pasek,

### DEBT LEVELS AND FISCAL FRAMEWORKS. Christian Kastrop Director of Policy Studies Branch Economics Department

DEBT LEVELS AND FISCAL FRAMEWORKS Christian Kastrop Director of Policy Studies Branch Economics Department Introduction OECD average gross government debt increased from 73% of GDP in 2007 to 111% in 2013.

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### Mathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades

Appendix A Mathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades To respond correctly to TIMSS test items, students need to be familiar with the mathematics

### The Alignment of Common Core and ACT s College and Career Readiness System. June 2010

The Alignment of Common Core and ACT s College and Career Readiness System June 2010 ACT is an independent, not-for-profit organization that provides assessment, research, information, and program management

### Problem of the Month: Fair Games

Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

### The Scientific Data Mining Process

Chapter 4 The Scientific Data Mining Process When I use a word, Humpty Dumpty said, in rather a scornful tone, it means just what I choose it to mean neither more nor less. Lewis Carroll [87, p. 214] In

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### It is time to prove some theorems. There are various strategies for doing

CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it

### CHAPTER 2. Logic. 1. Logic Definitions. Notation: Variables are used to represent propositions. The most common variables used are p, q, and r.

CHAPTER 2 Logic 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A proposition is a declarative sentence that is either true (denoted either T or 1) or false (denoted either F or 0). Notation:

### KNOWLEDGE FACTORING USING NORMALIZATION THEORY

KNOWLEDGE FACTORING USING NORMALIZATION THEORY J. VANTHIENEN M. SNOECK Katholieke Universiteit Leuven Department of Applied Economic Sciences Dekenstraat 2, 3000 Leuven (Belgium) tel. (+32) 16 28 58 09

### Connections to the Common Core State Standards for Literacy in Middle and High School Science

Connections to the Common Core State Standards for Literacy in Middle and High School Science This document is based on the Connections to the Common Core State Standards for Literacy in Science and Technical

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### FUNCTIONAL EXPLORATORY DATA ANALYSIS OF UNEMPLOYMENT RATE FOR VARIOUS COUNTRIES

QUANTITATIVE METHODS IN ECONOMICS Vol. XIV, No. 1, 2013, pp. 180 189 FUNCTIONAL EXPLORATORY DATA ANALYSIS OF UNEMPLOYMENT RATE FOR VARIOUS COUNTRIES Stanisław Jaworski Department of Econometrics and Statistics

### A Concrete Introduction. to the Abstract Concepts. of Integers and Algebra using Algebra Tiles

A Concrete Introduction to the Abstract Concepts of Integers and Algebra using Algebra Tiles Table of Contents Introduction... 1 page Integers 1: Introduction to Integers... 3 2: Working with Algebra Tiles...

### An Innocent Investigation

An Innocent Investigation D. Joyce, Clark University January 2006 The beginning. Have you ever wondered why every number is either even or odd? I don t mean to ask if you ever wondered whether every number

### Private Health insurance in the OECD

Private Health insurance in the OECD Benefits and costs for individuals and health systems Francesca Colombo, OECD AES, Madrid, 26-28 May 2004 http://www.oecd.org/health 1 Outline Q Background, method

### What do the recent regional GDP statistics tell us about Cohesion?

Analysis from the CPMR Secretariat July 2015 What do the recent regional GDP statistics tell us about Cohesion? Headline messages - Recent regional GDP statistics reveal that regional disparities are on

### (Refer Slide Time 6:09)

Digital Circuits and Systems Prof. S. Srinivasan Department of Electrical Engineering Indian Institute of Technology, Madras Lecture 3 Combinational Logic Basics We defined digital signal as a signal which

### Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary

Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:

### Voting in the EU Council A Scientific Approach

Voting in the EU Council A Scientific Approach Werner Kirsch a, Moshé Machover b, Wojciech Słomczyński c and Karol Życzkowski d In this note we propose a voting system for the EU Council of Ministers which

Flash Eurobarometer BUSINESS-TO-BUSINESS ALTERNATIVE DISPUTE RESOLUTION IN THE EU REPORT Fieldwork: March-April 22 Publication: November 22 This survey has been requested by Directorate-General for Justice

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Basic Concepts in Research and Data Analysis

Basic Concepts in Research and Data Analysis Introduction: A Common Language for Researchers...2 Steps to Follow When Conducting Research...3 The Research Question... 3 The Hypothesis... 4 Defining the

### Prentice Hall Connected Mathematics 2, 7th Grade Units 2009

Prentice Hall Connected Mathematics 2, 7th Grade Units 2009 Grade 7 C O R R E L A T E D T O from March 2009 Grade 7 Problem Solving Build new mathematical knowledge through problem solving. Solve problems

### The Quality of the Catalan and Spanish Education Systems: A Perspective from PISA

The Quality of the Catalan and Spanish Education Systems: A Perspective from PISA by Antonio Ciccone and Walter Garcia-Fontes* October 2008 * UPF AND ICREA (Ciccone) and UPF (Garcia-Fontes). Executive

### PRINCIPLES FOR EVALUATION OF DEVELOPMENT ASSISTANCE

PRINCIPLES FOR EVALUATION OF DEVELOPMENT ASSISTANCE DEVELOPMENT ASSISTANCE COMMITTEE PARIS, 1991 DAC Principles for Evaluation of Development Assistance Development Assistance Committee Abstract: The following

### Private Health insurance in the OECD

Private Health insurance in the OECD Benefits and costs for individuals and health systems Francesca Colombo, OECD AES, Madrid, 26-28 May 2003 http://www.oecd.org/health 1 Outline Background, method Overview

### Students in their first advanced mathematics classes are often surprised

CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are

### The equation for the 3-input XOR gate is derived as follows

The equation for the 3-input XOR gate is derived as follows The last four product terms in the above derivation are the four 1-minterms in the 3-input XOR truth table. For 3 or more inputs, the XOR gate

### MAT2400 Analysis I. A brief introduction to proofs, sets, and functions

MAT2400 Analysis I A brief introduction to proofs, sets, and functions In Analysis I there is a lot of manipulations with sets and functions. It is probably also the first course where you have to take

### Special Feature: Trends in personal income tax and employee social security contribution schedules

Special Feature: Trends in personal income tax and employee social security contribution schedules TAXING WAGES 211 OECD 212 27 1. Introduction Taxes on labour income including personal income taxes and

### A Note on the Optimal Supply of Public Goods and the Distortionary Cost of Taxation

A Note on the Optimal Supply of Public Goods and the Distortionary Cost of Taxation Louis Kaplow * Abstract In a recent article, I demonstrated that, under standard simplifying assumptions, it is possible

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### EUROPEAN YOUTH: PARTICIPATION IN DEMOCRATIC LIFE

Flash Eurobarometer EUROPEAN YOUTH: PARTICIPATION IN DEMOCRATIC LIFE REPORT Fieldwork: April 2013 Publication: May 2013 This survey has been requested by the European Commission, Directorate-General for

### Understanding the Progression of Math Courses in NEISD

Understanding the Progression of Math Courses in NEISD According to House Bill 1 (HB1), students in Texas are required to obtain credits for four courses in each subject area of the foundation curriculum

### Prentice Hall Mathematics: Algebra 1 2007 Correlated to: Michigan Merit Curriculum for Algebra 1

STRAND 1: QUANTITATIVE LITERACY AND LOGIC STANDARD L1: REASONING ABOUT NUMBERS, SYSTEMS, AND QUANTITATIVE SITUATIONS Based on their knowledge of the properties of arithmetic, students understand and reason

NOTE: IMMIGRATION TO CANADA 1928-1971 RANK COUNTRY OF ORIGIN TOTALS 1 United Kingdom * 1152415 2 United States of America 527346 3 Italy 471940 4 Germany 370641 5 Netherlands 185664 6 Poland 117244 7 Greece

### South Carolina College- and Career-Ready (SCCCR) Probability and Statistics

South Carolina College- and Career-Ready (SCCCR) Probability and Statistics South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR)

### NGSS Science & Engineering Practices Grades K-2

ACTIVITY 8: SCIENCE AND ENGINEERING SELF-ASSESSMENTS NGSS Science & Engineering Practices Grades K-2 1 = Little Understanding; 4 = Expert Understanding Practice / Indicator 1 2 3 4 NOTES Asking questions

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### HB 5 College Preparatory Math Content Framework

Target Students: This course is appropriate for 12 th grade students whose performance on measures outlined in TEC 28.014 indicates that the student is not ready to perform entry-level college coursework

### Chapter 2 Conceptualizing Scientific Inquiry

Chapter 2 Conceptualizing Scientific Inquiry 2.1 Introduction In order to develop a strategy for the assessment of scientific inquiry in a laboratory setting, a theoretical construct of the components

### COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

### CSI 24: How do Europeans differ in their attitudes to immigration?

Summary CSI 24: How do Europeans differ in their attitudes to immigration? Overall, from our sample of 21 European countries, Sweden has the most positive attitudes to immigration and the Czech Republic

### How close? An attempt at measuring the cultural distance between countries

Gustavo De Santis gustavo.desantis@unifi.it Mauro Maltagliati mauro.maltagliati@unifi.it How close? Silvana Salvini silvana.salvini@unifi.it An attempt at measuring the cultural distance between countries

### UNEMPLOYMENT BENEFITS WITH A FOCUS ON MAKING WORK PAY

EUROPEAN SEMESTER THEMATIC FICHE UNEMPLOYMENT BENEFITS WITH A FOCUS ON MAKING WORK PAY Thematic fiches are supporting background documents prepared by the services of the Commission in the context of the

### Lecture 14. Chapter 7: Probability. Rule 1: Rule 2: Rule 3: Nancy Pfenning Stats 1000

Lecture 4 Nancy Pfenning Stats 000 Chapter 7: Probability Last time we established some basic definitions and rules of probability: Rule : P (A C ) = P (A). Rule 2: In general, the probability of one event

### Data Analysis, Statistics, and Probability

Chapter 6 Data Analysis, Statistics, and Probability Content Strand Description Questions in this content strand assessed students skills in collecting, organizing, reading, representing, and interpreting

### Simple Random Sampling

Source: Frerichs, R.R. Rapid Surveys (unpublished), 2008. NOT FOR COMMERCIAL DISTRIBUTION 3 Simple Random Sampling 3.1 INTRODUCTION Everyone mentions simple random sampling, but few use this method for

### Simplifying Logic Circuits with Karnaugh Maps

Simplifying Logic Circuits with Karnaugh Maps The circuit at the top right is the logic equivalent of the Boolean expression: f = abc + abc + abc Now, as we have seen, this expression can be simplified

### Facts: Population Baldwin & Wyplosz 2006

Facts: Population Facts: Population 6 big nations: > 35 million (Germany, the UK, France, Italy, Spain and Poland). Netherlands: 16 million people. 8 small nations (size of a big city): 8 to 11 million:

### SOCIOECONOMIC INEQUALITIES IN HEALTH IN EUROPE. Johan Mackenbach Department of Public Health Erasmus MC

SOCIOECONOMIC INEQUALITIES IN HEALTH IN EUROPE Johan Mackenbach Department of Public Health Erasmus MC ECONOMIC PROSPERITY average income per inhabitant, 2002 World Bank 2004 LIFE EXPECTANCY at birth,

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### What is the wealth of the United States? Who s got it? And how

Introduction: What Is Data Analysis? What is the wealth of the United States? Who s got it? And how is it changing? What are the consequences of an experimental drug? Does it work, or does it not, or does

### Pan-European opinion poll on occupational safety and health

PRESS KIT Pan-European opinion poll on occupational safety and health Results across 36 European countries Press kit Conducted by Ipsos MORI Social Research Institute at the request of the European Agency

### In this section, you will develop a method to change a quadratic equation written as a sum into its product form (also called its factored form).

CHAPTER 8 In Chapter 4, you used a web to organize the connections you found between each of the different representations of lines. These connections enabled you to use any representation (such as a graph,

### Problem of the Month Through the Grapevine

The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems

### 4 Distribution of Income, Earnings and Wealth

4 Distribution of Income, Earnings and Wealth Indicator 4.1 Indicator 4.2a Indicator 4.2b Indicator 4.3a Indicator 4.3b Indicator 4.4 Indicator 4.5a Indicator 4.5b Indicator 4.6 Indicator 4.7 Income per

### INNOVATION IN THE PUBLIC SECTOR: ITS PERCEPTION IN AND IMPACT ON BUSINESS

Flash Eurobarometer INNOVATION IN THE PUBLIC SECTOR: ITS PERCEPTION IN AND IMPACT ON BUSINESS REPORT Fieldwork: February-March 22 Publication: June 22 This survey has been requested by the European Commission,

### Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets.

Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets. Logic is the study of reasoning. A language of propositions is fundamental to this study as well as true

### VULNERABILITY OF SOCIAL INSTITUTIONS

VULNERABILITY OF SOCIAL INSTITUTIONS 2 December 2014 Paris Seminar in Demographic Economics Falilou FALL Senior Economist OECD Economics Department 1 Introduction and outline Social institutions and the

### Research Briefing. The Best and the Brightest EU students at UK universities and as highly skilled graduate workers in the UK

Research Briefing The Best and the Brightest EU students at UK universities and as highly skilled graduate workers in the UK Academic performance and labour market outcomes of EU domiciled students in

### Transfer issues and directions for reform: Australian transfer policy in comparative perspective

Transfer issues and directions for reform: Australian transfer policy in comparative perspective Peter Whiteford, Social Policy Research Centre, University of New South Wales p.whiteford@unsw.edu.au 1

### How PRINCE2 Can Complement PMBOK and Your PMP Jay M. Siegelaub Impact Strategies LLC. Abstract. About PRINCE2

How PRINCE2 Can Complement PMBOK and Your PMP Jay M. Siegelaub Impact Strategies LLC Abstract PMBOK is the recognized (de facto) standard of project management knowledge. In the UK and Europe, PRINCE2

### The Logical Framework Approach An Introduction 1

The Logical Framework Approach An Introduction 1 1. What is the Logical Framework Approach? 1.1. The background The Logical Framework Approach (LFA) was developed in the late 1960 s to assist the US Agency

### THE ELECTRONIC CUSTOMS IMPLEMENTATION IN THE EU

Flash Eurobarometer THE ELECTRONIC CUSTOMS IMPLEMENTATION IN THE EU REPORT Fieldwork: April-May 214 Publication: October 214 This survey has been requested by the European Commission, Directorate-General

### VISUAL ALGEBRA FOR COLLEGE STUDENTS. Laurie J. Burton Western Oregon University

VISUAL ALGEBRA FOR COLLEGE STUDENTS Laurie J. Burton Western Oregon University VISUAL ALGEBRA FOR COLLEGE STUDENTS TABLE OF CONTENTS Welcome and Introduction 1 Chapter 1: INTEGERS AND INTEGER OPERATIONS