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**Chapter 14 Quantitative Data Analysis**

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Introduction Data analysis is an integral component of research methods, and it’s important that any proposal for quantitative research include a plan for the data analysis that will follow data collection. You have to anticipate your data analysis needs if you expect your research design to secure the requisite data.

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**Introducing Statistics**

Statistics play a key role in achieving valid research results, in terms of measurement, causal validity, and generalizability. Some statistics are useful primarily to describe the results of measuring single variables and to construct and evaluate multi-item scales. (Univariate Statistics) These statistics include frequency distributions, graphs, measures of central tendency and variation, and reliability tests.

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**Introducing Statistics, cont.**

Other statistics are useful primarily in achieving causal validity, by helping us to describe the association among variables and to control for, or otherwise take account of, other variables. (Bivariate and Multivariate Statistics) Crosstabulation is the technique for measuring association and controlling other variables. All of these statistics are termed descriptive statistics because they are used to describe the distribution of, and relationship among, variables.

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**Introducing Statistics, cont.**

It is possible to estimate the degree of confidence that can be placed in generalization from a sample to the population from which the sample was selected. The statistics used in making these estimates are termed inferential statistics. It is also important to choose statistics that are appropriate to the level of measurement of the variables to be analyzed.

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**Preparing Data for Analysis**

Using secondary data in this way has a major disadvantage: If you did not design the study yourself, it is unlikely that all the variables that you think should have been included actually were included and were measured in the way that you prefer. In addition, the sample may not represent just the population in which you are interested, and the study design may be only partially appropriate to your research question. It is the availability of secondary data that makes their use preferable for many purposes.

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**Preparing Data for Analysis, cont.**

If you have conducted your own survey or experiment, your quantitative data must be prepared in a format suitable for computer entry. Several options are available. Questionnaires or other data entry forms can be designed for scanning or direct computer entry. Once the computer database software is programmed to recognize the response codes, the forms can be fed through a scanner and the data will then be entered directly into the database.

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**Preparing Data for Analysis, cont.**

Whatever data entry method is used, the data must be checked carefully for errors—a process called data cleaning. Most survey research organizations now use a database management program to control data entry. Such programs as SPSS (Statistical Package for Social Sciences),SAS,CRISP,NCSS

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**Displaying Univariate Distributions**

Graphs and frequency distributions are the two most popular approaches; both allow the analyst to display the distribution of cases across the categories of a variable. Graphs have the advantage of providing a picture that is easier to comprehend, although frequency distributions are preferable when exact numbers of cases having particular values must be reported and when many distributions must be displayed in a compact form.

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**Displaying Univariate Distributions, cont.**

Three features of shape are important: Central tendency The most common value (for variables measured at the nominal level) or the value around which cases tend to center (for a quantitative variable). Types are Mean, Median and Mode Variability The extent to which cases are spread out through the distribution or clustered in just one location. Skewness The extent to which cases are clustered more at one or the other end of the distribution of a quantitative variable rather than in a symmetric pattern around its center.

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Graphs Even for the uninitiated, graphs can be easy to read, and they highlight a distribution’s shape. They are useful particularly for exploring data because they show the full range of variation and identify data anomalies that might be in need of further study. And good, professional-looking graphs can now be produced relatively easily with software available for personal computers. (EXCEL)

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**Graphs, cont. The most common types of graphs:**

A bar chart contains solid bars separated by spaces. It is a good tool for displaying the distribution of variables measured at the nominal level because there is, in effect, a gap between each of the categories. Histograms, in which the bars are adjacent, are used to display the distribution of quantitative variables that vary along a continuum that has no necessary gaps. In a frequency polygon, a continuous line connects the points representing the number or percentage of cases with each value.

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Exhibit 14.5

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Exhibit 14.6

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Exhibit 14.7

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**Frequency Distributions**

A frequency distribution displays the number of, percentage (the relative frequencies) of, or both cases corresponding to each of a variable’s values or group of values. Ungrouped Data—Constructing and reading frequency distributions for variables with few values is not difficult. Grouped Data—Many frequency distributions (and graphs) require grouping of some values after the data are collected.

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Exhibit 14.9

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Exhibit 14.10

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Exhibit 14.11

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**Combined and Compressed Distributions**

In a combined frequency display, the distributions for a set of conceptually similar variables having the same response categories are presented together. Compressed frequency displays can also be used to present crosstabular data and summary statistics more efficiently, by eliminating unnecessary percentages and by reducing the need for repetitive labels.

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Exhibit 14.14

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**Summarizing Univariate Distributions**

Summary statistics focus attention on particular aspects of a distribution and facilitate comparison among distributions. For example, if your purpose were to report variation in income by state in a form that is easy for most audiences to understand, you would usually be better off presenting average incomes. Of course, representing a distribution in one number loses information about other aspects of the distribution’s shape and so creates the possibility of obscuring important information.

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**Measures of Central Tendency**

Central tendency is usually summarized with one of three statistics: the mode, the median, or the mean. For any particular application, one of these statistics may be preferable, but each has a role to play in data analysis. To choose an appropriate measure of central tendency, the analyst must consider a variable’s level of measurement, the skewness of a quantitative variable’s distribution, and the purpose for which the statistic is used. In addition, the analyst’s personal experiences and preferences inevitably will play a role.

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**Measures of Central Tendency, cont.**

The mode is the most frequent value in a distribution. It is also termed the probability average because, being the most frequent value, it is the most probable. The mode is used much less often than the other two measures of central tendency because it can so easily give a misleading impression of a distribution’s central tendency. One problem with the mode occurs when a distribution is bimodal, in contrast to being unimodal. A bimodal (or trimodal, and so on) distribution has two or more categories with an equal number of cases and with more cases than any of the other categories.

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**Measures of Central Tendency, cont.**

The median is the position average, or the point that divides the distribution in half (the 50th percentile). The median is inappropriate for variables measured at the nominal level because their values cannot be put in order, and so there is no meaningful middle position. To determine the median, we simply array a distribution’s values in numerical order and find the value of the case that has an equal number of cases above and below it.

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**Measures of Central Tendency, cont.**

The mean, or arithmetic average, takes into account the values of each case in a distribution—it is a weighted average. The mean is computed by adding up the value of all the cases and dividing by the total number of cases, thereby taking into account the value of each case in the distribution: Mean = Sum of value of cases/Number of cases

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Median or Mean? Both the median and the mean are used to summarize the central tendency of quantitative variables, but their suitability for a particular application must be carefully assessed. The key issues to be considered in this assessment are the variable’s level of measurement, the shape of its distribution, and the purpose of the statistical summary. Consideration of these issues will sometimes result in a decision to use both the median and the mean and will sometimes result in neither measure being seen as preferable.

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Exhibit 14.16

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Measures of Variation You already have learned that central tendency is only one aspect of the shape of a distribution—the most important aspect for many purposes but still just a piece of the total picture. A summary of distributions based only on their central tendency can be very incomplete, even misleading.

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**Measures of Variation, cont.**

The way to capture these differences is with statistical measures of variation. Four popular measures of variation are the range, the interquartile range, the variance, and the standard deviation (which is the most popular measure of variability). To calculate each of these measures, the variable must be at the interval or ratio level (but many would argue that, like the mean, they can be used with ordinal-level measures, too). It’s important to realize that measures of variability are summary statistics that capture only part of what we need to be concerned with about the distribution of a variable.

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Range The range is a simple measure of variation, calculated as the highest value in a distribution minus the lowest value, plus 1: Range = Highest value - Lowest value + 1 It often is important to report the range of a distribution to identify the whole range of possible values that might be encountered. However, because the range can be drastically altered by just one exceptionally high or low value (termed an outlier), it does not do an adequate job of summarizing the extent of variability in a distribution.

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Interquartile Range A version of the range statistic, the interquartile range, avoids the problem created by outliers. Quartiles are the points in a distribution corresponding to the first 25% of the cases, the first 50% of the cases, and the first 75% of the cases. You already know how to determine the second quartile, corresponding to the point in the distribution covering half of the cases—it is another name for the median.

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**Interquartile Range, cont.**

The first and third quartiles are determined in the same way but by finding the points corresponding to 25% and 75% of the cases, respectively. The interquartile range is the difference between the first quartile and the third quartile (plus 1). Third quartile - first quartile + 1 = Interquartile range.

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Variance The variance is the average squared deviation of each case from the mean, so it takes into account the amount by which each case differs from the mean. The variance is used in many other statistics, although it is more conventional to measure variability with the closely related standard deviation than with the variance.

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Standard Deviation The standard deviation is simply the square root of the variance. It is the square root of the average squared deviation of each case from the mean.

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Exhibit 14.19

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**Analyzing Data Ethically: How Not to Lie with Statistics**

Using statistics ethically means first and foremost being honest and open. Findings should be reported honestly, and the researcher should be open about the thinking that guided her decision to use particular statistics. It is possible to distort social reality with statistics, and it is unethical to do so knowingly, even when the error is due more to carelessness than deceptive intent.

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**Analyzing Data Ethically: How Not to Lie with Statistics, cont.**

Summary statistics can easily be used unethically, knowingly or not. When we summarize a distribution in a single number, even in two numbers, we are losing much information. It is possible to mislead those who read statistical reports by choosing summary statistics that accentuate a particular feature of a distribution.

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Exhibit 14.20

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**Crosstabulating Variables**

Most data analyses focus on relationships among variables in order to test hypotheses or just to describe or explore relationships. For each of these purposes, we must examine the association among two or more variables. Crosstabulation (crosstab) is one of the simplest methods for doing so. A crosstabulation, or contingency table, displays the distribution of one variable for each category of another variable; it can also be termed a bivariate distribution.

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**Crosstabulating Variables, cont.**

You can also display the association between two variables in a graph In addition, crosstabs provide a simple tool for statistically controlling one or more variables while examining the associations among others.

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Graphing Association Graphs provide an efficient tool for summarizing relationships among variables.

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**Describing Association**

A crosstabulation table reveals four aspects of the association between two variables: Existence. Do the percentage distributions vary at all between categories of the independent variable? Strength. How much do the percentage distributions vary between categories of the independent variable?

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**Describing Association, cont.**

Direction. For quantitative variables, do values on the dependent variable tend to increase or decrease with an increase in value on the independent variable? Pattern. For quantitative variables, are changes in the percentage distribution of the dependent variable fairly regular (simply increasing or decreasing), or do they vary (perhaps increasing, then decreasing, or perhaps gradually increasing, then rapidly increasing)?

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Exhibit 14.27

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**Evaluating Association**

You will find when you read research reports and journal articles that social scientists usually make decisions about the existence and strength of association on the basis of more statistics than just a crosstabulation table. A measure of association is a type of descriptive statistics used to summarize the strength of an association. There are many measures of association, some of which are appropriate for variables measured at particular levels. One popular measure of association in crosstabular analyses with variables measured at the ordinal level is gamma.

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**Evaluating Association, cont.**

Inferential statistics are used in deciding whether it is likely that an association exists in the larger population from which the sample was drawn. Estimation of the probability that an association is not due to chance will be based on one of several inferential statistics, chi-square being the one used in most cross-tabular analyses. Chi-square An inferential statistic used to test hypotheses about relationships between two or more variables in a crosstabulation.

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**Evaluating Association, cont.**

When the analyst feels reasonably confident (at least 95% confident) that an association was not due to chance, it is said that the association is statistically significant. Statistical significance means that an association is not likely to be due to chance, according to some criterion set by the analyst.

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**Evaluating Association, cont.**

But statistical significance is not everything. Sampling error decreases as sample size increases. For this same reason, an association is less likely to appear on the basis of chance in a larger sample than in a smaller sample.

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**Controlling For a Third Variable**

Crosstabulation can also be used to study the relationship between two variables while controlling for other variables. Three different uses for three-variable crosstabulation:

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**Controlling For a Third Variable, cont.**

Testing a relationship for possible spuriousness helps to meet the nonspuriousness criterion for causality. Identifying an intervening variable can help to chart the causal mechanism by which variation in the independent variable influences variation in the dependent variable. Specifying the conditions when a relationship occurs can help to improve our understanding of the nature of that relationship.

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Exhibit 14.31

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Exhibit 14.34

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Regression Analysis In order to read most statistical reports and to conduct more sophisticated analyses of social data, you will have to extend your statistical knowledge. Many statistical reports and articles published in social science journals use a statistical technique called regression analysis or correlational analysis to describe the association between two or more quantitative variables. The terms actually refer to different aspects of the same technique.

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**Analyzing Data Ethically: How Not to Lie About Relationships**

When the data analyst begins to examine relationships among variables in some real data, social science research becomes most exciting. The moment of truth, it would seem, has arrived. Either the hypotheses are supported or they are not. But, in fact, this is also a time to proceed with caution and to evaluate the analyses of others with even more caution.

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**Analyzing Data Ethically: How Not to Lie About Relationships, cont.**

This range of possibilities presents a great hazard for data analysis. It becomes tempting to search around in the data until something interesting emerges. Rejected hypotheses are forgotten in favor of highlighting what’s going on in the data. It’s not wrong to examine data for unanticipated relationships; the problem is that inevitably some relationships among variables will appear just on the basis of chance association alone.

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**Analyzing Data Ethically: How Not to Lie About Relationships, cont.**

If you search hard and long enough, it will be possible to come up with something that really means nothing. Serendipitous findings do not need to be ignored, but they must be reported as such. Subsequent researchers can try to test deductively the ideas generated by our explorations. It is also important to understand the statistical techniques we are using and to use them appropriately.

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Conclusions We have demonstrated how a researcher can describe social phenomena, identify relationships among them, explore the reasons for these relationships, and test hypotheses about them. Statistics provide a remarkably useful tool for developing our understanding of the social world, a tool that we can use both to test our ideas and to generate new ones.

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Conclusions, cont. The numbers will be worthless if the methods used to generate the data are not valid; and the numbers will be misleading if they are not used appropriately, taking into account the type of data to which they are applied. And even assuming valid methods and proper use of statistics, there’s one more critical step, because the numbers do not speak for themselves. Ultimately, it is how we interpret and report the statistics that determines their usefulness.

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