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# Definitions of Key Terms in Statistical Data Analysis

One of the keys to understanding a specialized field is getting to know its technical vocabulary. As you continue to put together your research project, you might come across unfamiliar words. The following words often appear in quantitative studies. Remember to refer to them as needed.

Alternative hypothesis: The hypothesis that is accepted if the null hypothesis is rejected.

Analysis of variance (ANOVA): A statistical method for determining the significance of the differences among a set of sample means.

Aggregated data: Data for which individual scores on a measure are combined into a single group summary score.

Association: is the connection or relationship of two or more concepts or variables.  Two variables have an association if the distribution of one variable changes in concert with the other.  See also   causation,   correlation.

Bias is any situation in which the accuracy, reliability, validity, etc., of data, findings, or conclusions are distorted by the researcher’s or theorist’s methods or presuppositions (e.g., moral, political, religious beliefs or ideologies). In statistical analysis, bias is a technical term for a difference between a hypothetical true value of a variable in a population and the observed value in a particular sample.

Causation: the principle that one variable (X) produces change in another variable (Y). It is based on the assumption that events occur in a predictable, nonrandom way, and that one event leads to, or causes, another. To establish causation, the two variables must be associated or correlated with each other; the first variable (X) must precede the second variable (Y) in time and space; and alternative, noncausal explanations for the relationship (such as spurious ones) must be eliminated. Events in the physical and social worlds are generally too complex to be explained by any single factor. For this reason, scientists are guided by the principle of multiple causation, which states that one event occurs as a result of several factors operating or occurring in combination.

Central limit theorem: A mathematical conjecture that informs us that the sampling distribution of the mean approaches a normal curve as the sample size, n, gets larger.

Chi-square (2 ) distribution: A continuous probability distribution used directly or indirectly in many tests of significance. The most common use of the chi-square distribution is to test differences between proportions. Although this test is by no means the only test based on the chi-square distribution, it has come to be known as the chi-square test. The chi-square distribution has one parameter, its degrees of freedom (df). It has a positive skew; the skew is less with more degrees of freedom. The mean of a chi- square distribution is its df, the mode is df – 2, and the median is approximately df – 0.7.

Concept: a word or set of words that expresses a general idea about the nature of something.

Conceptualization: the mental process whereby ambiguous and imprecise notions are made clear and more precise. A conceptual definition states the meaning of a concept.

Confidence interval: A range of values used to estimate some population parameter with a specific level of confidence. In most statistical tests, confidence levels are 95% or 99%. The wider the confidence interval, the higher the confidence level will be.

Confidence level: A desired percentage of scores (often 95% or 99%) that the true parameter would fall within a certain range. If a study indicates that the Democratic candidate will capture 75% of the vote with a 3% margin of error (or confidence interval) at the 95% level of confidence, then the Democratic candidate can be 95% sure that she will capture between 72 and 75% of the votes.

Confounding variable: An extraneous variable that is not a focus of the study but is statistically related to (or correlated with) the independent variable. This means that as the independent variable changes, the confounding variable changes along with it.

Correlation: A mutual relationship or association of two or more concepts        or variables, such that when one changes in value, the other one doe also. Variables may be correlated positively (i.e., they change in the     same direction) or negatively (that is, they change in the opposite  direction). Correlation is necessary but not sufficient to demonstrate causation.

Correlation coefficient: A measurement between -1 and +1 indicating the strength of the relationship between two variables.

Critical region: The area of the sampling distribution that covers the value of the test statistic that is not due to chance variation. In most tests it represents between 1 and 5% of the graph of the distribution.

Critical value: The value from a sampling distribution that separates chance variation to variation that is not due to chance.

Cronbach’s alpha: This is a measure of internal reliability or consistency of the items in an index. Used often with tests that employ Likert-type scales. Values range from 0 to 1.0. Scores toward the high end of that range (above 0.70) suggest that the items in an index are measuring the same thing.

Data: Facts and figures collected through research. The word data is plural, just like the word “toys.” Data are us. 🙂 Datum is the singular form of data.

Deduction and Induction are processes of logical reasoning.  Deduction involves reasoning from general principles to particular instances.  In other words, it is drawing a conclusion from a set of premises or developing the specific expectations of hypotheses from a theory or theoretical perspective.  Induction involves reasoning from particular instances to general principles.  In other words, it is offering a premise or a theory about a category of events from observations of specific instances or from the results of hypothesis testing.  It is the process involved in empirical generalization. Although deduction is fundamental to the scientific method, sociological analyses are rarely strictly deductive, even if they may claim to be.

Degrees of freedom (df): The number of values free to vary after certain restrictions have been imposed on all values. The df depends on the sample size (n) and dimensionality (number of variables (k)).

Dependent variable: The variable that is measured and analyzed in an experiment. In traditional algebraic equations of the form y = x + , it is usually agreed that y is the dependent variable.

Dependent samples: The values in one sample are related to the values in another sample. Before and after results are dependent samples.

Descriptive statistics: The methods used to summarize the key characteristics of known population and sample data.

Descriptive Statistics: Procedures that summarize the distribution of a              variable or measure the relationship between two or more variables.

Effect size: The degree to which a practice, program, or policy has an effect based on research results, measured in units of standard deviation. If a researcher finds an effect size of d = .5 for the effect of a test preparation program on SAT scores, this means the average student who participates in the program will achieve one-half standard deviation above the average student who does not participate. If the standard deviation is 20 points, then the effect size translates into eight additional points, which will increase a student’s score on the test.

Empirical is a verifiable quality based on experience, experiment, or observation rather than on assumption, logic, inspiration, or any of the other ways by which we may understand the social world.  According to a well known story, purely rational considerations led to the conclusion that the bumblebee is aerodynamically incapable of flying. Empirical considerations force us to conclude, to the contrary, that bumblebees do a very good job of flying.  Sociologists frequently argue that much of what is wrong with our understanding of social behavior arises from the tendency to deal with this subject on the basis of reasoning rather than observation.  On the other hand, because social behavior is both very complex and generally symbolic in character, the application of purely empirical modes of investigation can only provide part of the social explanation for behavior.

Empirical generalization is the process by which the specific, observed results of research are held to apply to the general, unobserved category of events or population under study. It is a form of induction.

Experiment: A process that allows observations to be made. In probability an experiment can be repeated over and over under the same conditions.

External validity: The degree to which results from a study can be generalized to other participants, settings, treatments and measures.

Exploratory data analysis: Any of several methods, pioneered by John Tukey, of discovering unanticipated patterns and relationships but presenting quantitative data visually.

F distribution: A continuous probability distribution used in tests comparing two variances. It is used to compute probability values in the ANOVA. The F distribution has two parameters: degrees of freedom numerator (dfn) and degrees of freedom denominator (dfd).

Goodness of fit: Degree to which observed data coincide with theoretical expectations.

Histogram: A graph of connected vertical rectangles representing the frequency distribution of a set of data.

Hypothesis: A statement or claim that some characteristic of a population is true.

Hypothesis test: A method for testing claims made about populations; also called the test of significance.

Independent variable: The treatment variable. In traditional algebraic equations of the form y =       x + , it is usually agreed that x is the independent variable.

Inferential statistics: The methods of using sample data to make generalizations or inferences about a population.

Interval scale: A measurement scale in which equal differences between numbers stand for equal differences in the thing measured. The zero point is arbitrarily defined. Temperature is measured on an interval scale.

Kurtosis: The shape (degree of peakedness) of a curve that is a graphic representation of a unimodal frequency distribution. It indicates the degree to which data cluster around a central point for a given standard deviation. It can be expressed numerically and graphically.

Kruskal-Wallis test: A nonparametric hypothesis test used to compare three or more independent samples. It is the nonparametric version of a one-way ANOVA for ordinal data.

Left-tail test: Hypothesis test in which the critical region is located in the extreme left area of the probability distribution. The alternative hypothesis is the claim that a quantity is less than (<) a certain value.

Level of significance: The probability level at which the null hypothesis is rejected. Usually represented by the Greek letter alpha (α).

Linear Structural Relations (LISREL): A computer program developed by Jöreskog that is used for analyzing covariance structures through structural equation models. It can be used to analyze causal models with multiple indicators of latent variables and relationships between the latent variables. It goes beyond more typical factor analysis.

Mean: A measure of central tendency, the arithmetic average; the sum of scores divided by the number of scores.

Measurement: the process of determining the value or level (either qualitative or quantitative) of a particular attribute of a unit of analysis.  It refers to assigning numbers to concepts or variables.  These series of assigned numbers can be used to 1) classify or categorize at the nominal level of measurement; 2) rank or order at the ordinal level of measurement; or 3) assign a score at the interval level of measurement.

Median: A measure of central tendency that divides a distribution of scores into two equal halves so that half the scores are above the median and half are below it.

Methodology: the logic of scientific investigation, including analysis of the basic assumptions of science in general and of sociology in particular, processes of theory construction, interrelationships of theory and research, and procedures of empirical investigation.

Mode: A measure of central tendency that represents the most fashionable, or most frequently occurring, score.

Multiple regression: Study of linear relationships among three or more variables.

Nominal data: Data that are names only, with no real quantitative value. Often numbers are arbitrarily assigned to nominal data, such as male = 0, female = 1.

Nonparametric statistical methods: Statistical methods that do not require a normal distribution or that data be interval or rational.

Normal distribution: Gaussian curve. A theoretical bell-shaped, symmetrical distribution based on frequency of occurrence of chance events.

Null hypothesis: The null hypothesis is a hypothesis about a population parameter. It assumes no change or status quo (=). The purpose of hypothesis testing is to test the viability of the null hypothesis in the light of the data. Depending on the data, the null hypothesis either will or will not be rejected as a viable possibility. We do not use the term accept when referring to the results of a statistical test.

Odds in favor: The number of ways an event can happen compared to the number of ways that it cannot happen.

Ogive: A graphical method of representing cumulative frequencies.

One-tailed test: A statistical test in which the critical region lies in one tail of the distribution. If the alternative hypothesis has a <, then you will conduct a left-tailed test. If it contains a >, then it will be right-tailed test.

One-way ANOVA: Analysis of variance involving data classified into groups according to a single criterion.

Operational definition: A concise definition of a term characterized by the functional use of that term. Operational definitions focus on prototypical usage or usage in practice. Operational definitions need to be concise and no more than one to three sentences in length.

Ordinal scale: A rank-ordered scale of measurement in which equal differences between numbers do not represent equal differences between the things measured. A Likert-type scale is a common ordinal scale.

Outlier: A single observation far away from the rest of the data. One definition of “far away” is less than Q1 − 1.5 × IQR or greater than Q3 + 1.5 × IQR where Q1 and Q3 are the first and third quartiles, respectively, and IQR is the interquartile range (equal to Q3 − Q1). These values define the so-called inner fences, beyond which an observation would be labeled a mild outlier. Outliers can be indicative of the occurrence of a phenomenon that is qualitatively different than the typical pattern observed or expected in the sample; thus, the relative frequency of outliers could provide evidence of a relative frequency of departure from the process or phenomenon is typical for the majority of cases in a group.

Parameter: Some numerical characteristic of a population. If the mean score on a midterm exam for a statistics class was 87%, this score would be a parameter. It describes the population composed of all those who took the test. Population parameters are usually symbolized by Greek letters, such as μ for the mean and σ for standard deviation.

Parametric methods: Types of statistical procedures for testing hypotheses or estimating parameters based on population parameters that are measured on interval or rational scores. Data are usually normally distributed.

Pie chart: Graphical method of representing data in the form of a circle containing wedges. Population: All members of a specified group.

Probability: A measure of the likelihood that a given even will occur. Mathematical probabilities are expressed as numbers between 0 and 1.

Probability distribution: Collection of values of a random variable along with their corresponding probabilities.

Proposition: a statement or specification within a theory that describes a causal relationship between two or more concepts. A proposition may be translated into one or more testable hypotheses by operationalizing the concepts into measurable variables.

p value: The probability that a test statistic in a hypothesis test is at least as extreme as the one actually obtained. A p value is found after a test statistic is determined. It indicates how likely the results of an experiment were due to a chance happening.

Qualitative variable: A variable that is often measured with nominal data. Quantitative variable: A variable that is measured with interval and rational data.

Random sample: A subset of a population chosen in such a way that any member of the population has an equal chance of being selected.

Range: The difference between the highest and the lowest score.

Ratio scale: A scale that has equal differences and equal ratios between values and a true zero point. Heights, weights, and time are measured on rational scales.

Raw score: A score obtained in an experiment that has not been organized or analyzed. Regression line: The line of best fit that runs through a scatterplot.

Research methods: the procedures of studying a phenomenon, including ways of collecting and handling empirical observations and data.  Research methods commonly employed by educators include surveys, observation, and quasi-experimental.

Reliability and validity are evaluative qualities assigned to empirical research methods.  Reliability is the capacity of a research instrument to deliver an unchanged, dependable result or measurement when applied repeatedly to the same phenomenon.  Validity is the capacity of a research instrument to measure what it purports, or claims, to measure.    It generally is more difficult, both conceptually and practically, to establish validity than to establish reliability.  An instrument can be reliable but invalid; in that case, it will give consistent results that do not mean what they are supposed to mean. However, an instrument cannot he valid but unreliable. If it is unreliable, it cannot measure anything adequately. The difficulty of establishing the validity of an instrument sometimes can he bypassed (or at least minimized) with a good operational definition.

Right-tailed test: Hypothesis test in which the critical region is located in the extreme right area of the probability distribution. The alternative hypothesis is the claim that a quantity is greater than (>) a certain value.

Sample: A subset of a population.

Sampling error: Errors resulting from the sampling process itself.

Scattergram: The points that result when a distribution of paired values are plotted on a graph.

Sign test: A nonparametric hypothesis test used to compare samples from two populations.

Significance level: The probability that serves as a cutoff between results attributed to chance happenings and results attributed to significant differences.

Skewed distribution: An asymmetrical distribution.

Spearman’s rank correlation coefficient: Measure of the strength of the relationship between two variables.

Spearman’s rho: A correlation statistic for two sets of ranked data.

Standard deviation: The weighted average amount that individual scores deviate from the mean of a distribution of scores, which is a measure of dispersion equal to the square root of the variance. At least 75% of all scores will fall within the interval of two standard deviations from the mean. At least 89% of all scores will fall within three standard deviations from the mean. The 68, 95, 99.7 rule (applies generally to a variable X having normal (bell-shaped or mound-shaped) distribution with mean μ (the Greek letter mu) and standard deviation σ (the Greek letter sigma). However, this rule does not apply to distributions that are “very” nonnormal. The rule states: approximately 68% of the observations fall within one standard deviation of the mean; approximately 95% of the observations fall within two standard deviations of the mean; and approximately 99.7% of the observations fall within three standard deviations of the mean. Another general rule is this: If the distribution is approximately normal, the standard deviation is approximately equal to the range divided by 4.

Standard error of the mean: The standard deviation of all possible sample means.

Standard normal distribution: A normal distribution with a mean of 0 and a standard deviation equal to 1. Statistic: A measured characteristic of a sample.

Statistics: The collection, organization, analysis, interpretation, and prediction of data.

t distribution: Theoretical, bell-shaped distribution used to determine the significance of experimental results based on small samples. Also called the Student t distribution.

t test (Student t test): Significance test that uses the t distribution. A Student t test deals with the problems associated with inference based on small samples.

Test statistic: Used in hypothesis testing, it is the sample statistic based on the sample data. We obtain test statistics by plugging in data we gathered into a formula.

Theory: an explanation of some phenomenon.  More specifically, it is an explanation of the relationship between two or more concepts or variables.  A theory is not just a description of an empirical relationship; rather, it is an attempt to answer the question of why (and, sometimes, how) the relationship exists as it does.

Two-tailed test of significance: Any statistical test in which the critical region is divided into the two tails of the distribution. The null hypothesis is usually a variable equal to a certain quantity. When the alternative hypothesis is not equal, this implies < or > as alternatives. This yields a two-tailed test.

Type I error: The mistake of rejecting the null hypothesis when it is true.

Type II error: The mistake of failing to reject the null hypothesis when it is false.

Uniform distribution: A distribution of values evenly distributed over the range of possibilities.

Variable: Any measurable condition, event, characteristic, or behavior that is controlled or observed in a study. Variable is also something that can change or vary, so that its opposite is a constant. A variable occurs in different degrees (or has different values) among individuals, groups, objects, and events.  A dependent variable (Y) is an effect, result, or outcome; it is assumed to depend on or to be caused by at least one independent variable (X). A researcher or theorist uses the independent variable(s) to explain the dependent variable. In other words, changes in the independent variable(s) are theorized or hypothesized to be correlated with or to have caused the changes in the dependent variable. Researchers and theorists often specify their variables in the titles of their articles. For example, as you skim through a recent issue of Social Forces, you may come across an article titled “The Influences of Age, Sex, Income, and Marital Status on Church Attendance.” It is safe to conclude that five variables were examined in this study: age, sex, income, marital status, and church attendance.  In all likelihood, the dependent variable would be church attendance (Y), which would be presumed to be affected by the four independent variables of age (X1), sex (X2), income (X3), and marital status (X4).

[NOTE: The ability to identify and specify variables is essential for   all students of research methods, data analysis, and theory]

Variance: The square of the standard deviation; a measure of dispersion.

Verifiability is the principle of science by which any given piece of research and, especially, its results can be duplicated or replicated by other scientists.

Wilcoxon rank-sum test: A nonparametric hypothesis test used to compare two independent samples.

z score: Also known as a standard score. The z score indicates how far and in what direction an item deviates from its distribution’s mean, expressed in units of its distribution’s standard deviation. The mathematics of the z score transformation are such that if every item in a distribution is converted to its z score, the transformed scores will necessarily have a mean of 0 and a standard deviation of 1.

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