Robert B. Ewen

Measures of Central Tendency

The mean of a set of scores is computed by adding up all the scores and dividing the result by the number of scores. In symbols, X = ΣXN, X is the sample mean, ΣX is the sum of the X scores, and N is the total number of scores. This chapter illustrates computation from a regular frequency distribution. In general, means and other statistics should not be computed from grouped frequency distributions, which do not give the exact value of every score; however, they may be approximated by treating all the scores in any interval as if they fell at the midpoint of the interval. One important attribute of a set of scores is its location, that is, where in the possible range between minus infinity and plus infinity the scores tend to fall. This can be described in a single number by using either the mean, the best measure in most instances; or the median (the score corresponding to the 50th percentile), preferable when data are highly skewed or there are extreme data whose exact values are unknown, and when the objectives are purely descriptive.

Job attitudes of black and white workers: Male blue-collar workers in six companies.

Abstract A 74-item attitude questionnaire was administered in six companies to 101 black and 87 white male blue-collar employees holding similar jobs in the same company. Differences between the two ethnic groups were not marked, both in terms of job satisfaction and in other respects; Where there were differences the black workers were usually slightly more favorable. However, the picture was not uniform across the different companies.

introduction to power analysis

This chapter discusses the concepts involved in power analysis. It presents the methods for accomplishing the two major kinds of power analysis, which can be applied to the null hypothesis tests. There are four major parameters involved in power analysis, such as the significance criterion, the sample size, the population effect size, and power. These four parameters are mathematically related in such a way that any one of them is an exact function of the other three. The chapter discusses how power analysis of significance tests of the Pearson r of a sample proceeds quite simply. The test of the difference between the means of two independently drawn random samples is probably the most frequently performed test in the behavioral sciences. The chapter also presents the techniques for determining power that make possible the planning of efficient experiments designed to yield more conclusive statements if H O is not rejected.

Nonparametric and Distribution-Free Methods

This chapter discusses nonparametric and distribution-free methods. The main advantage of nonparametric and distribution-free statistical tests is that they do not require the population(s) being sampled to be normally distributed, and therefore, are applicable when gross nonnormality is suspected. The primary disadvantage of these methods is that when normality does exist, they are less powerful than the corresponding parametric tests. The power efficiency of a nonparametric or distribution-free test is almost always less than 100%, and sometimes much less. Thus it is wasteful to use these methods when parametric tests are applicable. The rank-sum test finds out the difference between the locations of tw o independent samples. The Kruskal-Wallis H test figures out differences among the locations of two or more independent samples. The Wilcoxon test finds out the difference between the locations of two matched samples. The median and sign tests are used primarily to obtain a quick approximation of the results of more powerful tests when samples are large. The median test is used to compare the locations of two or more independent samples. The sign test is used to compare the locations of two or more matched samples.

One-Way Analysis of Variance

This chapter presents the analysis of variance (ANOVA), which is a procedure for testing differences among three or more means for statistical significance. It is also used with just two samples. It permits null hypotheses to be tested that involve the means of three or more samples (groups); however, one-way ANOVA deals with one independent variable. The null hypothesis tested by ANOVA is the means of the populations from which the samples were randomly drawn are all equal; for example, the null hypothesis in the caffeine experiment is H O : μ 1 = μ 2 = μ 3 = μ 4 = μ 5. The alternative hypothesis states that H O is not true. The ANOVA procedure is based on a mathematical proof that the sample data can be made to yield two independent estimates of the population variance. The first step in the ANOVA design is to compute the sum of squares between groups (symbolized by SS B ), the sum of squares within groups (symbolized by SS W ), and the total sum of squares (symbolized by SS T ). A sum of squares is nothing more than a sum of squared deviations.

inferences about the mean of a single population

This chapter describes inferences about the mean of a single population. The chapter states the general considerations for hypothesis testing. The null hypothesis (denoted by the symbol H 0 ) and the alternative hypothesis (denoted by H 1 ) are stated. It is important to understand that it cannot be proved whether H 0 or H 1 is true because the entire population cannot be measured. It is assumed that H 0 is true. Then the data is obtained and the assumption is tested out. If this assumption is unlikely to be true, it is abandoned and the “bets should be switched” to H 1 , otherwise the assumption is retained. Before collecting any data, it is necessary to define in numerical terms what is meant by “unlikely to be true.” In statistical terminology, this is called selecting a criterion (or level) of significance, represented by the symbol a. Having selected a criterion of significance, data is selected and the appropriate statistical test is computed.

Probability and the General Strategy of Inferential Statistics

This chapter defines probability and discusses the general strategy of inferential statistics. The probability of the occurrence of an event is calculated by dividing the number of ways in which the specified event can occur by the total number of possible events. The odds against an event are the number of unfavorable events against the number of favourable events. For the general strategy of inferential statistics, it is necessary to draw an inference about a population based on results obtained from a sample. The chapter presents an example involving the probability of obtaining six heads in six flips.

14 – One-Way Analysis of Variance

Publisher Summary This chapter presents one-way analysis of variance. In statistics, one-way analysis of variance is a technique used to compare means of two or more samples, using the F distribution. This technique can be used only for numerical data. The chapter explains that one measure for determining the strength of the relationship between the independent and dependent variables is ɛ, known as epsilon.

11 – Linear Correlation and Prediction

Publisher Summary This chapter presents linear correlation and prediction. Statistical researchers often use a linear relationship to predict the average numerical value of Y for a given value of X using a straight line called the regression line. If one knows the slope and the y-intercept of that regression line, then one can plug in a value for X and predict the average value for Y. In other words, one predicts the average Y from X.

13 – Introduction to Power Analysis

Publisher Summary This chapter presents introduction to power analysis. Power is the probability of getting a significant result in a statistical test. Power is equal to 1 — β, where β= the probability of a Type II error. If power is not known, a researcher can waste a great deal of time by conducting an experiment that has little chance to produce significance even if H0 is false. Worse, a promising line of research can be abandoned because the researcher does not know that he should have relatively little confidence concerning his failure to reject H0, that is, the probability of a Type II error is large. The four major parameters of power analysis are: the significance criterion, α; the sample size, N; the population “effect” size, γ; and power, 1 — β. The general procedure for the two most important kinds of power analysis are: power determination and sample size determination.