Summary - Quantitative Data

Statistics versus Parameters

• A parameter is a characteristic of a population. It is a numerical or graphic way to summarize data obtained from the population.
• A statistic, on the other hand, is a characteristic of a sample. It is a numerical or graphic way to summarize data obtained from a sample.

Types of Numerical Data

• There are two fundamental types of numerical data a researcher can collect. Quantitative data are obtained by determining placement on a scale that indicates amount or degree. Categorical data are data obtained by determining the frequency of occurrences in each of several categories.

Types for Summarizing Quantitative Data

• There are 4 types of summarizing Qualitative Data which are frequency polygon, skewed polygons, histograms and stem leaf plots and the normal curve.

Correlation

• A correlation coefficient is a numerical index expressing the degree of relationship that exists between two quantitative variables. The one most commonly used in educational research is the Pearson r.
• A scatter plot is a graphic way to describe a relationship between two quantitative variables.

Techniques for Summarizing Categorical Data

• Researchers use various graphic techniques to summarize categorical data, including frequency tables, bar graphs, and pie charts.
• A cross break table is a graphic way to report a relationship between two or more categorical variables.

Chapter 11 Inferential Statistics

What Are Inferential Statistics?

• Inferential statistics refer to certain procedures that allow researchers to make inferences about a population based on data obtained from a sample.
• The term probability, as used in research, refers to the predicted relative frequency with which a given event will occur.

Sampling Error

• The term sampling error refers to the variations in sample statistics that occur as a result of repeated sampling from the same population.

The Distribution of Sample Means

• A sampling distribution of means is a frequency distribution resulting from plotting the means of a very large number of samples from the same population.
• The standard error of the mean is the standard deviation of a sampling distribution of means. The standard error of the difference between means is the standard deviation of a sampling distribution of differences between sample means.

Confidence Intervals

• A confidence interval is a region extending both above and below a sample statistic (such as a sample mean) within which a population parameter (such as the population mean) may be said to fall with a specified probability of being wrong.

Hypothesis Testing

• Statistical hypothesis testing is a way of determining the probability that an obtained sample statistic will occur, given a hypothetical population parameter.
• A research hypothesis specifies the nature of the relationship the researcher thinks exists in the population.
• The null hypothesis typically specifies that there is no relationship in the population.

Significance Levels

• The term significance level (or level of significance), as used in research, refers to the probability of a sample statistic occurring as a result of sampling error.
• The significance levels most commonly used in educational research are the .05 and .01 levels.
• Statistical significance and practical significance are not necessarily the same. Even if a result is statistically significant, it may not be practically (i.e., educationally) significant.

Tests of Statistical Significance

• A one-tailed test of significance involves the use of probabilities based on one-half of a sampling distribution because the research hypothesis is a directional hypothesis.
• A two-tailed test, on the other hand, involves the use of probabilities based on both sides of a sampling distribution because the research hypothesis is a nondirectional hypothesis.

Parametric Tests for Quantitative Data

• A parametric statistical test requires various kinds of assumptions about the nature of the population from which the samples involved in the research study were taken.
• Some of the commonly used parametric techniques for analyzing quantitative data include the t-test for means, ANOVA, ANCOVA, MANOVA, MANCOVA, and the t-test for r.

Parametric Tests for Categorical Data

• The most common parametric technique for analyzing categorical data is the t-test for differences in proportions.

Nonparametric Tests for Quantitative Data

• A nonparametric statistical technique makes few, if any, assumptions about the nature of the population from which the samples in the study were taken.
• Some of the commonly used nonparametric techniques for analyzing quantitative data are the Mann-Whitney U test, the Kruskal-Wallis one-way analysis of variance, the sign test, and the Friedman two-way analysis of variance.

Nonparametric Tests for Categorical Data

• The chi-square test is the nonparametric technique most commonly used to analyze categorical data.
• The contingency coefficient is a descriptive statistic indicating the degree of relationship that exists between two categorical variables.

Power of a Statistical Test

• The power of a statistical test for a particular set of data is the likelihood of identifying a difference, when it in fact exists, between population parameters.
• Parametric tests are generally, but not always, more powerful than nonparametric tests.

Chapter 12 – Statistic in Perspectives

Approaches to Research

• A good deal of educational research is done in one of two ways: either two or more groups are compared, or variables within one group are related.
• The data in a study may be either quantitative or categorical.

Comparing Groups Using Quantitative Data

• When comparing two or more groups using quantitative data, researchers can compare them through frequency polygons, calculation of averages, and calculation of spreads.
• We recommend, therefore, constructing frequency polygons, using data on the means of known groups, calculating effect sizes, and reporting confidence intervals when comparing quantitative data from two or more groups.

Comparing Groups When the Data Involved Are Categorical

• When the data are categorical, groups can be compared by reporting either percentages of frequencies in cross break tables.
• It is a good idea to report both the percentage and the number of cases in a crossbreak table, as percentages alone can be misleading.
• Therefore, we recommend constructing crossbreak tables and calculating contingency coefficients when comparing categorical data involving two or more groups.

Comparing Groups Using Categorical Data

• When you are examining relationships among categorical data within one group, we again recommend constructing crossbreak tables and calculating contingency coefficients.