Overview of Statistical Concepts and Procedures Introduction to Educational Research (6th ed.) Craig A. Mertler & C.M. Charles Appendix A Overview of Statistical Concepts and Procedures This multimedia product and its contents are protected under copyright law. The following are prohibited by law: • Any public performance or display, including transmission of any image over a network; • Preparation of any derivative work, including the extraction, in whole or in part, of any images; • Any rental, lease, or lending of the program.

Copyright © Allyn & Bacon 2008 The Nature and Use of Statistics Statistics are used for the following reasons: To summarize data and reveal what is typical and atypical within a group To show relative standing of individuals in a group To show relationships among variables To show similarities and differences among groups To estimate error that may have occurred in sample selection To test for significance of findings Copyright © Allyn & Bacon 2008

Populations and Samples Relationships among population, sample, parameters, and statistics Population—the totality of individuals or objects that correspond to a particular description Parameters—numerical values that describe populations Sample—smaller subgroup selected from a population Statistics—numerical values that describe samples

Copyright © Allyn & Bacon 2008 Parametric and Nonparametric Statistics Parametric statistics—used for analyzing traits that are normally distributed in the population—that is, in a manner that approximates the normal probability curve Nonparametric statistics—used to describe and analyze data that are not assumed to be normally distributed in the population Copyright © Allyn & Bacon 2008

Copyright © Allyn & Bacon 2008 The Calculation and Interpretation of Descriptive Statistics Measures of central tendency: Mean—the arithmetic average Median—the score that divides the distribution into two equal halves Mode—the most frequently occurring score Copyright © Allyn & Bacon 2008

Copyright © Allyn & Bacon 2008 The Calculation and Interpretation of Descriptive Statistics (cont’d.) Measures of variability: Range—the distance from the highest to the lowest score Standard deviation—the average distance of the scores from the mean Copyright © Allyn & Bacon 2008

Copyright © Allyn & Bacon 2008 The Calculation and Interpretation of Descriptive Statistics (cont’d.) Relative position: Percentile rank—the percentage of individuals who scored at or below a particular score Converted scores—transforming scores to a standard deviation-based scale; a z-score is such an example: Copyright © Allyn & Bacon 2008

The Calculation and Interpretation of Descriptive Statistics (cont’d.) Relationships: Coefficient of correlation—measure of the covarying relationship between two or more variables -1.00 -.70 -.30 0 +.30 +.70 +1.00 | - - - - - - - | - - - - - - - | - - - - - - - | - - - - - - - | - - - - - - - | - - - - - - - | perfect relationship perfect relationship absence of a relationship Copyright © Allyn & Bacon 2008

Copyright © Allyn & Bacon 2008 The Calculation and Interpretation of Descriptive Statistics (cont’d.) Relationships (cont’d.): Many types of correlation coefficients exist; most common is the Pearson r where Copyright © Allyn & Bacon 2008

The Calculation and Interpretation of Descriptive Statistics (cont’d.) The normal curve: Copyright © Allyn & Bacon 2008

Copyright © Allyn & Bacon 2008 The Calculation and Interpretation of Descriptive Statistics (cont’d.) Relative standing associated with the normal curve: Percentile ranks Stanines z-scores: T-scores: Copyright © Allyn & Bacon 2008

The Calculation and Interpretation of Inferential Statistics Error estimates: Indicate the range within which a given measure probably lies Confidence intervals: Indicate the probability that a population value lies within certain specified boundaries Tests of significance: Indicate whether the finding is ‘real’ or simply due to chance Significance of correlation (e.g., r, etc.) Significance of mean differences (e.g., t-test, F-test, etc.) Copyright © Allyn & Bacon 2008

The Calculation and Interpretation of Inferential Statistics (cont’d.) Chi-square analysis: A nonparametric test for significance of frequency distributions Standard error (of the mean): Estimate of how closely a statistic matches its corresponding population parameter Copyright © Allyn & Bacon 2008

Copyright © Allyn & Bacon 2008 The Calculation and Interpretation of Inferential Statistics (cont’d.) Standard error (of measurement): Estimate of the standard error of a single measurement Standard error (of the difference between two means): Estimate of the standard error between two separate measurements where Copyright © Allyn & Bacon 2008

The Calculation and Interpretation of Inferential Statistics (cont’d.) Testing for significance: Degrees of freedom—the number of scores in a sample that are free to vary (with respect to the mean) t-test—for two means F-test (ANOVA)—for more than two means, or in place of t-test when samples are large and unequal Copyright © Allyn & Bacon 2008

The Calculation and Interpretation of Inferential Statistics (cont’d.) Errors in statistical testing: Type I error—You conclude that there is a correlation (or significant difference, etc.) when in reality there is not … You’ve wrongly rejected the null hypothesis. Type II error—You conclude that there is not a correlation (or significant difference, etc.) when in reality there is … You’ve wrongly failed to reject the null hypothesis. Copyright © Allyn & Bacon 2008