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Introduction to Educational Research (4th ed.) C. M. Charles/Craig A. Mertler Appendix Overview of Statistical Concepts and Procedures 1 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 2 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 3 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 4 The Calculation and Interpretation of Descriptive Statistics • Measures of central tendency: » Mean—the arithmetic average X X n » Median—the score that divides the distribution into two equal halves » Mode—the most frequently occurring score 5 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 X X 2 SD N1 6 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 deviationbased scale; a z-score is such an example: z X 7 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 |-------|-------|-------|-------|-------|-------| 8 The Calculation and Interpretation of Descriptive Statistics (cont’d.) • Relationships (cont’d.): » Many types of correlation coefficients exist; most common is the Pearson r r SP SSX SSY where SP X X Y Y 9 The Calculation and Interpretation of Descriptive Statistics (cont’d.) • The normal curve: 10 The Calculation and Interpretation of Descriptive Statistics (cont’d.) • Relative standing associated with the normal curve: » Percentile ranks » Stanines » z-scores: » T-scores: z X T 50 10z 11 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.) 12 The Calculation and Interpretation of Inferential Statistics (cont’d.) • Chi-square analysis: » A nonparametric test for significance of frequency distributions 2 f o f e 2 fe • Standard error (of the mean): » Estimate of how closely a statistic matches its corresponding population parameter SE X SD N 1 13 The Calculation and Interpretation of Inferential Statistics (cont’d.) • Standard error (of measurement): » Estimate of the standard error of a single measurement SE M SD 1 r • Standard error (of the difference between two means): » Estimate of the standard error between two separate measurements SE dM SE SE 2 X 2 Y 14 The Calculation and Interpretation of Inferential Statistics (cont’d.) • Standard error (of correlation): » Estimate of the standard error of a correlation coefficient SE r 1 N 1 15 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 X Y t SE dM » F-test (ANOVA)—for more than two means s2 between groups F 2 swi thin groups 16 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. 17