1. 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
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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
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3. 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

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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
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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
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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
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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:
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8. The Calculation and Interpretation of Descriptive Statistics (cont’d.)
Relationships:
Coefficient of correlation—measure of the covarying relationship between two or more variables
| | | | | | |
perfect relationship
perfect relationship
absence of a relationship
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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
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10. The Calculation and Interpretation of Descriptive Statistics (cont’d.)
The normal curve:
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The Calculation and Interpretation of Descriptive Statistics (cont’d.)
Relative standing associated with the normal curve:
Percentile ranks
Stanines
z-scores:
T-scores:
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12. 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.)
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13. 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
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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
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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
F-test (ANOVA)—for more than two means, or in place of t-test when samples are large and unequal
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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.
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