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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
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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
X

X
n
» 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
 X  X 
2
SD 
N1
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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 deviationbased scale; a z-score is such an example:
z
X

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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
|-------|-------|-------|-------|-------|-------|
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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
r
SP
SSX SSY
where
SP   X  X Y  Y 
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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:
z
X

T  50 10z
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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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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
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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
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
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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
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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
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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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