Download The Calculation and Interpretation of Descriptive Statistics (cont`d.)

Introduction to Educational Research (5th ed.)
Craig A. Mertler & C.M. Charles
Appendix A
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 
(N 1)
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 deviation-based 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:
z
» T-scores:
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, Ftest, etc.)
12
The Calculation and Interpretation of
Inferential Statistics (cont’d.)
• Chi-square analysis:
» A nonparametric test for significance of frequency
distributions
•
f o  f e

2
 
fe
2
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
s2p s2p


nx ny
where
SSx  SSy
s 
df x  df y
2
p
14
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, or in
place of t-test when samples are large and unequal
s2 betweengroups
F 2
swithingroups
15
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.
16