Fundamentals of statistical reasoning in education /
Fundamentals of statistical reasoning in education /
Theodore Coladarci
- 2nd ed.
- Hoboken, NJ : John Wiley & Sons, c2008.
- xiv, 482 p. ill. ; 24 cm. + 1 CD-ROM (4 3/4 in.)
Includes bibliographical references (p. 419-420) and index.
Contents
Chapter 1 introduction 1
Why statistics? 1
Descriptive statistics 2
Inferential statistics 3
The role of statistics in
Educational research 4
Variables and their
Measurement 5
Some tips on studying
Statistics 9
Part i
Descriptive statistics 13
Chapter 2 frequency
Distributions 15
2.1 why organize data? 15
2.2 frequency distributions for
Quantitative variables 15
2.3 grouped scores 17
2.4 some guidelines for forming
Class intervals 18
2.5 constructing a grouped-data
Frequency distribution 19
2.6 the relative frequency
Distribution 21
2.7 exact limits 22
2.8 the cumulative percentage
Frequency distribution 24
2.9 percentile ranks 25
2.10 frequency distributions for
Qualitative variables 27
2.11 summary 28
Chapter 3 graphic
Representation 37
3.1 why graph data? 37
3.2 graphing qualitative data: the
Bar chart 37
3.3 graphing quantitative data: the
Histogram 38
3.4 the frequency polygon 42
3.5 comparing different
Distributions 43
3.6 relative frequency and
Proportional area 44
3.7 characteristics of frequency
Distributions 46
3.8 the box plot 49
3.9 summary 51
Chapter 4 central tendency 59
4.1 the concept of central tendency 59
4.2 the mode 59
4.3 the median 60
4.4 the arithmetic mean 62
4.5 central tendency and
Distribution symmetry 64
4.6 which measure of central
Tendency to use? 66
4.7 summary 67
Ix
Revised
78001 f04.3d ggs 3/15/07 19:9
Chapter 5 variability 75
5.1 central tendency is not enough:
The importance of variability 75
5.2 the range 76
5.3 variability and deviations from
The mean 77
5.4 the variance 78
5.5 the standard deviation 79
5.6 the predominance of the
Variance and standard
Deviation 81
5.7 the standard deviation and the
Normal distribution 81
5.8 comparing means of two
Distributions: the relevance of
Variability 82
5.9 in the denominator: n vs. N 21 85
5.10 summary 85
Chapter 6 normal distributions
And standard scores 91
6.1 a little history: sir francis
Galton and the normal curve 91
6.2 properties of the normal curve 92
6.3 more on the standard deviation
And the normal distribution 93
6.4 z scores 95
6.5 the normal curve table 97
6.6 finding area when the score is
Known 99
6.7 reversing the process: finding
Scores when the area is known 102
6.8 comparing scores from different
Distributions 104
6.9 interpreting effect size 105
6.10 percentile ranks and the normal
Distribution 107
6.11 other standard scores 108
6.12 standard scores do not
??Normalize?? A distribution 109
6.13 the normal curve and
Probability 110
6.14 summary 110
Chapter 7 correlation 119
7.1 the concept of association 119
7.2 bivariate distributions and
Scatterplots 119
7.3 the covariance 124
7.4 the pearson r 130
7.5 computation of r: the calculating
Formula 133
7.6 correlation and causation 135
7.7 factors influencing pearson r 136
7.8 judging the strength of
Association: r 2 139
7.9 other correlation coefficients 141
7.10 summary 141
Chapter 8 regression and
Prediction 149
8.1 correlation versus prediction 149
8.2 determining the line of
Best fit 150
8.3 the regression equation in
Terms of raw scores 153
8.4 interpreting the raw-score
Slope 156
8.5 the regression equation in
Terms of z scores 157
8.6 some insights regarding
Correlation and prediction 158
8.7 regression and sums of squares 161
8.8 measuring the margin of
Prediction error: the standard
Error of estimate 163
8.9 correlation and causality
(revisited) 168
8.10 summary 169
X contents
Revised
78001 f04.3d ggs 3/15/07 19:9
Part 2
Inferential statistics 179
Chapter 9 probability and
Probability
Distributions 181
9.1 statistical inference: accounting
For chance in sample results 181
9.2 probability: the study of chance 182
9.3 definition of probability 183
9.4 probability distributions 185
9.5 the or/addition rule 187
9.6 the and/multiplication rule 188
9.7 the normal curve as a
Probability distribution 189
9.8 ??So what??? Probability
Distributions as the basis for
Statistical inference 192
9.9 summary 192
Chapter 10 sampling
Distributions 197
10.1 from coins to means 197
10.2 samples and populations 198
10.3 statistics and parameters 199
10.4 random sampling model 200
10.5 random sampling in practice 201
10.6 sampling distributions of means 202
10.7 characteristics of a sampling
Distribution of means 204
10.8 using a sampling distribution
Of means to determine
Probabilities 207
10.9 the importance of sample
Size (n) 211
10.10 generality of the concept of a
Sampling distribution 212
10.11 summary 213
Chapter 11 testing statistical
Hypotheses about m
When s is known:
The one-sample
Z test 221
11.1 testing a hypothesis about m:
Does ??Homeschooling?? Make a
Difference? 221
11.2 dr. Meyer?s problem in a
Nutshell 222
11.3 the statistical hypotheses:
H0 and h1 223
11.4 the test statistic z 225
11.5 the probability of the test
Statistic: the p value 226
11.6 the decision criterion: level of
Significance (a) 227
11.7 the level of significance and
Decision error 229
11.8 the nature and role of h0 and h1 231
11.9 rejection versus retention of h0 232
11.10 statistical significance versus
Importance 233
11.11 directional and nondirectional
Alternative hypotheses 235
11.12 prologue: the substantive versus
The statistical 237
11.13 summary 239
Chapter 12 estimation 247
12.1 hypothesis testing versus
Estimation 247
12.2 point estimation versus interval
Estimation 248
12.3 constructing an interval estimate
Of m 249
12.4 interval width and level of
Confidence 252
12.5 interval width and sample size 253
Contents xi
Revised
78001 f04.3d ggs 3/15/07 19:9
12.6 interval estimation and
Hypothesis testing 253
12.7 advantages of interval estimation 255
12.8 summary 256
Chapter 13 testing statistical
Hypotheses about m
When s is not
Known: the
One-sample t test 263
13.1 reality: s often is unknown 263
13.2 estimating the standard error of
The mean 264
13.3 the test statistic t 266
13.4 degrees of freedom 267
13.5 the sampling distribution of
Student?s t 268
13.6 an application of student?s t 270
13.7 assumption of population
Normality 272
13.8 levels of significance versus
P values 273
13.9 constructing a confidence interval
For m when s is not known 275
13.10 summary 275
Chapter 14 comparing the
Means of two
Populations:
Independent
Samples 283
14.1 from one mu to two 283
14.2 statistical hypotheses 284
14.3 the sampling distribution of
Differences between means 285
14.4 estimating sx12x2 288
14.5 the t test for two independent
Samples 289
14.6 testing hypotheses about two
Independent means: an example 290
14.7 interval estimation of m1 2 m2 293
14.8 appraising the magnitude of a
Difference: measures of effect
Size for x12x2 295
14.9 how were groups formed?
The role of randomization 299
14.10 statistical inferences and
Nonstatistical generalizations 300
14.11 summary 301
Chapter 15 comparing the
Means of dependent
Samples 309
15.1 the meaning of ??Dependent?? 309
15.2 standard error of the difference
Between dependent means 310
15.3 degrees of freedom 312
15.4 the t test for two dependent
Samples 312
15.5 testing hypotheses about two
Dependent means: an example 315
15.6 interval estimation of md 317
15.7 summary 318
Chapter 16 comparing the
Means of three or
More independent
Samples: one-way
Analysis of
Variance 327
16.1 comparing more than two
Groups: why not multiple t tests? 327
16.2 the statistical hypotheses in
One-way anova 328
16.3 the logic of one-way anova:
An overview 329
16.4 alison?s reply to gregory 332
16.5 partitioning the sums of squares 333
16.6 within-groups and between-
Groups variance estimates 337
Xii contents
Revised
78001 f04.3d ggs 3/15/07 19:9
16.7 the f test 337
16.8 tukey?s ??Hsd?? Test 339
16.9 interval estimation of mi 2 mj 342
16.10 one-way anova: summarizing
The steps 343
16.11 estimating the strength of the
Treatment effect: effect size (o? 2) 345
16.12 anova assumptions (and
Other considerations) 346
16.13 summary 347
Chapter 17 inferences about
The pearson
Correlation
Coefficient 357
17.1 from m to r 357
17.2 the sampling distribution of r
When r 5 0 357
17.3 testing the statistical hypothesis
That r 5 0 359
17.4 an example 359
17.5 table e 361
17.6 the role of n in the statistical
Significance of r 363
17.7 statistical significance versus
Importance (again) 364
17.8 testing hypotheses other than
R 5 0 364
17.9 interval estimation of r 365
17.10 summary 367
Chapter 18 making inferences
From frequency
Data 375
18.1 frequency data versus score data 375
18.2 a problem involving frequencies:
The one-variable case 376
18.3 x2: a measure of discrepancy
Between expected and observed
Frequencies 377
18.4 the sampling distribution of x2 379
18.5 completion of the voter survey
Problem: the x2 goodness-of-fit
Test 380
18.6 the x2 test of a single proportion 381
18.7 interval estimate of a
Single proportion 383
18.8 when there are two variables:
The x2 test of independence 385
18.9 the null hypothesis of
Independence 387
18.10 calculating the two-variable x2 388
18.11 the x2 test of independence:
Summarizing the steps 391
18.12 the 2 _ 2 contingency table 392
18.13 testing a difference between
Two proportions 393
18.14 the independence of
Observations 393
18.15 x2 and quantitative variables 394
18.16 other considerations 395
18.17 summary 395
Chapter 19 statistical ??Power??
(and how to
Increase it) 403
19.1 the power of a statistical test 403
19.2 power and type ii error 404
19.3 effect size (revisited) 405
19.4 factors affected power:
The effect size 406
19.5 factors affecting power:
Sample size 407
19.6 additional factors affecting
Power 408
19.7 significance versus importance 410
19.8 selecting an appropriate
Sample size 410
19.9 summary 414
References 419
Contents xiii
Revised
78001 f04.3d ggs 3/15/07 19:9
Appendix a review of basic
Mathematics 421
A.1 introduction 421
A.2 symbols and their meaning 421
A.3 arithmetic operations involving
Positive and negative numbers 422
A.4 squares and square roots 422
A.5 fractions 423
A.6 operations involving parentheses 424
A.7 approximate numbers,
Computational accuracy, and
Rounding 425
Appendix b answers to selected
End-of-chapter
Problems 426
Appendix c statistical tables 448
Index 461
Xiv contents
9780470084069 (paper/CDROM) 0470084065 (paper/CDROM)
Educational statistics.
370.2/1 / C683
Includes bibliographical references (p. 419-420) and index.
Contents
Chapter 1 introduction 1
Why statistics? 1
Descriptive statistics 2
Inferential statistics 3
The role of statistics in
Educational research 4
Variables and their
Measurement 5
Some tips on studying
Statistics 9
Part i
Descriptive statistics 13
Chapter 2 frequency
Distributions 15
2.1 why organize data? 15
2.2 frequency distributions for
Quantitative variables 15
2.3 grouped scores 17
2.4 some guidelines for forming
Class intervals 18
2.5 constructing a grouped-data
Frequency distribution 19
2.6 the relative frequency
Distribution 21
2.7 exact limits 22
2.8 the cumulative percentage
Frequency distribution 24
2.9 percentile ranks 25
2.10 frequency distributions for
Qualitative variables 27
2.11 summary 28
Chapter 3 graphic
Representation 37
3.1 why graph data? 37
3.2 graphing qualitative data: the
Bar chart 37
3.3 graphing quantitative data: the
Histogram 38
3.4 the frequency polygon 42
3.5 comparing different
Distributions 43
3.6 relative frequency and
Proportional area 44
3.7 characteristics of frequency
Distributions 46
3.8 the box plot 49
3.9 summary 51
Chapter 4 central tendency 59
4.1 the concept of central tendency 59
4.2 the mode 59
4.3 the median 60
4.4 the arithmetic mean 62
4.5 central tendency and
Distribution symmetry 64
4.6 which measure of central
Tendency to use? 66
4.7 summary 67
Ix
Revised
78001 f04.3d ggs 3/15/07 19:9
Chapter 5 variability 75
5.1 central tendency is not enough:
The importance of variability 75
5.2 the range 76
5.3 variability and deviations from
The mean 77
5.4 the variance 78
5.5 the standard deviation 79
5.6 the predominance of the
Variance and standard
Deviation 81
5.7 the standard deviation and the
Normal distribution 81
5.8 comparing means of two
Distributions: the relevance of
Variability 82
5.9 in the denominator: n vs. N 21 85
5.10 summary 85
Chapter 6 normal distributions
And standard scores 91
6.1 a little history: sir francis
Galton and the normal curve 91
6.2 properties of the normal curve 92
6.3 more on the standard deviation
And the normal distribution 93
6.4 z scores 95
6.5 the normal curve table 97
6.6 finding area when the score is
Known 99
6.7 reversing the process: finding
Scores when the area is known 102
6.8 comparing scores from different
Distributions 104
6.9 interpreting effect size 105
6.10 percentile ranks and the normal
Distribution 107
6.11 other standard scores 108
6.12 standard scores do not
??Normalize?? A distribution 109
6.13 the normal curve and
Probability 110
6.14 summary 110
Chapter 7 correlation 119
7.1 the concept of association 119
7.2 bivariate distributions and
Scatterplots 119
7.3 the covariance 124
7.4 the pearson r 130
7.5 computation of r: the calculating
Formula 133
7.6 correlation and causation 135
7.7 factors influencing pearson r 136
7.8 judging the strength of
Association: r 2 139
7.9 other correlation coefficients 141
7.10 summary 141
Chapter 8 regression and
Prediction 149
8.1 correlation versus prediction 149
8.2 determining the line of
Best fit 150
8.3 the regression equation in
Terms of raw scores 153
8.4 interpreting the raw-score
Slope 156
8.5 the regression equation in
Terms of z scores 157
8.6 some insights regarding
Correlation and prediction 158
8.7 regression and sums of squares 161
8.8 measuring the margin of
Prediction error: the standard
Error of estimate 163
8.9 correlation and causality
(revisited) 168
8.10 summary 169
X contents
Revised
78001 f04.3d ggs 3/15/07 19:9
Part 2
Inferential statistics 179
Chapter 9 probability and
Probability
Distributions 181
9.1 statistical inference: accounting
For chance in sample results 181
9.2 probability: the study of chance 182
9.3 definition of probability 183
9.4 probability distributions 185
9.5 the or/addition rule 187
9.6 the and/multiplication rule 188
9.7 the normal curve as a
Probability distribution 189
9.8 ??So what??? Probability
Distributions as the basis for
Statistical inference 192
9.9 summary 192
Chapter 10 sampling
Distributions 197
10.1 from coins to means 197
10.2 samples and populations 198
10.3 statistics and parameters 199
10.4 random sampling model 200
10.5 random sampling in practice 201
10.6 sampling distributions of means 202
10.7 characteristics of a sampling
Distribution of means 204
10.8 using a sampling distribution
Of means to determine
Probabilities 207
10.9 the importance of sample
Size (n) 211
10.10 generality of the concept of a
Sampling distribution 212
10.11 summary 213
Chapter 11 testing statistical
Hypotheses about m
When s is known:
The one-sample
Z test 221
11.1 testing a hypothesis about m:
Does ??Homeschooling?? Make a
Difference? 221
11.2 dr. Meyer?s problem in a
Nutshell 222
11.3 the statistical hypotheses:
H0 and h1 223
11.4 the test statistic z 225
11.5 the probability of the test
Statistic: the p value 226
11.6 the decision criterion: level of
Significance (a) 227
11.7 the level of significance and
Decision error 229
11.8 the nature and role of h0 and h1 231
11.9 rejection versus retention of h0 232
11.10 statistical significance versus
Importance 233
11.11 directional and nondirectional
Alternative hypotheses 235
11.12 prologue: the substantive versus
The statistical 237
11.13 summary 239
Chapter 12 estimation 247
12.1 hypothesis testing versus
Estimation 247
12.2 point estimation versus interval
Estimation 248
12.3 constructing an interval estimate
Of m 249
12.4 interval width and level of
Confidence 252
12.5 interval width and sample size 253
Contents xi
Revised
78001 f04.3d ggs 3/15/07 19:9
12.6 interval estimation and
Hypothesis testing 253
12.7 advantages of interval estimation 255
12.8 summary 256
Chapter 13 testing statistical
Hypotheses about m
When s is not
Known: the
One-sample t test 263
13.1 reality: s often is unknown 263
13.2 estimating the standard error of
The mean 264
13.3 the test statistic t 266
13.4 degrees of freedom 267
13.5 the sampling distribution of
Student?s t 268
13.6 an application of student?s t 270
13.7 assumption of population
Normality 272
13.8 levels of significance versus
P values 273
13.9 constructing a confidence interval
For m when s is not known 275
13.10 summary 275
Chapter 14 comparing the
Means of two
Populations:
Independent
Samples 283
14.1 from one mu to two 283
14.2 statistical hypotheses 284
14.3 the sampling distribution of
Differences between means 285
14.4 estimating sx12x2 288
14.5 the t test for two independent
Samples 289
14.6 testing hypotheses about two
Independent means: an example 290
14.7 interval estimation of m1 2 m2 293
14.8 appraising the magnitude of a
Difference: measures of effect
Size for x12x2 295
14.9 how were groups formed?
The role of randomization 299
14.10 statistical inferences and
Nonstatistical generalizations 300
14.11 summary 301
Chapter 15 comparing the
Means of dependent
Samples 309
15.1 the meaning of ??Dependent?? 309
15.2 standard error of the difference
Between dependent means 310
15.3 degrees of freedom 312
15.4 the t test for two dependent
Samples 312
15.5 testing hypotheses about two
Dependent means: an example 315
15.6 interval estimation of md 317
15.7 summary 318
Chapter 16 comparing the
Means of three or
More independent
Samples: one-way
Analysis of
Variance 327
16.1 comparing more than two
Groups: why not multiple t tests? 327
16.2 the statistical hypotheses in
One-way anova 328
16.3 the logic of one-way anova:
An overview 329
16.4 alison?s reply to gregory 332
16.5 partitioning the sums of squares 333
16.6 within-groups and between-
Groups variance estimates 337
Xii contents
Revised
78001 f04.3d ggs 3/15/07 19:9
16.7 the f test 337
16.8 tukey?s ??Hsd?? Test 339
16.9 interval estimation of mi 2 mj 342
16.10 one-way anova: summarizing
The steps 343
16.11 estimating the strength of the
Treatment effect: effect size (o? 2) 345
16.12 anova assumptions (and
Other considerations) 346
16.13 summary 347
Chapter 17 inferences about
The pearson
Correlation
Coefficient 357
17.1 from m to r 357
17.2 the sampling distribution of r
When r 5 0 357
17.3 testing the statistical hypothesis
That r 5 0 359
17.4 an example 359
17.5 table e 361
17.6 the role of n in the statistical
Significance of r 363
17.7 statistical significance versus
Importance (again) 364
17.8 testing hypotheses other than
R 5 0 364
17.9 interval estimation of r 365
17.10 summary 367
Chapter 18 making inferences
From frequency
Data 375
18.1 frequency data versus score data 375
18.2 a problem involving frequencies:
The one-variable case 376
18.3 x2: a measure of discrepancy
Between expected and observed
Frequencies 377
18.4 the sampling distribution of x2 379
18.5 completion of the voter survey
Problem: the x2 goodness-of-fit
Test 380
18.6 the x2 test of a single proportion 381
18.7 interval estimate of a
Single proportion 383
18.8 when there are two variables:
The x2 test of independence 385
18.9 the null hypothesis of
Independence 387
18.10 calculating the two-variable x2 388
18.11 the x2 test of independence:
Summarizing the steps 391
18.12 the 2 _ 2 contingency table 392
18.13 testing a difference between
Two proportions 393
18.14 the independence of
Observations 393
18.15 x2 and quantitative variables 394
18.16 other considerations 395
18.17 summary 395
Chapter 19 statistical ??Power??
(and how to
Increase it) 403
19.1 the power of a statistical test 403
19.2 power and type ii error 404
19.3 effect size (revisited) 405
19.4 factors affected power:
The effect size 406
19.5 factors affecting power:
Sample size 407
19.6 additional factors affecting
Power 408
19.7 significance versus importance 410
19.8 selecting an appropriate
Sample size 410
19.9 summary 414
References 419
Contents xiii
Revised
78001 f04.3d ggs 3/15/07 19:9
Appendix a review of basic
Mathematics 421
A.1 introduction 421
A.2 symbols and their meaning 421
A.3 arithmetic operations involving
Positive and negative numbers 422
A.4 squares and square roots 422
A.5 fractions 423
A.6 operations involving parentheses 424
A.7 approximate numbers,
Computational accuracy, and
Rounding 425
Appendix b answers to selected
End-of-chapter
Problems 426
Appendix c statistical tables 448
Index 461
Xiv contents
9780470084069 (paper/CDROM) 0470084065 (paper/CDROM)
Educational statistics.
370.2/1 / C683
