Basic Statistical Analysis

indexerrt.qxd 8/21/2002 9:47 AM Page 1 Corrected index pages for Sprinthall Basic Statistical Analysis Seventh Edition

indexerrt.qxd 8/21/2002 9:47 AM Page 656 Index Abscissa, 24 AB-STAT, vii ADD-OR rule, and probability, 126 After-only design, 204 05 Alpha level, 174 78 confidence levels and, 245 46 increasing, 256 levels of, 174 78 Alternative hypothesis (Ha), 156, 242 one-tail, 249 for t test, 242 Analysis of variance (ANOVA) advantages of, 313 14 applications, 330 31 between-group variability, 332 experimental research, 330 factorial ANOVA, 331 Friedman ANOVA by ranks, 465 67 null hypothesis and, 315 one-way F ratio, 325 27 post-facto research, 331 significant interaction, 332 sum of squares, 317 18 Tukey s HSD, 328 30, 444 two-way ANOVA, 332 36, 342 43 A priori hypothesis, 357 Association research, 270 Association (See Correlation) Averages, law of, 120 Bar graphs, 25 27 Before-after (B/A) design (See also Repeated Measures) and paired t ratio 429 34 and within-subjects F ratio, 441 46 experimental research, 199, 205 07 problems of, 427 29 Beta coefficient, 396 Beta level, power of statistical test, 254 Between-group variability, 332 Between-Subjects Experimental Design, 204 05 Bias political polling, 144 46 sampling, 141 Bimodal distributions, 38 39 Binomial distribution compared to normal distributions, 560 continuous distributions, 559 discrete distributions, 559 Binomial probability, 557 Binomial proportions, 564 z test and, 564 Bivariate scatter plot, 389 91 regression equation, 392 96 regression line, 390 92 Bonferroni test, 315 Canonical correlation, 416 Categorical Data (See also nominal data), 195 Cause and effect cause, problems isolating, 271 correlation and, 273, 277 as trap in research, 193 94 Ceiling effect, 209 Census, 136 Central limit theorem, 151 Central tendency, measures of mean, 30 35 median, 35 36 mode, 36 39 Chance defying chance, 174 frequency expected due to, 355 laws of, 118 20 Chance hypothesis (See Null hypothesis) Chi square and proportions, 567 coefficient of contingency, 371 73 dependent samples and, 368 70 locating differences and, 364 McNemar test, 369 71 1 by k Chi square (goodness of fit), 356 percentages and, 367 requirements for, 375 76 in research simulations, 532 34, 538, 551, 552 r k chi square, 358 2 2 chi square, 362, 365, 367 z score and, 368 Coefficient of contingency, 371 73 limitations, 373 nominal data, 371 Coefficient of determination, 287 Collinearity, 411 Combination research, 215 Combinations and probability, 121 23 Computers bugs in software, 521 caution, words of, 524 common problems, 518 22 computer literacy, 516 18 logic checkpoints, 524 types of statistical programs, 518 Concurrent validity, 491 Conditional probability, 118 Confidence intervals, 180 82, 402 04 for differences between independent samples, 245 for paired differences, 434 long run and, 182 precision and, 180 single sample t and, 183 standard error of estimate and, 401 03 two-sample t and, 245 Confidence levels, alpha and, 245 46 Confounding variables, 217 Constants, nature of, 190 91 Construct validity, 495 Continuous distributions, 559 Control Groups repeated-measures design, 205 change in, 436 37 experimental research, 199, 204 07 failure to use, examples, 217 18 inadequate group, examples, 218 19 paired t, 439 41 Correlated samples problems of, 436 Correlation (See also Regression analysis) cause and effect, 270 73, 278 coefficient of, 273 74, 300 decision about choosing/using tests, 299 300 interclass and intraclass, 274 656

indexerrt.qxd 8/21/2002 9:47 AM Page 657 Index 657 interpreting values, 276 negative correlations, 274, 277, 279 Pearson r, 273 positive correlations, 274, 277, 278 scatter plots and, 277 80 significance, 282 83 Spearman, 293 Spearman-Brown, 481 zero correlations, 274, 277 Correlation matrix, 291 Correlation strength (See also Test reliability) determination of, 287 Covariate, 412 Cox, Gertrude, 14 15 Criterion referencing, 489 and reliability, 477 Cronbach s alpha, 487 Cyril Hoyt reliability method, 488 Curve (See also Distributions; Normal curve) leptokurtic, 58 mesokurtic, 58 platykurtic, 58 Data categorical data, 195 interval data, 196 97 measurement data, 196 97 nominal data, 195 96 ordinal data, 196 ranked data, 196 ratio data, 197 Deciles, 48 Degrees of freedom chi square, 356, 360 61 1 k chi square (goodness of fit), 356 paired t ratio, 429 34 r k chi square (r by k), 358 sum of squares, 322 t ratio, 169 70, 243 44 Dependent (correlated) selection, 427 Dependent samples, chi square and, 368 70 Dependent Variables, 191 93 Descriptive statistics, 17 18 Deviation method, standard deviation, 50 54 Deviation score, 50 Difference chi square and, 364 distribution of differences, 232 33 Difference, hypothesis of alpha and confidence levels, 245 estimated standard error of, 233 experimental research and, 214 post-facto research and, 214 power, 254 sample groups, independent vs. correlated, 427 29 significance, 239 t ratio, 235 39, 242, 244 Discrete distributions, 559 Discriminant analysis, 343 Dispersion (See Variability) Distribution free tests (See Nonparametric tests) Distribution of differences, 230 33 mean of, 230 31 null hypothesis, 241 42 standard deviation of, 232 33 two populations, 232 Distributions bimodal distributions, 38 39 binomial distributions, 559 61 of differences, 230 33 frequency distributions, 23 27 skewed distributions, 33 36, 39 42 unimodal distribution, 38 Double-blind research, 200 02 Effect Size, 222 chi square, 374 factorial ANOVA, 331 one-way ANOVA, 327 paired t, 434 35 single-sample t, 178 two-sample t, 255 within subjects ANOVA, 444 Error, meaning in statistics, 151 52 (See also Specific types of errors) Estimated standard deviation, 162 67 calculation of, 162 65 Estimated standard error of difference, 233 35 paired t ratio, corrected equation for, 429 Estimated standard error of the mean, 166 68 Eta square, 327, 338, 339 Exit polling, 145 Experimental research, 190, 199, 202 between-subjects, 204 05 analysis of variance (ANOVA), 330 31 repeated-measures design, 205 07 combination research, 215 16 control groups, 208 09, 210 dependent selection, 203 04 double-blind research, 200 02 equivalent groups, creating, 202 experimental group, 208 09 hypothesis of difference and, 229 matched-group design, 211 matched-subjects design, 205 07, 210 11 quasi-experimental design, 212 randomized assignments, 203 random samples, 138 40 repeated measures design, 429 representative sampling, 138 research simulations, 532 54 validity, external/internal, 202 External Validity, 202 Face validity, 490 Factor, nature of, 332 Factorial ANOVA, 331 between-group variability, 332 calculations, 332 38 compared to within-subjects F ratio, 441 46 graphs and, 340 41 in research simulations, 541 theory of, 331 Fisher, Sir Ronald, 316 Floor effect, 210 F ratio requirements, 327 in research simulations, 538, 541, 552, 553 sum of squares, 322 23 within-subjects F ratio, 441 46 Frequency distribution curve, 66 Frequency of distributions, 23 27 Frequency of Error, Law of, 151 52 Frequency polygons, 25 27 Friedman ANOVA by ranks, 465 67 calculations, 465 67 in research simulations, 542 sample size, 156, 467 Frustration-regression hypothesis, research design, 217 18 F table, one-way F ratio, 327 Gallup poll, 144 Galton, Sir Francis, 151, 276, 398 99 Gambler s fallacy, 119 20 Gauss, Karl Friedrich, 70 Goodness of fit (See 1 k chi square) Gossett, William Sealy, 16, 240 Grade-equivalent scores (GEs), 108 Graphs correlations and, 277 factorial ANOVA, 331 frequency distributions, 23 27 frequency polygons, 25 27 histograms, 25 27 scatter plot, 277 80 variability and, 57 60 zero as base of ordinate, 27 30 Grouped-data techniques, 55 57 Guilford, J.P., 287 Halo effect, 219 20 Hawthorne effect, 207 Histograms, 25 27 Homogeneity of variance, 327 Homoscedasticity, 292 Honestly Significant Difference test (See Tukey s HSD) Hypothesis, types for statistical testing, 531 Independent variables, 191 93 Inferential statistics, 18 (See also Parameter estimates) key concepts in, 136 38 Informed consent, 215 Interaction effects, 332 Interdecile range, 50 Interval data, 196 97 Interval estimate, 168 69, 179 82 confidence interval, 182 83, 402 04 Interval scale, 196 97

indexerrt.qxd 8/21/2002 9:47 AM Page 658 658 Index www.ablongman.com/sprinthall7e Item analysis, 496 item difficulty, 496 97 item discrimination, 497 Kruskal-Wallis H test, 461 62 calculation, 461 62 sample size, 462 Kuder-Richardson (K-R 21) reliability, 483 Kurtosis, 58 60 1/6 rule, 59 standard deviation/range relationship, 58 60 Law of averages, 120 Law of Frequency of Error, 151 52 Laws of chance, 120 Leptokurtic curve, 58 Levene s test for equality of variances, 631 Linearity, 404 Literary Digest poll, 142 44 Long-run relative frequency, 115, 118 Lovelace, Ada, 520 McNemar test, 370 71 dependent samples, 369 70 matched-subjects design, 370 in research simulations, 540 Yates correction, 371 Manipulated independent variable, 199 200 MANOVA, 343 Mann-Whitney U, 458 60 calculations, 458 60 interpretation, 460 in research simulations, 543 sample size, 460 Matched-subjects (M/S) design (See also Paired t ratio; Withinsubjects F ratio) experimental research, 210 11 McNemar test, 370 71 paired t ratio, 429 34 problems of, 427 29 Maximum correlation (test and criterion), 492 93 Mean, 31 33 calculation, 31 33 confidence interval, 168 69 distribution of differences, 230 31 of distribution of means, 146 48 formula, 31 33 interpretation, 31 skewed distribution, 31 34 standard error of, 150 51 from z score, 98 99 Mean square, sum of squares, 322 Measurement, nature of, 194 95 Measurement data, 196 97 Measurement scaling, 195 98 interval scale, 196 97 nominal scale, 195 96 ordinal scale, 196 ratio scale, 197 Measurement theory, 194 Median (Mdn), 35 36 calculation, 35 skewed distributions, 36 Mere presence phenomenon, 364 Mesokurtic curve, 58 Meta-analysis, 221 23 Minimum difference for t ratio, 248 Mode (Mo), 36 39 bimodal distributions, 38 39 finding modes, 36 39 interpretation, 38 39 unimodal distributions, 38 MULT-AND rule and probability, 126 27 Multiple R, 404 10 components of, 406 07 equation for, 406 multiple regression, 408 09 in research simulations, 546 Negative correlations, 274, 277, 279 Nominal data, 195 96 (See also Chi square) coefficient of contingency, 371 73 Nominal scale, 195 96 Nonparametric tests (See also Chi square) advantages/disadvantages of, 467 Friedman ANOVA by ranks, 465 67 Kruskal-Wallis H test, 461 62 Mann-Whitney U, 458 60 ordinal data and, 457 58 Wilcoxon T test, 463 65 Normal curve areas of, 80 82 equation, 79 80 features of, 66 68, 85 as frequency distribution curve, 65 68 normal curve equivalents (NCEs), 104 05 Norm referencing, 474 Null hypothesis, 156, 172, 173 78 analysis of variance (ANOVA) and, 315 distribution of differences, 243 nature of, 241 for t test, 242 Pearson r, 283 Odds, 120 Oh boy! graph, 29 1 k chi square (goodness of fit), 355 calculation, 355 degrees of freedom, 356 57 interpretation, 356 testing a priori hypothesis, 357 One-tail t table, 249 52, 578 advantages/disadvantages of, 251 alternative hypothesis and, 242, 249 negative t ratio, 251 sign of, 251 One-tail t test, 249 alternative hypothesis, 242, 249 One-way ANOVA (See F ratio) One-way F ratio, 324 27 calculation between subjects, 325 F table, 327 requirements of, 327 summary of results, 326 Ordinal data scale, 196 (See also nonparametric tests) Spearman and, 293 Ordinate, 27 29 Outliers, 141, 286 Paired t ratio, 429 31 advantages of, 430 before-after design, 436 cautions about, 436 control group changes and, 436 41 degrees of freedom, 431 34 and matched control group, 439 power and, 435 in research simulations, 550 standard error of difference, corrected equation for 429 Parameter estimates alpha level, 174 78 as hypothesis, 171 72 interval estimates, 179 82 point estimates, 168 69 of population standard deviation, 162 66 of standard error of the mean, 166 67 t ratio, 169 73 z scores, 154 56 Parameters nature of, 137 sampling distributions, 152 54 Partial correlation, 412 equation, 412 2 variable or covariate identification, 412 Pascal, Blaise, 16, 116 17 Path analysis, 411 12 Pearson, Karl, 275, 400 Pearson r, 273 93 coefficient of determination, 287 interpretation of, 287 limitations, 293 null hypothesis and, 283 Pearson r table, 283, 579 reliability, 477 requirements for, 292 restricted range, 285 86 significance, 282 83, 284 85 z score method, 280 Percentages chi square and, 367 68 converted to probability statements, 120 21 Percentile ranks, 48 49 Percentiles, 48 49 to raw scores, 91 93 table, 91, 576 from z scores, 82 to z scores, 91 Permutations and probability, 121 23 Platykurtic curve, 58 Point biserial, 498

indexerrt.qxd 8/21/2002 9:47 AM Page 659 Index 659 Point estimate of population mean, 168 Political polling, 142 44 bias, 144 46 Gallup poll, 144 Literary Digest poll, 142 44 Population nature of, 139 standard deviation, 162 66 Positive correlations, 274, 277, 278 Post-facto research, 190, 212 14 analysis of variance (ANOVA), 331 combination research, 215 ethical issues, 214 hypothesis of association and, 270 hypothesis of difference and, 229 nature of, 212 14 post-hoc fallacy, 213 research simulations, 532 54 Post-hoc fallacy, 213 Power, 254, 435 paired t ratio and, 435 Power of statistical tests, 254 beta level, 254 Predictive validity, 477 Probability ADD-OR rule, 126 binomial probability, 557 combining probabilities, 126 28 conditional probability, 118 gambler s fallacy, 119 20 independent events and, 115 long run relative frequency, 115, 118 MULT-AND rule, 126 27 versus odds, 120 percentage areas of normal curve and, 120 26 percentages converted to probability statements, 120 21 z scores, 123 25 Proportions binomial proportions, 564 65 chi square and, 567 68 difference, testing, 566 67 Qualitative research, 216 Quartile deviation, 49 50 Quartiles, 48 49 Quasi-experimental design, 212 Quota (stratified) sampling, 140 Randomized assignment, 203 Random sampling, 138 40 Range (R), 48 50 interdecile range, 50 interquartile range, 49 percentiles, 48 49 relationship to standard deviation, 58 59 restricted range, 285 86 Ranked data, 196 (See also Ordinal data) Ratio data, 197 Raw scores from percentiles, 91 93 from T scores, 102 03 from z scores, 90 91 to T scores, 100 02 to z scores, 78 82 Regression analysis beta coefficient, 396 bivariate scatter plot, 389 confidence interval equation, 402 03 multiple R, 404 06 regression equation, 392 96 standard error of estimate, 401 03 theory of regression, 398 Regression line, 390 92 extent of scatter around, 390 91 slope of, 393 Y intercept of, 392 Reliability (See Test reliability) Repeated measures design, 208 Representative sampling, 138 Research, 199 (See also Experimental research; Post-facto research) burden of proof and, 555 cause and effect trap, 193 94 combination research, 215 dependent variables, 191 93 experimental research, 190, 199, 202 fitting statistical test to, 256 independent variables, 191 93 key characteristics, 554 post-facto research, 190, 199, 212 15 qualitative, 216 simulations, 532 54 variables/constants in, 190 91 Research errors, 216 17 case examples, 217 21 confounding variables, 217 control group related, 217 halo effect, 219 20 Hawthorne effect, 207 Research simulations checklist questions for, 530 32 critical decision points, 532 examples of, 532 methodology, 529 30 Restricted range, 285 86 r k chi square (r by k), 358 calculation, 359 60 contingency table, 358 59 degrees of freedom, 356 57 interpretation, 360 variations of, 361 Robustness, 331 Rulon formula, 503 Sample, nature of, 137 Sample size, 156 57 Sample Standard deviation, 162 66 calculation, 162 66 Sampling bias, 141 outliers, 141 political polling, 142 46 random sampling, 138 40 representative sample, 138 sampling distributions, 146 54 sampling error, 140 41 stratified (quota) sampling, 140 Sampling distributions central limit theorem, 151 of difference, 230 infinite vs. finite sampling, 148 49 mean of distribution of means, 146 48 parameters, importance of, 152 54 standard error of the mean, 150 51, 153 Scatter plot, 277 bivariate scatter plot, 389 91 configurations, 279 80 Secondary variance, 217 Significance correlation, 284 hypothesis of difference and, 229 30, 239, 242 nature of, 173 Pearson r and, 182 83, 284 85 t ratio, 172 two samples, evaluating, 235 38 Significant interaction, 341 Single-sample t ratio, 175 78 confidence intervals and, 183 t comparison, 175 78 Skewed distributions mean, 33 median, 36 skewness assessing, 42, 570 working with, 39 Spearman, Charles, 293 Spearman r, 293 300 calculating for non-normal distributions of interval data, 297 calculating with interval data, 295 calculating with ordinal data, 294 95 requirements for, 299 in research simulations, 537 Spearman-Brown prophecy formula, 481 Split-half reliability, 482 Squares (See Sum of squares) SPSS (statistical Analysis Package for the Social Scientist, 517 19, 586 Standard deviation, 50 54 computational method, 52 53 deviation method, 50 54 estimated standard deviation, 162 65 outliers, 141 relationship to range, 58 59 sample standard deviation, 166 68 unbiased estimator, 163 from z scores to, 95 99 Standard error, meaning in statistics, 151 Standard error of difference, 233 Standard error of estimate, 401 03 calculation, 402 confidence interval, 402 03 Standard error of measurement, 500 Standard error of the mean, 150 51, 153 estimated standard error of the mean, 166 68

indexerrt.qxd 8/21/2002 9:47 AM Page 660 660 Index www.ablongman.com/sprinthall7e Stanines, 105 08 normal curve, 105 08 table, 105 08 z scores, 105 08 Statistics nature of, 17, 137 38 common stumbling blocks related to, 4 12 descriptive statistics, 17 18 history of, 14 17 inferential statistics, 18 Stepwise regression, 416 Stevens, S.S., 195 Stochastic model, 277 Stratified (quota) sampling, 140 Subject independent variables, 191 Sum of squares, 317, 322 components of variability, 318 computational method, 318 21 converting to variance estimates, 322 24 degrees of freedom, 324 F ratio, 324 26 interpretation of, 321 mean square, 325 Test bias, 476 Test reliability, 477 Alternate or parallel form, 480 Correlation strength, determination of, 284 86 Internal consistency, 480 81 Spearman-Brown prophecy, 481 Split-half method, 482 Techniques for increasing reliability, 488 Test-retest method, 478 Test Validity, 477 Concurrent, 491 Construct, 495 Content, 491 Face, 490 Predictive, 492 93 t ratio, 169 73 calculation, 236 degrees of freedom, 169 70, 243, 247 equal size samples, 236 for independent samples, 236 38, 243 limitations, 253 54 negative, 251 one-tail t test, 249, 251 paired t ratio, 429 34 requirements for, 254 significance, 173, 239 sign of, 172, 244, 251 single-sample t ratio, 170 t comparison, 244 two samples, value of, 253 two-tail t table, 170, 243, 577 two-tail t test, 243 unequal size samples, 238 39 z score and, 234 35 T score applications, 101 04 calculations, 100 04 from raw scores, 101 04 to raw scores, 102 04 t table decisions about using tables, 251 52 one-tail t table, 249 52, 578 two-tail t table, 170, 577 t test alternative hypothesis for, 242, 249 vs. correlation coefficient, 300 null hypothesis for, 241 42 in research simulations, 536, 543 successive, drawbacks of, 314 315 two-tail t test, 243 True experiment, 202, 204 Tukey s HSD, 328 30, 444 applications, 329 30 calculation, 328 29, 444 interpretation, 329 within-subjects F ratio, 441 46 2 2 chi square, 362, 365, 367 Yates correction, 362, 366 Two-tail t table, 170, 243, 577 sign of t ratio, 172 t comparison, 173 Two-tail t test, 243 Two-way ANOVA, 332 36, 342 43 Type 1 error, 173 78, 254 (See also Alpha level) Type 2 error, 254, 435 (See also Beta level) Unbiased estimate, 163 Unimodal distributions, 38 39 U test (See Mann-Whitney U test) Validity (See Test validity) Variability, measures of graphs and, 57 kurtosis, 58 range, 48 50 standard deviation, 50 54 variance, 54 zero, value in, 54 55 Variables, 190 93 confounding variables, 217 dependent variables, 191 93 independent variables, 191 93 Variance, 54 (See also Analysis of variance) calculation, 54 55 homogeneity of variance, 327 Variance estimates, sum of squares converted to, 322 24 Wilcoxon T test, 463 65 procedure, 463 65 in research simulations, 549 sample size, 464 Wilks Lambda, 637 Within-subjects design, 205, 441 Within-subjects F ratio, 441 46 calculation, 441 46 correlation within subjects, importance of, 445 compared to factorial ANOVA, 446 interpretation, 446 in research simulations, 532 Tukey s HSD, 444 Wow! graph, 29 30 Yates correction, 362 McNemar test, 369 71 2 2 chi square, 362 Zero correlations, 274, 277 z scores, 69 86 applications, 78 80 areas of normal curve, 80 82 chi square and, 368 equation, 78 79 to mean, 98 99 parameter estimates and, 154 56 Pearson r, 280 percentage rules, 86 from percentiles, 91 96 to percentiles, 73 78 z test, 155 56