STATISTICAL REFERENCE TABLE OF CONTENTS STATISTICS WITH UNWEIGHTED DATA .............................................................. 1 Mean ...................................................................................................................... 1 Variance ................................................................................................................ 1 Technical Comment: Variance of a Sample Proportion .................. 2 Standard Deviation .............................................................................................. 2 Standard Error..................................................................................................... 3 Grouped Median .................................................................................................. 3 Skewness and Kurtosis ........................................................................................ 3 STATISTICS WITH WEIGHTED DATA .................................................................... 5 Weighted Mean .................................................................................................... 5 Weighted Variance............................................................................................... 5 Weighted Standard Deviation ............................................................................. 6 Weighted Sample Size.......................................................................................... 6 Effective Sample Size ........................................................................................... 6 Weighted Standard Error ................................................................................... 7 CONFIDENCE INTERVALS FOR MEANS ............................................................... 8 SIGNIFICANCE TESTING ...........................................................................................13 T-TESTS – INDEPENDENT ..........................................................................................18 UNWEIGHTED DATA .......................................................................................18 Assume equal variances ...........................................................................18 Assume unequal variances ......................................................................19 Technical Comment: A Note on Degrees of Freedom ...................19 Letting WinCross determine whether variances are equal or not ......20 Part-Whole Comparisons ........................................................................20 SINGLY and MULTIPLY WEIGHTED DATA ..............................................21 Assume equal variances ...........................................................................22 Assume unequal variances ......................................................................22 Part-Whole Comparisons ........................................................................23 T-TESTS – DEPENDENT PAIRED/OVERLAP ( LOC+/VAR+) ..............................24 UNWEIGHTED DATA .......................................................................................25 t-Test for Means with Partial Pairing .....................................................25 Technical Comment: On Calculating Covariances .......................26 Technical Comment: A Note on Perfect Pairing ............................27 Part-Whole Comparisons ........................................................................28 SINGLY WEIGHTED DATA ............................................................................30 t-Test for Means with Partial Pairing ....................................................30 Technical Comment: A Note on Perfect Pairing .............................32 Part-Whole Comparisons ........................................................................32 MULTIPLY WEIGHTED DATA ......................................................................35 t-Test for Means with Partial Pairing ....................................................35 T-TESTS – DEPENDENT PAIRED/OVERLAP (MULTI) ........................................37 UNWEIGHTED DATA .......................................................................................37 Part-Whole Comparisons ........................................................................39 SINGLY WEIGHTED DATA ............................................................................41 Part-Whole Comparisons ........................................................................43 MULTIPLY WEIGHTED DATA ......................................................................44 Z-TESTS - INDEPENDENT ..........................................................................................46 UNWEIGHTED DATA .......................................................................................46 Using unpooled proportions ....................................................................46 Using pooled proportions ........................................................................47 Technical Comment: Testing for Equality of Two Multinomial Proportions .................................................................47 Part-Whole Comparisons ........................................................................48 SINGLY and MULTIPLY WEIGHTED DATA ..............................................50 Using unpooled proportions ......................................................................50 Using pooled proportions ..........................................................................50 Part-Whole Comparisons .........................................................................51 Z-TESTS – DEPENDENT PAIRED/OVERLAP ( LOC+/VAR+) ..............................53 UNWEIGHTED DATA .......................................................................................53 z-Test for Proportions with Partial Pairing...........................................53 Technical Comment: A Note on Perfect Pairing ..........................54 Part-Whole Comparisons ........................................................................55 SINGLY WEIGHTED DATA ............................................................................56 z-Test for Proportions with Partial Pairing...........................................57 Technical Comment: A Note on Perfect Pairing ...........................58 Part-Whole Comparisons ........................................................................58 MULTIPLY WEIGHTED DATA ......................................................................60 Z-TESTS – DEPENDENT PAIRED/OVERLAP (MULTI) ........................................62 UNWEIGHTED DATA .......................................................................................62 Part-Whole Comparisons ........................................................................63 SINGLY WEIGHTED DATA ............................................................................64 Part-Whole Comparisons ........................................................................65 MULTIPLY WEIGHTED DATA ......................................................................66 COMPARING VOLUMETRIC PERCENTAGES ......................................................68 DEPENDENT PAIRED/OVERLAP ( LOC+/VAR+) ............................................69 UNWEIGHTED DATA .......................................................................................69 SINGLY WEIGHTED DATA ............................................................................72 MULTIPLY WEIGHTED DATA ......................................................................73 DEPENDENT PAIRED/OVERLAP (MULTI) ........................................................74 UNWEIGHTED DATA .......................................................................................74 SINGLY WEIGHTED DATA ............................................................................76 MULTIPLY WEIGHTED DATA .....................................................................78 COMPARISON WITH TOTAL ...............................................................................79 LOC+/VAR+: UNWEIGHTED & SINGLY WEIGHTED .............................80 MULTI: UNWEIGHTED & SINGLY WEIGHTED ........................................82 ONE-WAY ANOVA ........................................................................................................85 CHI-SQUARE ..................................................................................................................90 FACTOR ANALYSIS .....................................................................................................91 SAMPLE BALANCING ................................................................................................96 Goodness-of-fit minimization technique .............................................................98 Iterative proportional fitting technique .............................................................99 REGRESSION ...............................................................................................................104 Appendix I ......................................................................................................................111 Critical Value for t-Distribution Table ........................................................................112 STATISTICAL REFERENCE General Notation Unless otherwise specified n will denote the number of observations in the data set, and the observations will be denoted by x1, x2, …, xn. If the observations can only take on a specific set of possible values, k will