Standard Normal Distribution 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.0000 0.0040 0.0080 0.0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359 0.1 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753 0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.1026 0.1064 0.1103 0.1141 0.3 0.1179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517 0.4 0.1554 0.1591 0.1628 0.1664 0.1700 0.1736 0.1772 0.1808 0.1844 0.1879 0.5 0.1915 0.1950 0.1985 0.2019 0.2054 0.2088 0.2123 0.2157 0.2190 0.2224 0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2517 0.2549 0.7 0.2580 0.2611 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852 0.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133 0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389 1.0 0.3413 0.3438 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621 1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830 1.2 0.3849 0.3869 0.3888 0.3907 0.3925 0.3944 0.3962 0.3980 0.3997 0.4015 1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4115 0.4131 0.4147 0.4162 0.4177 1.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319 1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441 1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545 1.7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.4633 1.8 0.4641 0.4649 0.4656 0.4664 0.4671 0.4678 0.4686 0.4693 0.4699 0.4706 1.9 0.4713 0.4719 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767 2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817 2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857 2.2 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890 2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.4911 0.4913 0.4916 2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.4931 0.4932 0.4934 0.4936 2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952 2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964 2.7 0.4965 0.4966 0.4967 0.4968 0.4969 0.4970 0.4971 0.4972 0.4973 0.4974 2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4981 2.9 0.4981 0.4982 0.4982 0.4983 0.4984 0.4984 0.4985 0.4985 0.4986 0.4986 3.0 0.4987 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990 t distribution One-tail α two-tail α df 0.4 0.8 0.25 0.5 0.1 0.2 1.00 0.325 1.000 3.078 6.314 12.706 2.00 0.289 0.816 1.886 2.920 3.00 0.277 0.765 1.638 2.353 4.00 0.271 0.741 1.533 5.00 0.267 0.727 6.00 0.265 7.00 0.005 0.01 0.001 0.002 0.0005 0.001 31.821 63.657 318.310 636.619 4.303 6.965 9.925 22.326 31.599 3.182 4.541 5.841 10.213 12.924 2.132 2.776 3.747 4.604 7.173 8.610 1.476 2.015 2.571 3.365 4.032 5.893 6.869 0.718 1.440 1.943 2.447 3.143 3.707 5.208 5.959 0.263 0.711 1.415 1.895 2.365 2.998 3.499 4.785 5.408 8.00 0.262 0.706 1.397 1.860 2.306 2.896 3.355 4.501 5.041 9.00 0.261 0.703 1.383 1.833 2.262 2.821 3.250 4.297 4.781 10.00 0.260 0.700 1.372 1.812 2.228 2.764 3.169 4.144 4.587 11.00 0.260 0.697 1.363 1.796 2.201 2.718 3.106 4.025 4.437 12.00 0.259 0.695 1.356 1.782 2.179 2.681 3.055 3.930 4.318 13.00 0.259 0.694 1.350 1.771 2.160 2.650 3.012 3.852 4.221 14.00 0.258 0.692 1.345 1.761 2.145 2.624 2.977 3.787 4.141 15.00 0.258 0.691 1.341 1.753 2.131 2.602 2.947 3.733 4.073 16.00 0.258 0.690 1.337 1.746 2.120 2.583 2.921 3.686 4.015 17.00 0.257 0.689 1.333 1.740 2.110 2.567 2.898 3.646 3.965 18.00 0.257 0.688 1.330 1.734 2.101 2.552 2.878 3.610 3.922 19.00 0.257 0.688 1.328 1.729 2.093 2.539 2.861 3.579 3.883 20.00 0.257 0.687 1.325 1.725 2.086 2.528 2.845 3.552 3.850 21.00 0.257 0.686 1.323 1.721 2.080 2.518 2.831 3.527 3.819 22.00 0.256 0.686 1.321 1.717 2.074 2.508 2.819 3.505 3.792 23.00 0.256 0.685 1.319 1.714 2.069 2.500 2.807 3.485 3.768 24.00 0.256 0.685 1.318 1.711 2.064 2.492 2.797 3.467 3.745 25.00 0.256 0.684 1.316 1.708 2.060 2.485 2.787 3.450 3.725 26.00 0.256 0.684 1.315 1.706 2.056 2.479 2.779 3.435 3.707 27.00 0.256 0.684 1.314 1.703 2.052 2.473 2.771 3.421 3.690 28.00 0.256 0.683 1.313 1.701 2.048 2.467 2.763 3.408 3.674 29.00 0.256 0.683 1.311 1.699 2.045 2.462 2.756 3.396 3.659 30.00 0.256 0.683 1.310 1.697 2.042 2.457 2.750 3.385 3.646 40.00 0.681 1.303 1.684 2.021 2.423 2.704 3.307 3.551 50.00 0.679 1.299 1.676 2.009 2.403 2.678 3.261 3.496 60.00 0.679 1.296 1.671 2.000 2.390 2.660 3.232 3.460 90.00 0.677 1.291 1.662 1.987 2.369 2.632 3.185 3.403 1.289 1.658 1.980 2.358 2.617 3.160 3.373 1.282 1.645 1.960 2.326 2.576 3.090 3.291 120.00 inf 0.253 0.674 0.05 0.025 0.01 0.1 0.05 0.02 Critical values ofstudent's t MGSC2207 LEE EQUATIONS All equations on this page (except #6) are slight variations of just one equation (a) which have been designed to handle different real-life situations. In each row below are two variations of exactly the same equation (i.e., statistical test). a σ known Parametric statistical tests σ unknown one-sample t-test for means  s  1  X  t    n indep-groups t-test for diff b/t means 2 1 −  2 (X1 − X 2 )  t s2 s2 + n1 n2 b σ known z = Sep 2019 X−  1 - 2  s d  t d  n  d     n Tests of Means 1 t 2 s (X t = SS X = X X− = n 1 - X 2 ) − (1 −  2 ) s2 s2 + n1 n2 dep-groups t-test for diff b/t means 3 c σ unknown z = 3 t = d − d sd p1 q1 p 2 q 2 + n1 n2 s= (n1 − 1) s12 + (n 2 − 1) s 22 n1 + n 2 − 2 df = n - 1 = ( weighted ) avg std dev df = n1 + n2 - 2 SS d sd = nd − 1 (d ) 2 SS d = d − nd 2 nd 5a if H 0 : has the form  1 =  2 ( p1 − p 2 ) − ( 1 −  2 ) pq pq + n1 n2 d d = X1 − X 2 total no. of successes total sample size = weighted avg p p n + p 2 n2 = 1 1 n1 + n2 p= 5b if H 0 : has the form  1 =  2 + ? % ( p1 − p 2 ) − ( 1 −  2 ) z = Non-parametric statistical tests (for testing weak numbers) (no σ – there is no such thing as a population std dev for weak numbers) n Tests of Proportions (Binomial) p − 4 z=  (1 −  ) n z = 5  1 −  2 ( p1 − p 2 )  z n (X )2 SS X n −1 nd df = nd – 1 two-group z-test for diff b/t props − s s = d= one-sample z-test for prop pq pz 4  n 2 X− 6  p1 q1 p 2 q 2 + n1 n2 2 (O − E ) 2 =  E dfchisquare = (r – 1) (c – 1) multiply m arg inals of cell E cell = grand total Correlation/Regression r= SS XY SS X SSY n = number of cases k = number of predictors ρ = population correlation (this is the Greek letter rho, not a “p”)  Y = b0 + b1 X Y df = n – 2 b1 −  1 sb1 for any k SS ERROR = SSY − b1 * SS XY df = n – 2 Y SS ERROR n − k −1 1 SS X k = number of predictors e 1 ( X − X )2 + n SS X  t se 1 + ANOVA FEffect = = s ERROR = e s for any k  Y ts s for any k SS XY SS X  YX 1− r2 n−2 t SLOPE = b0 = Y − b1 X b1 = r− t COR = FREG = 1 (X − X )2 + n SS X MS EFFECT MS ERROR To compute the total variation or SSTOTAL, pretend that all N scores are in a single group and compute the variation there is among all N scores. In independent groups or BG designs, the error variation is the variation within groups SSWG. Any variation within a column is always error variation. Why? Because all individuals in a column received the same treatment and should, therefore, be identical. Adding up the variation in each of the treatment groups (i.e., within each column) gives the SSWG. In WG or RM designs, the error variation is the residual error variation SSRESIDUAL. Multiple comparison test q = studentized range statistic value SS   2 SS RESIDUAL =  (Y − Y ) df REGRESSION = k SS df RESIDUAL = n − k − 1 =X2 − TOTAL ( X ) TOTAL =  (Y − Y ) 2 Differences between Means: Dep Groups (or RM) 2 N df TOTAL = N −1 = se 2 SS REGRESSION =  (Y − Y ) MS REGRESSION MS RESIDUALl Differences between Means: Indep Groups b1 = SS1 + SS 2 +... + SSk df WG = k (n − 1) SS TOTAL = SS BG + SSWG For BG variation, make all scores in the first column equal to mean for that column. Do the same for each column. These scores have no variation within a group so the total variation among these scores must equal the variation between groups SSBG. Compute the SSBG using the equation above for SSTOTAL on these new scores (this new total variation contains no error variation and is, therefore, between-group variation only). An alternative is to use the equation below to compute the SSBG. df TOTAL = N −1 = SS1 + SS 2 +... SSk df W G = k (n − 1) For BG variation, see comment to left. Alternatively, SS BG = n( X1 − X ) 2 + n( X 2 − X ) 2 + ... df BG = k −1 Compute variation due to individual differences by making all scores in a row = mean for individual in that row (or use eqn below). Compute SSIND DIFFS using either the equation for SSTOTAL on these new scores or use equation below. SS IND DIFFS = k ( X A − X ) 2 + k ( X B − X ) 2 + ... df IND DIFFS = n −1 SS = SSWG − SS IND DIFFS 2 2 df RESIDUAL = ( k − 1)(n − 1) SS BG = n(X1 − X ) + n(X 2 − X ) + ... SS TOTAL = SS BG + SS IND DIFFS + SS RESIDUAL df BG = k −1 where N = total no. of scores n = no. of individuals k = no. of scores per person where N = total no. of scores n = no. of individuals k = no. of groups X= grand mean (i.e., mean of all scores) Min diff = q * RESIDUAL MS W G ERROR n X= grand mean (i.e., mean of all scores) Min diff = q * MS RESIDUAL ERROR n