Frequently Used Statistics Formulas and Tables Cheat Sheet, Cheat Sheet of Statistics

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Frequently Used Statistics Formulas and Tables Chapter 2 highest value - lowest valueClass Width = (increase to next integer) number classes upper limit + lower limitClass Midpoint = 2 Chapter 3 sample size population size frequency n N f sum w weight = = = Σ = = Sample mean: Population mean: ( )Weighted mean: ( )Mean for frequency table: highest value + lowest valueMidrange 2 xx n x N w xx w f xx f µ ∑ = ∑ = ∑ • = ∑ ∑ • = ∑ = 2 2 2 2 Range = Highest value - Lowest value ( )Sample standard deviation: 1 ( )Population standard deviation: Sample variance: Population variance: x xs n x N s µσ σ ∑ − = − ∑ − = Chapter 3 Limits for Unusual Data Below : - 2 Above: 2 µ σ µ σ+ Empirical Rule About 68%: - to About 95%: -2 to 2 About 99.7%: -3 to 3 µ σ µ σ µ σ µ σ µ σ µ σ + + + 2 2 Sample coefficient of variation: 100% Population coefficient of variation: 100% Sample standard deviation for frequency table: [ ( ) ] [ ( ) ] ( 1) sCV x CV n f x f xs n n σ µ = = ∑ • − ∑ • = −   Sample z-score: Population z-score: x xz s xz µ σ − = − = 3 1 1 3 Interquartile Range: (IQR) Modified Box Plot Outliers lower limit: Q - 1.5 (IQR) upper limit: Q + 1.5 (IQR) Q Q= − 2 Chapter 4 Probability of the complement of event ( ) = 1 - ( ) Multiplication rule for independent events ( ) ( ) ( ) General multiplication rules ( ) ( ) ( , ) A P not A P A P A and B P A P B P A and B P A P B given A = • = • ( ) ( ) ( , ) Addition rule for mutually exclusive events ( ) ( ) + ( ) General addition rule ( ) ( ) + ( ) ( ) P A and B P A P A given B P A or B P A P B P A or B P A P B P A and B = • = = − !Permutation rule: ( )!n r nP n r = − !Combination rule: !( )!n r nC r n r = − Permutation and Combination on TI 83/84 n Math PRB nPr enter r n Math PRB nCr enter r Note: textbooks and formula sheets interchange “r” and “x” for number of successes Chapter 5 Discrete Probability Distributions: 2 2 Mean of a discrete probability distribution: [ ( )] Standard deviation of a probability distribution: [ ( )] x P x x P x µ σ µ = ∑ • = ∑ • − Binomial Distributions number of successes (or x) probability of success = probability of failure 1 = 1 Binomial probability distribution ( ) Mean: Standard deviation: r n r n r r p q q p p q P r C p q np npq µ σ − = = = − + = = = Poisson Distributions 2 number of successes (or ) = mean number of successes (over a given interval) Poisson probability distribution ( ) ! 2.71828 (over some interval) r r x eP r r e mean µ µ µ µ σ µ σ µ − = = ≈ = = = 5 Chapter 10 Regression and Correlation 2 2 2 2 2 Linear Correlation Coefficient (r) ( )( ) ( ) ( ) ( ) ( ) OR ( ) where z score for x and z score for y 1 explained variationCoefficient of Determination: total v x y x y n xy x yr n x x n y y z z r z z n r ∑ − ∑ ∑ = ∑ − ∑ ∑ − ∑ Σ = = = − = 2 2 0 1 2 0 / 2 2 2 2 ariation ˆ( )Standard Error of Estimate: s 2 or s 2 ˆ ˆPrediction Interval: ( )1where 1 ( ) ( ) Sample test statistic for with 1 2 e e e y y n y b y b xy n y E y y E n x x E t s n n x x r rt r n α ∑ − = − ∑ − ∑ − ∑ = − − < < + − = + + Σ − Σ = − − . . 2d f n= − Least-Squares Line (Regression Line or Line of Best Fit) 0 1 0 1 1 12 2 2 0 0 12 2 ˆ note that is the y-intercept and is the slope ( )( ) where or ( ) ( ) ( )( ) ( )( ) where or ( ) ( ) y x y b b x b b sn xy x yb b r sn x x and y x x xyb b y b x n x x = + ∑ − ∑ ∑ = = ∑ − ∑ ∑ ∑ − ∑ ∑ = = − ∑ − ∑ 0 0 0 0 2 / 2 2 2 1 1 1 1 / 2 2 2 Confidence interval for y-intercept 1 where E = ( ) Confidence interval for slope where E = ( ) e e b E b E xt s n xx n b E b E s t xx n α α β β β β − < < + + ∑ ∑ − − < < + • ∑ ∑ − Chapter 11 2 2 ( ) (row total)(column total) where sample size Tests of Independence . . ( 1)( 1) Goodness of fit . . (number of categories) 1 O E E E d f R C d f χ −= ∑ = = − − = − Chapter 12 One Way ANOVA 2 2 2 2 all groups 2 2 all groups number of groups; total sample size ( ) ( ) ( ) ( ) where . . 1 . . TOT TOT TOT i TOT BET i i W i i TOT BET W BET BET BET BET W W k N x SS x N x x SS n N x SS x n SS SS SS SSMS d f k d f SS MS d = = ∑ = ∑ −  ∑ ∑ = −      ∑ = ∑ −     = + = = − = ∑ ∑ where . . . . WW d f N k f = − where . . numerator = . . 1 . . denominator = . . number of rows; number of columns row factor Row factor : error column factor Column factor : BET BET W W MSF d f d f k MS d f d f N k r c MSF MS MSF MS = = − = − = = Two - Way ANOVA error interaction Interaction : error with degrees of freedom for row factor = 1 column factor = 1 interaction = ( 1)( 1) error = ( 1) MSF MS r c r c rc n − − − − − NEGATIVE z Scores Standard Normal (z) Distribution: Cumulative Area from the LEFT Zz | 00 .01 02 03 04 05 .06 07 08 09 0003 0003. --.0003.—S «0003S «0003S «0003._—S «0003. —S«0008_~—S 0003-0002 -32 0007 0007. .0006_-—«.0006._—S=——«0006._—S«0006._~—S«0006._—S0005_~— 0005-0005, 0013 0013 0013 .0012 0012 .00n1 .00n oon 0010 ~—-.0010 -28 0026 0025 0024-0028. 0023._ = 0022S 0021_~—S0021_~—S 00200019 0047 0045 0044. 0043. = 0041_~— 004000390038 ~— 00370036 0082 0080 0078 = 0075. .0073.—0071 0069 .0068 -2.2 0139 086 0132 0129 0125 0122 ong oe 0228 102220217 0212 0207. 02020197 0192 | -18 | 0359 0351 0344 0336. = 0329-0322, 0314 .0307 | -16 | 0548 0537 0526 .0516 0505 * 0495 0485 0475 | -14 | .o808 .0793 0778 .0764 .0749 m51 M31 m2 1093 1075 1056 1038 1020 | -10 | 1587 1562 1539 1515 1492 1469 1446 1423 | -os | .2n9 2090 2061 .2033 2005 | 1977 1949 1922 | -o6 | 2743 27092676 = 26432611 | -04 | 3446 3409 33723336. = 3300 4207 A168 4129 4090 .4052 -0.0 -5000 4960 4920 -4880 4840 | 4801 4761 4721 -4681 4641 NOTE: For values of z below —3.49, use 0.0001 for the area. *Use these common values that result from interpolation: Z score Area =1.645 = 0.0500, << -2.575 0.0050 POSITIVE z Scores (continued) Cumulative Area from the LEFT 00 1 .02 03 04 OS 06 07 08 .09 j 04 | jos | .5000 .5040 .5080 .5120 5160 5199 5793 5832 .5871 5910 5948 5987 6554 6591 6628 6664 .6700 6736 7257 7291 7324 7357 7389 7422 5239 5279 6026 6064 6772 6808 7454 7486 5319 5359 6103 6141 6844 6879 7517 7549 | og | .7881 p10 | p14 | p16 | ps | .9861 9960 7910 .7939 .7967 7995 8023 8051 8078 8106 .8133 8413 8438 8461 .8485 8508 8531 8554 .8577 8599 .8621 .8849 8869 8888 .8907 8925, 8944 8962 8980 .8997 9015, 9192 .9207 9222 9236 .9251 9265 9279 9292 .9306 9319 .9452 .9463 9474 9484 9495 * 9505 9515 .9525, 9535, 9545 9641 .9649 .9656 .9664 .9671 .9678 9686 .9693 9699 .9706 9772 .9778 .9783 .9788 9793 .9798 9803 .9808 9812 .9817 .9864 .9868 9871 9875 .9878 9881 .9884 .9887 9890 9918 .9920 .9922 9925 .9927 .9929 9931 9932 9934 .9936 .9953 .9955 .9956 .9957 .9959 9961 9962 .9963 .9964 9974 9975 9976 9977 9977 .9993 .9993 .9994 .9994 .9994 .9997 .9997 .9997 9997 .9997 9997 .9997 9997 NOTE: For values of z above 3.49, use 0.9999 for the area. *Use these common values that result from interpolation: Z score Area Common Critical Values Confidence Critical Level Value 1.645, .9500 0.90 1.645, 2.575 0.9950 0.95 1.96 0.99 2575 Formulas and Tables by Mario F. Triola Copyright 2010 Pearson Education, Inc. yes") Chi-Square (x) Distribution Area to the Right of the Critical Value Degrees of Freedom | [0995 099 0975 095 090] | o10 0.05 0025 001 0.008 | 1 = = 0001 0004 0016 2706 3841 5024 6635 7879 0072 ONS 0216 §=0.352. 0.584 6.251 7815 9348 1345 12.838 0412 0554 0831 1145 1610 9236 11.071 12833 15.086 16.750 0.989 1239 1690 2167 2833 12017 14067 16.013 18475 20278 | 9 | 1735 2088 2700 3325 4168 14684 16.919 19023 21666 23.589 | 1 | 2603 3.053 3816 4575 5578 17275 19675 21920 24.725 26.757 3565 4107 5.009 5892 7042 19812 22.362 24.736 27688 29.819 4601 5.229 6.262 7261 «8547-22307 24.996 27.488 30.578 32.801 5.697 6408 7564 8672 10.085 24769 27587 30191 33.409 35.718 |} 19 | 6844 7633 8907 1017 11.651 27204 30144 32.852 36191 38.582 8034 8897 10.283 11591 13.240 29615 32.671 35.479 38.932 41.401 9.260 10196 11689 13.091 14848 32007 35172 38.076 = 41638 44.181 10.520 11.524 13120 14.61 16.473 34382 37652 40646 «44.314 46.928 1.808 12879 14573 16151 18114 36.741 = 40.113 43194 46.963 49.645 1312114257 -16.047 17708 19.768 39.087 42.557 45.722 49.588 52.336 | 40 | 20.707 22164 24.433 26.509 29.051 51805 55758 59342 63.691 66.766 | 6o | 35534 37485 40482 43188 46.459 74.397 79.082 83.298 + 88.379——91.952 | so | 51172 53540 57153 60.391 64.278 96.578 101879 106.629 112329 116.321 100 67.328 70.065 74.222 77929 82358 118.498 124.342—*129.561_—«*135.807 140.169 From Donald 8. Owen, Handbook of Statistical Tables, © 1962 Addison-Wesley Publishing Co., Reading, MA. Reprinted with permission of the publisher. Degrees of Freedom n-1 for confidence intervals or hypothesis tests with a standard deviation or variance k-l for goodness-of-fit with k categories (r—1)(¢ — 1) for contingency tables with r rows and c columns k-l for Kruskal-Wallis test with k samples Determining Sample Size for Population Variance or Standard Deviation Table 7-2 SE ele ee Tobe 9 Peete ma 409 50% To be 99 eg Pu ee ae the sample size n should be at least 3.149 806 ai 38 a7 38 of the value of a ee Butte 133,449 5,458 1,402 369 172 101 68 [table 7-2 from page 350, Triola 4" edition) Se Red oes! BL) ee te sis within Sh 10%, 30% 40% 50% To be 99% es eA) 10% 20% 30% 40% 50% eee eee should be at least 768 2 48 21 2 gl) a am Meu Pu Lat d 33,218 1,336 336 38 22 14 TABLE A-& (Critical Values of the 0.306 0.361 0.335 0.312 0.204 o.279 0.254 0.236 0.220 0.207 0.196 SSsasssshSus Pearson Correlation Coefficient r alpha = .01 0.990 0.959 007 NOTE: To test Hk p= 0 against H1: p # 0, reject HO if the absolute value of ris greater than the critical value in the table Greek Alphabet Greek Letter Mame Equivalent | Sound vvhen Spoken A «a | Alpha A al-fah B Bp Beta B bay-tah T ¥ Gamma G gam-ah A 3 Delta D del-tah Eo é Epsilon E ep-si-lon Zz ¢ Zeta Z zay-tah H Eta E ay-tay @ 86 Theta Th thay-tah To. lota | eye-o-tah E « Kappa K cap-ah A od Lambda L lamb-dah Mi wu Mu M mew N v Nu N new = €& Xi x zzEye Oo Omicron oO om-ah-cron II «a Pi P pie P p Rho R row Eo Sigma 3 sig-ma T Tt Tau T tawh Y ov Upsilon U oop-si-lon D 6 Phi Ph figh or fie x Y Chi Ch kigh Yow | Psi Ps sigh Qo Omega 0 o-may-gah http://www.keyway.ca/htm2002/greekal.htm