FORMULAS FOR STATISTICS 1

FORMULAS FOR STATISTICS 1 X = 1 n Sample statistics X i or x = 1 n x i (sample mean) S 2 = 1 n 1 s 2 = 1 n 1 (X i X) 2 = 1 n 1 (x i x) 2 = 1 n 1 Xi 2 n n 1 X 2 x 2 i n n 1 x 2 or (sample variance) E(X) = µ, var(x) = σ2 n, E(S 2 ) = σ 2 Sample statistics for normal distribution Z = X µ σ/ n T = X µ S/ n N(0, 1) t with n 1 degrees of freedom V = (n 1)S2 σ 2 = (X i X) 2 σ 2 χ 2 with n 1 degrees of freedom Z = (X 1 X 2 ) (µ 1 µ 2 ) σ 2 1 /n 1 + σ 2 2/n 2 N(0, 1) T = (X 1 X 2 ) (µ 1 µ 2 ) S p 1/n1 + 1/n 2 t with n 1 + n 2 2 degrees of freedom, where S 2 p = (n 1 1)S 2 1 + (n 2 1)S 2 2 n 1 + n 2 2 W = (X 1 X 2 ) (µ 1 µ 2 ) S 2 1 /n 1 + S 2 2/n 2 t with assuming σ 1 = σ 2 (a 1 + a 2 ) 2 a 2 1/(n 1 1) + a 2 2/(n 2 1) degrees of freedom, where a 1 = s2 1 n 1 and a 2 = s2 2 n 2 (Welch Satterthwaite approximation) F = S2 1/σ 2 1 S 2 2/σ 2 2 F with n 1 1 and n 2 1 degrees of freedom i

Point estimation Parameter θ Estimate ˆθ Estimator ˆΘ µ ˆµ = x X σ 2 σ 2 = s 2 S 2 m ˆm = q(0.5) Q(0.5) Interval estimation of expectation Z = X µ σ/ n : x ± z σ α/2 n T = X µ S/ n : x ± t s α/2 n (with n 1 degrees of freedom) Z = (X 1 X 2 ) (µ 1 µ 2 ) σ1 2 : (x 1 x 2 ) ± z α/2 + σ2 2 σ 2 1 /n 1 + σ2/n 2 2 n 1 n 2 T = (X 1 X 2 ) (µ 1 µ 2 ) 1 : (x 1 x 2 ) ± t α/2 s p + 1 S p 1/n1 + 1/n 2 n 1 n 2 (with n 1 + n 2 2 degrees of freedom) W = (X 1 X 2 ) (µ 1 µ 2 ) s 2 1 : (x 1 x 2 ) ± t α/2 + s2 2 (Welch Satterthwaite) S 2 1 /n 1 + S2/n 2 2 n 1 (with = (a 1 + a 2 ) 2 a 2 1/(n 1 1) + a 2 2/(n 2 1) degrees of freedom, where a 1 = s2 1 n 1 and a 2 = s2 2 n 2 ) n 2 Estimation of proportion for the binomial distribution ( ) n P(X = x) = p x (1 p) n x x E(X) = np, var(x) = np(1 p) ˆp = x n i=x ( ) n ˆp i i L(1 ˆp L ) n i = α 2, x i=0 ( ) n ˆp i i U(1 ˆp U ) n i = α 2 V = F = S2 1/σ 2 1 S 2 2/σ 2 2 (n 1)S2 σ 2 : : s 2 1 s 2 2 Interval estimation of variance (n 1)s 2 h 2,α/2 and 1 f 2,α/2 and s2 1 s 2 2 1 f 1,α/2 (n 1)s2 h 1,α/2 (with n 1 degrees of freedom) (with n 1 1 and n 2 1 degrees of freedom) ii

Testing expectations z = x µ 0 σ/ n and H 0 : µ = µ 0 : H 1 Critical region P-probability µ > µ 0 z z α 1 Φ(z) µ < µ 0 z z α Φ(z) µ µ 0 z z α/2 2 min ( Φ(z), 1 Φ(z) ) Φ is the standard normal cumulative distribution function. t = x µ 0 s/ n and H 0 : µ = µ 0 : H 1 Critical region P-probability µ > µ 0 t t α 1 F (t) µ < µ 0 t t α F (t) µ µ 0 t t α/2 2 min ( F (t), 1 F (t) ) F is the cumulative t-distribution function with n 1 degrees of freedom. z = x 1 x 2 d 0 and H 0 : µ 1 µ 2 = d 0 : σ 2 1 /n 1 + σ2/n 2 2 H 1 Critical region P-probability µ 1 µ 2 > d 0 z z α 1 Φ(z) µ 1 µ 2 < d 0 z z α Φ(z) µ 1 µ 2 d 0 z z α/2 2 min ( Φ(z), 1 Φ(z) ) Φ is the standard normal cumulative distribution function. t = x 1 x 2 d 0, where s 2 p = (n 1 1)s 2 1 + (n 2 1)s 2 2, and H 0 : µ 1 µ 2 = d 0 : s p 1/n1 + 1/n 2 n 1 + n 2 2 H 1 Critical region P-probability µ 1 µ 2 > d 0 t t α 1 F (t) µ 1 µ 2 < d 0 t t α F (t) µ 1 µ 2 d 0 t t α/2 2 min ( F (t), 1 F (t) ) F is the cumulative t-distribution function with n 1 + n 2 2 degrees of freedom. t = x 1 x 2 d 0 and H 0 : µ 1 µ 2 = d 0 (Welch Satterthwaite) : s 2 1 /n 1 + s 2 2/n 2 H 1 Critical region P-probability µ 1 µ 2 > d 0 t t α 1 F (t) µ 1 µ 2 < d 0 t t α F (t) µ 1 µ 2 d 0 t t α/2 2 min ( F (t), 1 F (t) ) F is approximatively the cumulative t-distribution function with (a 1 + a 2 ) 2 a 2 1/(n 1 1) + a 2 2/(n 2 1) degrees of freedom, where a 1 = s2 1 and a 2 = s2 2. n 1 n 2 iii

v = Testing variances (n 1)s2 σ 2 0 and H 0 : σ 2 = σ 2 0 : H 1 Critical region P-probability σ 2 > σ0 2 v h 2,α 1 F (v) σ 2 < σ0 2 v h 1,α F (v) σ 2 σ0 2 v h 1,α/2 or v h 2,α/2 2 min ( F (v), 1 F (v) ) F is the cumulative χ 2 -distribution function with n 1 degrees of freedom. f = 1 k s 2 1 s 2 2 and H 0 : σ 2 1 = kσ 2 2 : H 1 Critical region P-probability σ1 2 > kσ2 2 f f 2,α 1 G(f) σ1 2 < kσ2 2 f f 1,α G(f) σ1 2 kσ2 2 f f 1,α/2 tai f f 2,α/2 2 min ( G(f), 1 G(f) ) G is the cumulative F-distribution function with n 1 1 and n 2 1 degrees of freedom. Goodness-of-fit test H o : P(T 1 ) = p 1,..., P(T k ) = p k H = k (F i np i ) 2 np i χ 2 with k 1 degrees of freedom Independence test. Contingence tables P(T 1 ) = p 1,..., P(T k ) = p k and P(S 1 ) = q 1,..., P(S l ) = q l P(T i S j ) = p i,j (i = 1,..., k and j = 1,..., l) H 0 : p i,j = p i q j (i = 1,..., k and j = 1,..., l) H = k l (F i,j F i G j /n) 2 j=1 F i G j /n χ 2 with (k 1)(l 1) degrees of freedom Contingence table: S 1 S 2 S l Σ T 1 f 1,1 f 1,2 f 1,l f 1 T 2 f 2,1 f 2,2 f 2,l f 2........ T k f k,1 f k,2 f k,l f k Σ g 1 g 2 g l n iv

Maximum likelihood estimation L(θ 1,..., θ m ; x 1,..., x n ) = f(x 1 ; θ 1,..., θ m ) f(x n ; θ 1,..., θ m ) l(θ 1,..., θ m ; x 1,..., x n ) = ln f(x 1 ; θ 1,..., θ m ) + + ln f(x n ; θ 1,..., θ m ) (ˆθ 1,..., ˆθ m ) = argmax L(θ 1,..., θ m ; x 1,..., x n ) or θ 1,...,θ m (ˆθ 1,..., ˆθ m ) = argmax θ 1,...,θ m l(θ 1,..., θ m ; x 1,..., x n ) Model: Data: Linear regression y = β 0 + β 1 x 1 + + β k x k + ɛ x 1 x 2 x k y x 1,1 x 1,2 x 1,k y 1 x 2,1 x 2,2 x 2,k y 2.... x n,1 x n,2 x n,k y n 1 x 1,1 x 1,2 x 1,k y 1 ɛ 1 β 0 1 x 2,1 x 2,2 x 2,k X =......., y = y 2., ɛ = ɛ 2., β = β 1. 1 x n,1 x n,2 x n,k y n Data model: y = Xβ + ɛ. ɛ n β k ˆβ = b = (X T X) 1 X T y = β + (X T X) 1 X T ɛ C = (c ij ) = (X T X) 1 H = XCX T (hat matrix) P = I n H E(b i ) = β i, var(b i ) = c ii σ 2 and cov(b i, b j ) = c ij σ 2 σ 2 = ŷ = Xb e = y ŷ = Py = Pɛ SSE = e 2 = e 2 i = (y i ŷ i ) 2 SSE n k 1 = MSE, MSE = RMSE v

SST = (y i y) 2, SSR = (ŷ i y) 2 SST = SSE + SSR MST = SST n 1, MSR = SSR k ANOVA-table: Source of variation Degrees of freedom Sums of squares Mean squares F Regression Residual Total variation k n k 1 n 1 SSR SSE SST MSR σ 2 = MSE (MST) F = MSR MSE H 0 : β 1 = = β k = 0 : F = MSR MSE F with k and n k 1 degrees of freedom H 0 : β i = β 0,i ; T i = b i β 0,i RMSE c ii t with n k 1 degrees of freedom R 2 = SSR SST = 1 SSE SST Radj 2 = 1 MSE MST = 1 n 1 n k 1 Categorical regressors: z i z i,1 z i,2 z i,mi 1 A i,1 1 0 0 A i,2 0 1 0.... A i,mi 1 0 0 1 A i,mi 0 0 0 SSE SST, y = β 0 + β 1 x 1 + + β k x k + l (β i,1 z i,1 + + β i,mi 1z i,mi 1) + ɛ Logistical regression P(y = A) = 1 1 + e β 0 β 1 x 1 β k x k L(β 0,..., β k ) = L 1 (β 0,..., β k ) L n (β 0,..., β k ), where 1 p i = 1 + e β 0 β 1 x i,1 β k x i,k, if y i = A L i (β 0,..., β k ) = 1 p i = e β 0 β 1 x i,1 β k x i,k 1 + e β 0 β 1 x i,1 β k x i,k, if y i = B vi

(b 0, b 1,..., b k ) = argmax β 0,β 1...,β k L(β 0, β 1,..., β k ) ˆp 0 = 1 1 + e b 0 b 1 x 0,1 b k x 0,k. H = 12 n(n + 1) k j=1 Kruskal Wallis-test H 0 : µ 1 = = µ k W 2 j n j 3(n + 1) χ 2 with k 1 degrees of freedom Spearman rank correlation coefficient Samples: x 1,1,...,x 1,n and x 2,1,...,x 2,n Sequence numbers: r 1,1,...,r 1,n and r 2,1,...,r 2,n r S = r S = 1 (r 1,i r)(r 2,i r) (r 1,i r) 2 6 n(n 2 1) (r 2,i r) 2, where r = d 2 i, where d i = r 1,i r 2,i n + 1 2 (Assuming there are no duplicate numbers in the samples!) vii

Standard normal distribution Quantiles of the standard normal distribution z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5190 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7969 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8513 0.8554 0.8577 0.8529 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9215 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9492 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998 viii

χ 2 -distribution Quantiles of the chi-square distribution (left tail) d.o.f Quantile (left tail) 0.005 0.010 0.025 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.975 0.99 0.995 1 0.39E-4 0.00016 0.00098 0.0039 0.0158 0.102 0.455 1.32 2.71 3.84 5.02 6.63 7.88 2 0.0100 0.0201 0.0506 0.103 0.211 0.575 1.39 2.77 4.61 5.99 7.38 9.21 10.6 3 0.0717 0.115 0.216 0.352 0.584 1.21 2.37 4.11 6.25 7.81 9.35 11.3 12.8 4 0.207 0.297 0.484 0.711 1.06 1.92 3.36 5.39 7.78 9.49 11.1 13.3 14.9 5 0.412 0.554 0.831 1.15 1.61 2.67 4.35 6.63 9.24 11.1 12.8 15.1 16.7 6 0.676 0.872 1.24 1.64 2.20 3.45 5.35 7.84 10.6 12.6 14.4 16.8 18.5 7 0.989 1.24 1.69 2.17 2.83 4.25 6.35 9.04 12.0 14.1 16.0 18.5 20.3 8 1.34 1.65 2.18 2.73 3.49 5.07 7.34 10.2 13.4 15.5 17.5 20.1 22.0 9 1.73 2.09 2.70 3.33 4.17 5.9 8.34 11.4 14.7 16.9 19.0 21.7 23.6 10 2.16 2.56 3.25 3.94 4.87 6.74 9.34 12.5 16.0 18.3 20.5 23.2 25.2 11 2.60 3.05 3.82 4.57 5.58 7.58 10.3 13.7 17.3 19.7 21.9 24.7 26.8 12 3.07 3.57 4.40 5.23 6.30 8.44 11.3 14.8 18.5 21.0 23.3 26.2 28.3 13 3.57 4.11 5.01 5.89 7.04 9.3 12.3 16.0 19.8 22.4 24.7 27.7 29.8 14 4.07 4.66 5.63 6.57 7.79 10.2 13.3 17.1 21.1 23.7 26.1 29.1 31.3 15 4.60 5.23 6.26 7.26 8.55 11.0 14.3 18.2 22.3 25.0 27.5 30.6 32.8 16 5.14 5.81 6.91 7.96 9.31 11.9 15.3 19.4 23.5 26.3 28.8 32.0 34.3 17 5.70 6.41 7.56 8.67 10.1 12.8 16.3 20.5 24.8 27.6 30.2 33.4 35.7 18 6.26 7.01 8.23 9.39 10.9 13.7 17.3 21.6 26.0 28.9 31.5 34.8 37.2 19 6.84 7.63 8.91 10.1 11.7 14.6 18.3 22.7 27.2 30.1 32.9 36.2 38.6 20 7.43 8.26 9.59 10.9 12.4 15.5 19.3 23.8 28.4 31.4 34.2 37.6 40.0 21 8.03 8.90 10.3 11.6 13.2 16.3 20.3 24.9 29.6 32.7 35.5 38.9 41.4 22 8.64 9.54 11.0 12.3 14.0 17.2 21.3 26.0 30.8 33.9 36.8 40.3 42.8 23 9.26 10.2 11.7 13.1 14.8 18.1 22.3 27.1 32.0 35.2 38.1 41.6 44.2 24 9.89 10.9 12.4 13.8 15.7 19.0 23.3 28.2 33.2 36.4 39.4 43.0 45.6 25 10.5 11.5 13.1 14.6 16.5 19.9 24.3 29.3 34.4 37.7 40.6 44.3 46.9 26 11.2 12.2 13.8 15.4 17.3 20.8 25.3 30.4 35.6 38.9 41.9 45.6 48.3 27 11.8 12.9 14.6 16.2 18.1 21.7 26.3 31.5 36.7 40.1 43.2 47.0 49.6 28 12.5 13.6 15.3 16.9 18.9 22.7 27.3 32.6 37.9 41.3 44.5 48.3 51.0 29 13.1 14.3 16.0 17.7 19.8 23.6 28.3 33.7 39.1 42.6 45.7 49.6 52.3 30 13.8 15.0 16.8 18.5 20.6 24.5 29.3 34.8 40.3 43.8 47.0 50.9 53.7 31 14.5 15.7 17.5 19.3 21.4 25.4 30.3 35.9 41.4 45.0 48.2 52.2 55.0 32 15.1 16.4 18.3 20.1 22.3 26.3 31.3 37.0 42.6 46.2 49.5 53.5 56.3 33 15.8 17.1 19.0 20.9 23.1 27.2 32.3 38.1 43.7 47.4 50.7 54.8 57.6 34 16.5 17.8 19.8 21.7 24.0 28.1 33.3 39.1 44.9 48.6 52.0 56.1 59.0 35 17.2 18.5 20.6 22.5 24.8 29.1 34.3 40.2 46.1 49.8 53.2 57.3 60.3 36 17.9 19.2 21.3 23.3 25.6 30.0 35.3 41.3 47.2 51.0 54.4 58.6 61.6 37 18.6 20.0 22.1 24.1 26.5 30.9 36.3 42.4 48.4 52.2 55.7 59.9 62.9 38 19.3 20.7 22.9 24.9 27.3 31.8 37.3 43.5 49.5 53.4 56.9 61.2 64.2 39 20.0 21.4 23.7 25.7 28.2 32.7 38.3 44.5 50.7 54.6 58.1 62.4 65.5 40 20.7 22.2 24.4 26.5 29.1 33.7 39.3 45.6 51.8 55.8 59.3 63.7 66.8 41 21.4 22.9 25.2 27.3 29.9 34.6 40.3 46.7 52.9 56.9 60.6 65.0 68.1 42 22.1 23.7 26.0 28.1 30.8 35.5 41.3 47.8 54.1 58.1 61.8 66.2 69.3 43 22.9 24.4 26.8 29.0 31.6 36.4 42.3 48.8 55.2 59.3 63.0 67.5 70.6 44 23.6 25.1 27.6 29.8 32.5 37.4 43.3 49.9 56.4 60.5 64.2 68.7 71.9 45 24.3 25.9 28.4 30.6 33.4 38.3 44.3 51.0 57.5 61.7 65.4 70.0 73.2 0.005 0.010 0.025 0.05 0.10 0.25 0.50 0.75 0.90 0.95 0.975 0.99 0.995 This table was produced using APL programs written by William Knight. ix

t-distribution Quantiles of the t-distribution 0.10 0.05 0.025 0.01 0.005 0.001 0.0005 (right tail) 0.20 0.10 0.05 0.02 0.01 0.002 0.001 (both tails) -------+---------------------------------------------------------+----- D 1 3.078 6.314 12.71 31.82 63.66 318.3 637 1 e 2 1.886 2.920 4.303 6.965 9.925 22.330 31.6 2 g 3 1.638 2.353 3.182 4.541 5.841 10.210 12.92 3 r 4 1.533 2.132 2.776 3.747 4.604 7.173 8.610 4 e 5 1.476 2.015 2.571 3.365 4.032 5.893 6.869 5 e 6 1.440 1.943 2.447 3.143 3.707 5.208 5.959 6 s 7 1.415 1.895 2.365 2.998 3.499 4.785 5.408 7 8 1.397 1.860 2.306 2.896 3.355 4.501 5.041 8 o 9 1.383 1.833 2.262 2.821 3.250 4.297 4.781 9 f 10 1.372 1.812 2.228 2.764 3.169 4.144 4.587 10 11 1.363 1.796 2.201 2.718 3.106 4.025 4.437 11 f 12 1.356 1.782 2.179 2.681 3.055 3.930 4.318 12 r 13 1.350 1.771 2.160 2.650 3.012 3.852 4.221 13 e 14 1.345 1.761 2.145 2.624 2.977 3.787 4.140 14 e 15 1.341 1.753 2.131 2.602 2.947 3.733 4.073 15 d 16 1.337 1.746 2.120 2.583 2.921 3.686 4.015 16 o 17 1.333 1.740 2.110 2.567 2.898 3.646 3.965 17 m 18 1.330 1.734 2.101 2.552 2.878 3.610 3.922 18 19 1.328 1.729 2.093 2.539 2.861 3.579 3.883 19 20 1.325 1.725 2.086 2.528 2.845 3.552 3.850 20 21 1.323 1.721 2.080 2.518 2.831 3.527 3.819 21 22 1.321 1.717 2.074 2.508 2.819 3.505 3.792 22 23 1.319 1.714 2.069 2.500 2.807 3.485 3.768 23 24 1.318 1.711 2.064 2.492 2.797 3.467 3.745 24 25 1.316 1.708 2.060 2.485 2.787 3.450 3.725 25 26 1.315 1.706 2.056 2.479 2.779 3.435 3.707 26 27 1.314 1.703 2.052 2.473 2.771 3.421 3.690 27 28 1.313 1.701 2.048 2.467 2.763 3.408 3.674 28 29 1.311 1.699 2.045 2.462 2.756 3.396 3.659 29 30 1.310 1.697 2.042 2.457 2.750 3.385 3.646 30 32 1.309 1.694 2.037 2.449 2.738 3.365 3.622 32 34 1.307 1.691 2.032 2.441 2.728 3.348 3.601 34 36 1.306 1.688 2.028 2.434 2.719 3.333 3.582 36 38 1.304 1.686 2.024 2.429 2.712 3.319 3.566 38 40 1.303 1.684 2.021 2.423 2.704 3.307 3.551 40 42 1.302 1.682 2.018 2.418 2.698 3.296 3.538 42 44 1.301 1.680 2.015 2.414 2.692 3.286 3.526 44 46 1.300 1.679 2.013 2.410 2.687 3.277 3.515 46 48 1.299 1.677 2.011 2.407 2.682 3.269 3.505 48 50 1.299 1.676 2.009 2.403 2.678 3.261 3.496 50 55 1.297 1.673 2.004 2.396 2.668 3.245 3.476 55 60 1.296 1.671 2.000 2.390 2.660 3.232 3.460 60 65 1.295 1.669 1.997 2.385 2.654 3.220 3.447 65 70 1.294 1.667 1.994 2.381 2.648 3.211 3.435 70 80 1.292 1.664 1.990 2.374 2.639 3.195 3.416 80 100 1.290 1.660 1.984 2.364 2.626 3.174 3.390 100 150 1.287 1.655 1.976 2.351 2.609 3.145 3.357 150 200 1.286 1.653 1.972 2.345 2.601 3.131 3.340 200 -------+---------------------------------------------------------+----- 0.20 0.10 0.05 0.02 0.01 0.002 0.001 (both tails) 0.10 0.05 0.025 0.01 0.005 0.001 0.0005 (right tail) This table was calculated by APL programs written by William Knight. x

F-distribution Right tail quantiles f 2, for F-distribution for = 0.05, 0.025, 0.01 and degrees of freedom v 1 and v 2. The inverses are the left tail quantiles f 1, for the same :s, and degrees of freedom v 2 and v 1. v 2 v 1 v 2 xi

Signed-rank test Critical Values of the Wilcoxon Signed Ranks Test n Two-Tailed Test One-Tailed Test! =.05! =.01! =.05! =.01 5 -- -- 0 -- 6 0 -- 2 -- 7 2 -- 3 0 8 3 0 5 1 9 5 1 8 3 10 8 3 10 5 11 10 5 13 7 12 13 7 17 9 13 17 9 21 12 14 21 12 25 15 15 25 15 30 19 16 29 19 35 23 17 34 23 41 27 18 40 27 47 32 19 46 32 53 37 20 52 37 60 43 21 58 42 67 49 22 65 48 75 55 23 73 54 83 62 24 81 61 91 69 25 89 68 100 76 26 98 75 110 84 27 107 83 119 92 28 116 91 130 101 29 126 100 140 110 30 137 109 151 120 xii

Mann Whitney-testi Critical Values of the Wilcoxon Ranked-Sums Test (Two-Tailed Testing) n! m 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3.05 -- -- 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14.01 -- -- -- -- -- -- 6 6 6 7 7 7 8 8 8 8 9 9 4.05 -- 10 11 12 13 14 14 15 16 17 18 19 20 21 21 22 23 24.01 -- -- -- 10 10 11 11 12 12 13 13 14 15 15 16 16 17 18 5.05 15 16 17 18 20 21 22 23 24 26 27 28 29 30 32 33 34 35.01 -- -- 15 16 16 17 18 19 20 21 22 22 23 24 25 26 27 28 6.05 22 23 24 26 27 29 31 32 34 35 37 38 40 42 43 45 46 48.01 -- 21 22 23 24 25 26 27 28 30 31 32 33 34 36 37 38 39 7.05 29 31 33 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62.01 -- 28 29 31 32 34 35 37 38 40 41 43 44 46 47 49 50 52 8.05 38 40 42 44 46 49 51 53 55 58 60 62 65 67 70 72 74 77.01 -- 37 38 40 42 43 45 47 49 51 53 54 56 58 60 62 64 66 9.05 47 49 52 55 57 60 62 65 68 71 73 76 79 82 84 87 90 93.01 45 46 48 50 52 54 56 58 61 63 65 67 69 72 74 76 78 81 10.05 58 60 63 66 69 72 75 78 81 84 88 91 94 97 100 103 107 110.01 55 57 59 61 64 66 68 71 73 76 79 81 84 86 89 92 94 97 11.05 69 72 75 79 82 85 89 92 96 99 103 106 110 113 117 121 124 128.01 66 68 71 73 76 79 82 84 87 90 93 96 99 102 105 108 111 114 12.05 82 85 89 92 96 100 104 107 111 115 119 123 127 131 135 139 143 147.01 79 81 84 87 90 93 96 99 102 105 109 112 115 119 122 125 129 132 13.05 95 99 103 107 111 115 119 124 128 132 136 141 145 150 154 158 163 167.01 92 94 98 101 104 108 111 115 118 122 125 129 133 136 140 144 147 151 14.05 110 114 118 122 127 131 136 141 145 150 155 160 164 169 172 179 183 188.01 106 109 112 116 120 123 127 131 135 139 143 147 151 155 159 163 168 172 15.05 125 130 134 139 144 149 154 159 164 169 174 179 184 190 195 200 205 210.01 122 125 128 132 136 140 144 149 153 157 162 166 171 175 180 184 189 193 16.05 142 147 151 157 162 167 173 178 183 189 195 200 206 211 217 222 228 234.01 138 141 145 149 154 158 163 167 172 177 181 186 191 196 201 206 210 215 17.05 159 164 170 175 181 187 192 198 204 210 216 220 228 234 240 246 252 258.01 155 159 163 168 172 177 182 187 192 197 202 207 213 218 223 228 234 239 18.05 178 183 189 195 201 207 213 219 226 232 238 245 251 257 264 270 277 283.01 173 177 182 187 192 197 202 208 213 218 224 229 235 241 246 252 258 263 19.05 197 203 209 215 222 228 235 242 248 255 262 268 275 282 289 296 303 309.01 193 197 202 207 212 218 223 229 235 241 246 253 259 264 271 277 283 289 20.05 218 224 230 237 244 251 258 265 272 279 286 293 300 308 315 322 329 337.01 213 218 223 228 234 240 246 252 258 264 270 277 283 289 296 302 309 315 Note: n is the number of scores in the group with the smallest sum of ranks; m is the number of scores in the other group. xiii

Two-sided tolerance interval table The table gives the coefficient k for a two-sided tolerance interval. k: γ = 0.1 γ = 0.05 γ = 0.01 n α = 0.1 α = 0.05 α = 0.01 α = 0.1 α = 0.05 α = 0.01 α = 0.1 α = 0.05 α = 0.01 5 3.4993 4.1424 5.3868 4.2906 5.0767 6.5977 6.6563 7.8711 10.222 6 3.1407 3.7225 4.8498 3.7325 4.4223 5.7581 5.3833 6.3656 8.2910 7 2.9129 3.4558 4.5087 3.3895 4.0196 5.2409 4.6570 5.5198 7.1907 8 2.7542 3.2699 4.2707 3.1560 3.7454 4.8892 4.1883 4.9694 6.4812 9 2.6367 3.1322 4.0945 2.9864 3.5459 4.6328 3.8596 4.5810 5.9803 10 2.5459 3.0257 3.9579 2.8563 3.3935 4.4370 3.6162 4.2952 5.6106 11 2.4734 2.9407 3.8488 2.7536 3.2727 4.2818 3.4286 4.0725 5.3243 12 2.4139 2.8706 3.7591 2.6701 3.1748 4.1555 3.2793 3.8954 5.0956 13 2.3643 2.8122 3.6841 2.6011 3.0932 4.0505 3.1557 3.7509 4.9091 14 2.3219 2.7624 3.6200 2.5424 3.0241 3.9616 3.0537 3.6310 4.7532 15 2.2855 2.7196 3.5648 2.4923 2.9648 3.8852 2.9669 3.5285 4.6212 16 2.2536 2.6822 3.5166 2.4485 2.9135 3.8189 2.8926 3.4406 4.5078 17 2.2257 2.6491 3.4740 2.4102 2.8685 3.7605 2.8277 3.3637 4.4084 18 2.2007 2.6197 3.4361 2.3762 2.8283 3.7088 2.7711 3.2966 4.3213 19 2.1784 2.5934 3.4022 2.3460 2.7925 3.6627 2.7202 3.2361 4.2433 20 2.1583 2.5697 3.3715 2.3188 2.7603 3.6210 2.6758 3.1838 4.1747 21 2.1401 2.5482 3.3437 2.2941 2.7312 3.5832 2.6346 3.1360 4.1125 22 2.1234 2.5285 3.3183 2.2718 2.7047 3.5490 2.5979 3.0924 4.0562 23 2.1083 2.5105 3.2951 2.2513 2.6805 3.5176 2.5641 3.0528 4.0044 24 2.0943 2.4940 3.2735 2.2325 2.6582 3.4888 2.5342 3.0169 3.9580 25 2.0813 2.4786 3.2538 2.2151 2.6378 3.4622 2.5060 2.9836 3.9147 26 2.0693 2.4644 3.2354 2.1990 2.6187 3.4375 2.4797 2.9533 3.8751 27 2.0581 2.4512 3.2182 2.1842 2.6012 3.4145 2.4560 2.9247 3.8385 28 2.0477 2.4389 3.2023 2.1703 2.5846 3.3933 2.4340 2.8983 3.8048 29 2.0380 2.4274 3.1873 2.1573 2.5693 3.3733 2.4133 2.8737 3.7721 30 2.0289 2.4166 3.1732 2.1450 2.5548 3.3546 2.3940 2.8509 3.7426 31 2.0203 2.4065 3.1601 2.1337 2.5414 3.3369 2.3758 2.8299 3.7148 32 2.0122 2.3969 3.1477 2.1230 2.5285 3.3205 2.3590 2.8095 3.6885 33 2.0045 2.3878 3.1360 2.1128 2.5167 3.3048 2.3430 2.7900 3.6638 34 1.9973 2.3793 3.1248 2.1033 2.5053 3.2901 2.3279 2.7727 3.6405 35 1.9905 2.3712 3.1143 2.0942 2.4945 3.2761 2.3139 2.7557 3.6185 36 1.9840 2.3635 3.1043 2.0857 2.4844 3.2628 2.3003 2.7396 3.5976 37 1.9779 2.3561 3.0948 2.0775 2.4748 3.2503 2.2875 2.7246 3.5782 38 1.9720 2.3492 3.0857 2.0697 2.4655 3.2382 2.2753 2.7105 3.5593 39 1.9664 2.3425 3.0771 2.0623 2.4568 3.2268 2.2638 2.6966 3.5414 40 1.9611 2.3362 3.0688 2.0552 2.4484 3.2158 2.2527 2.6839 3.5244 41 1.9560 2.3301 3.0609 2.0485 2.4404 3.2055 2.2424 2.6711 3.5085 42 1.9511 2.3244 3.0533 2.0421 2.4327 3.1955 2.2324 2.6593 3.4927 43 1.9464 2.3188 3.0461 2.0359 2.4254 3.1860 2.2228 2.6481 3.4780 44 1.9419 2.3134 3.0391 2.0300 2.4183 3.1768 2.2137 2.6371 3.4638 45 1.9376 2.3083 3.0324 2.0243 2.4117 3.1679 2.2049 2.6268 3.4502 46 1.9334 2.3034 3.0260 2.0188 2.4051 3.1595 2.1964 2.6167 3.4370 47 1.9294 2.2987 3.0199 2.0136 2.3989 3.1515 2.1884 2.6071 3.4245 48 1.9256 2.2941 3.0139 2.0086 2.3929 3.1435 2.1806 2.5979 3.4125 49 1.9218 2.2897 3.0081 2.0037 2.3871 3.1360 2.1734 2.5890 3.4008 50 1.9183 2.2855 3.0026 1.9990 2.3816 3.1287 2.1660 2.5805 3.3899 55 1.9022 2.2663 2.9776 1.9779 2.3564 3.0960 2.1338 2.5421 3.3395 60 1.8885 2.2500 2.9563 1.9599 2.3351 3.0680 2.1063 2.5094 3.2968 65 1.8766 2.2359 2.9378 1.9444 2.3166 3.0439 2.0827 2.4813 3.2604 70 1.8662 2.2235 2.9217 1.9308 2.3005 3.0228 2.0623 2.4571 3.2282 75 1.8570 2.2126 2.9074 1.9188 2.2862 3.0041 2.0442 2.4355 3.2002 80 1.8488 2.2029 2.8947 1.9082 2.2735 2.9875 2.0282 2.4165 3.1753 85 1.8415 2.1941 2.8832 1.8986 2.2621 2.9726 2.0139 2.3994 3.1529 90 1.8348 2.1862 2.8728 1.8899 2.2519 2.9591 2.0008 2.3839 3.1327 95 1.8287 2.1790 2.8634 1.8820 2.2425 2.9468 1.9891 2.3700 3.1143 100 1.8232 2.1723 2.8548 1.8748 2.2338 2.9356 1.9784 2.3571 3.0977 xiv

One-sided tolerance interval table The table gives the coefficient k for a one-sided tolerance interval. k: γ = 0.1 γ = 0.05 γ = 0.01 n α = 0.1 α = 0.05 α = 0.01 α = 0.1 α = 0.05 α = 0.01 α = 0.1 α = 0.05 α = 0.01 5 2.7423 3.3998 4.6660 3.4066 4.2027 5.7411 5.3617 6.5783 8.9390 6 2.4937 3.0919 4.2425 3.0063 3.7077 5.0620 4.4111 5.4055 7.3346 7 2.3327 2.8938 3.9720 2.7554 3.3994 4.6417 3.8591 4.7279 6.4120 8 2.2186 2.7543 3.7826 2.5819 3.1873 4.3539 3.4972 4.2852 5.8118 9 2.1329 2.6499 3.6414 2.4538 3.0312 4.1430 3.2404 3.9723 5.3889 10 2.0656 2.5684 3.5316 2.3546 2.9110 3.9811 3.0479 3.7383 5.0737 11 2.0113 2.5026 3.4434 2.2753 2.8150 3.8523 2.8977 3.5562 4.8290 12 1.9662 2.4483 3.3707 2.2101 2.7364 3.7471 2.7767 3.4099 4.6330 13 1.9281 2.4024 3.3095 2.1554 2.6705 3.6592 2.6770 3.2896 4.4720 14 1.8954 2.3631 3.2572 2.1088 2.6144 3.5845 2.5931 3.1886 4.3372 15 1.8669 2.3289 3.2118 2.0684 2.5660 3.5201 2.5215 3.1024 4.2224 16 1.8418 2.2990 3.1720 2.0330 2.5237 3.4640 2.4594 3.0279 4.1233 17 1.8195 2.2724 3.1369 2.0017 2.4862 3.4144 2.4051 2.9627 4.0367 18 1.7995 2.2486 3.1054 1.9738 2.4530 3.3703 2.3570 2.9051 3.9604 19 1.7815 2.2272 3.0771 1.9487 2.4231 3.3308 2.3142 2.8539 3.8924 20 1.7652 2.2078 3.0515 1.9260 2.3960 3.2951 2.2757 2.8079 3.8316 21 1.7503 2.1901 3.0282 1.9053 2.3714 3.2628 2.2408 2.7663 3.7766 22 1.7366 2.1739 3.0069 1.8864 2.3490 3.2332 2.2091 2.7285 3.7268 23 1.7240 2.1589 2.9873 1.8690 2.3283 3.2061 2.1801 2.6940 3.6812 24 1.7124 2.1451 2.9691 1.8530 2.3093 3.1811 2.1535 2.6623 3.6395 25 1.7015 2.1323 2.9524 1.8381 2.2917 3.1579 2.1290 2.6331 3.6011 26 1.6914 2.1204 2.9367 1.8242 2.2753 3.1365 2.1063 2.6062 3.5656 27 1.6820 2.1092 2.9221 1.8114 2.2600 3.1165 2.0852 2.5811 3.5326 28 1.6732 2.0988 2.9085 1.7993 2.2458 3.0978 2.0655 2.5577 3.5019 29 1.6649 2.0890 2.8958 1.7880 2.2324 3.0804 2.0471 2.5359 3.4733 30 1.6571 2.0798 2.8837 1.7773 2.2198 3.0639 2.0298 2.5155 3.4465 31 1.6497 2.0711 2.8724 1.7673 2.2080 3.0484 2.0136 2.4963 3.4214 32 1.6427 2.0629 2.8617 1.7578 2.1968 3.0338 1.9984 2.4782 3.3977 33 1.6361 2.0551 2.8515 1.7489 2.1862 3.0200 1.9840 2.4612 3.3754 34 1.6299 2.0478 2.8419 1.7403 2.1762 3.0070 1.9703 2.4451 3.3543 35 1.6239 2.0407 2.8328 1.7323 2.1667 2.9946 1.9574 2.4298 3.3343 36 1.6182 2.0341 2.8241 1.7246 2.1577 2.9828 1.9452 2.4154 3.3155 37 1.6128 2.0277 2.8158 1.7173 2.1491 2.9716 1.9335 2.4016 3.2975 38 1.6076 2.0216 2.8080 1.7102 2.1408 2.9609 1.9224 2.3885 3.2804 39 1.6026 2.0158 2.8004 1.7036 2.1330 2.9507 1.9118 2.3760 3.2641 40 1.5979 2.0103 2.7932 1.6972 2.1255 2.9409 1.9017 2.3641 3.2486 41 1.5934 2.0050 2.7863 1.6911 2.1183 2.9316 1.8921 2.3528 3.2337 42 1.5890 1.9998 2.7796 1.6852 2.1114 2.9226 1.8828 2.3418 3.2195 43 1.5848 1.9949 2.7733 1.6795 2.1048 2.9141 1.8739 2.3314 3.2059 44 1.5808 1.9902 2.7672 1.6742 2.0985 2.9059 1.8654 2.3214 3.1929 45 1.5769 1.9857 2.7613 1.6689 2.0924 2.8979 1.8573 2.3118 3.1804 46 1.5732 1.9813 2.7556 1.6639 2.0865 2.8903 1.8495 2.3025 3.1684 47 1.5695 1.9771 2.7502 1.6591 2.0808 2.8830 1.8419 2.2937 3.1568 48 1.5661 1.9730 2.7449 1.6544 2.0753 2.8759 1.8346 2.2851 3.1457 49 1.5627 1.9691 2.7398 1.6499 2.0701 2.8690 1.8275 2.2768 3.1349 50 1.5595 1.9653 2.7349 1.6455 2.0650 2.8625 1.8208 2.2689 3.1246 55 1.5447 1.9481 2.7126 1.6258 2.0419 2.8326 1.7902 2.2330 3.0780 60 1.5320 1.9333 2.6935 1.6089 2.0222 2.8070 1.7641 2.2024 3.0382 65 1.5210 1.9204 2.6769 1.5942 2.0050 2.7849 1.7414 2.1759 3.0039 70 1.5112 1.9090 2.6623 1.5812 1.9898 2.7654 1.7216 2.1526 2.9739 75 1.5025 1.8990 2.6493 1.5697 1.9765 2.7481 1.7040 2.1321 2.9474 80 1.4947 1.8899 2.6377 1.5594 1.9644 2.7326 1.6883 2.1137 2.9237 85 1.4877 1.8817 2.6272 1.5501 1.9536 2.7187 1.6742 2.0973 2.9024 90 1.4813 1.8743 2.6176 1.5416 1.9438 2.7061 1.6613 2.0824 2.8832 95 1.4754 1.8675 2.6089 1.5338 1.9348 2.6945 1.6497 2.0688 2.8657 100 1.4701 1.8612 2.6009 1.5268 1.9265 2.6839 1.6390 2.0563 2.8496 xv