FORMULAS FOR STATISTICS 1

Sample statistics

X =1

n

n∑i=1

Xi or x =1

n

n∑i=1

xi (sample mean)

S2 =1

n− 1

n∑i=1

(Xi −X)2=1

n− 1

n∑i=1

X2i −

n

n− 1X

2or

s2 =1

n− 1

n∑i=1

(xi − x)2 =1

n− 1

n∑i=1

x2i −n

n− 1x 2 (sample variance)

E(X) = µ , var(X) =σ2

n, E(S2) = σ2

Sample statistics for normal distribution

Z =X − µσ/√n∼ N(0, 1)

T =X − µS/√n∼ t with n− 1 degrees of freedom

V =(n− 1)S2

σ2=

n∑i=1

(Xi −X)2

σ2∼ χ2 with n− 1 degrees of freedom

Z =(X1 −X2)− (µ1 − µ2)√

σ21/n1 + σ2

2/n2

∼ N(0, 1)

T =(X1 −X2)− (µ1 − µ2)

Sp

√1/n1 + 1/n2

∼ t with n1 + n2 − 2 degrees of freedom, where

S2p =

(n1 − 1)S21 + (n2 − 1)S2

2

n1 + n2 − 2assuming σ1 = σ2

W =(X1 −X2)− (µ1 − µ2)√

S21/n1 + S2

2/n2

≈ t with(a1 + a2)

2

a21/(n1 − 1) + a22/(n2 − 1)degrees of freedom, where

a1 =s21n1

and a2 =s22n2

(Welch–Satterthwaite approximation)

F =S21/σ

21

S22/σ

22

∼ F with n1 − 1 and n2 − 1 degrees of freedom

i

Point estimation

Parameter θ Estimate θ Estimator Θ

µ µ = x X

σ2 σ2 = s2 S2

m m = q(0.5) Q(0.5)

Interval estimation of expectation

Z =X − µσ/√n

: x± zα/2σ√n

T =X − µS/√n

: x± tα/2s√n

(with n− 1 degrees of freedom)

Z =(X1 −X2)− (µ1 − µ2)√

σ21/n1 + σ2

2/n2

: (x1 − x2)± zα/2

√σ21

n1

+σ22

n2

T =(X1 −X2)− (µ1 − µ2)

Sp

√1/n1 + 1/n2

: (x1 − x2)± tα/2sp√

1

n1

+1

n2

(with n1 + n2 − 2 degrees of freedom)

W =(X1 −X2)− (µ1 − µ2)√

S21/n1 + S2

2/n2

: (x1 − x2)± tα/2

√s21n1

+s22n2

(Welch–Satterthwaite)

(with ∼=(a1 + a2)

2

a21/(n1 − 1) + a22/(n2 − 1)degrees of freedom, where a1 =

s21n1

and a2 =s22n2

)

Estimation of proportion for the binomial distribution

P(X = x) =

(n

x

)px(1− p)n−x

E(X) = np , var(X) = np(1− p)

p =x

n

n∑i=x

(n

i

)piL(1− pL)n−i =

α

2,

x∑i=0

(n

i

)piU(1− pU)n−i =

α

2

Interval estimation of variance

V =(n− 1)S2

σ2:

(n− 1)s2

h2,α/2and

(n− 1)s2

h1,α/2(with n− 1 degrees of freedom)

F =S21/σ

21

S22/σ

22

:s21s22

1

f2,α/2and

s21s22

1

f1,α/2(with n1 − 1 and n2 − 1 degrees of freedom)

ii

Testing expectations

z =x− µ0

σ/√n

and H0 : µ = µ0 :

H1 Critical region P-probability

µ > µ0 z ≥ zα 1− Φ(z)µ < µ0 z ≤ −zα Φ(z)µ 6= µ0 |z| ≥ zα/2 2 min

(Φ(z), 1− Φ(z)

)Φ is the standard normal cumulative distribution function.

t =x− µ0

s/√n

and H0 : µ = µ0 :

H1 Critical region P-probability

µ > µ0 t ≥ tα 1− F (t)µ < µ0 t ≤ −tα F (t)µ 6= µ0 |t| ≥ tα/2 2 min

(F (t), 1− F (t)

)F is the cumulative t-distribution function with n− 1 degrees of freedom.

z =x1 − x2 − d0√σ21/n1 + σ2

2/n2

and H0 : µ1 − µ2 = d0 :

H1 Critical region P-probability

µ1 − µ2 > d0 z ≥ zα 1− Φ(z)µ1 − µ2 < d0 z ≤ −zα Φ(z)µ1 − µ2 6= d0 |z| ≥ zα/2 2 min

(Φ(z), 1− Φ(z)

)Φ is the standard normal cumulative distribution function.

t =x1 − x2 − d0

sp√

1/n1 + 1/n2

, where s2p =(n1 − 1)s21 + (n2 − 1)s22

n1 + n2 − 2, and H0 : µ1−µ2 = d0 :

H1 Critical region P-probability

µ1 − µ2 > d0 t ≥ tα 1− F (t)µ1 − µ2 < d0 t ≤ −tα F (t)µ1 − µ2 6= d0 |t| ≥ tα/2 2 min

(F (t), 1− F (t)

)F is the cumulative t-distribution function with n1 + n2 − 2 degrees of freedom.

t =x1 − x2 − d0√s21/n1 + s22/n2

and H0 : µ1 − µ2 = d0 (Welch–Satterthwaite) :

H1 Critical region P-probability

µ1 − µ2 > d0 t ≥ tα 1− F (t)µ1 − µ2 < d0 t ≤ −tα F (t)µ1 − µ2 6= d0 |t| ≥ tα/2 2 min

(F (t), 1− F (t)

)F is approximatively the cumulative t-distribution function with

(a1 + a2)2

a21/(n1 − 1) + a22/(n2 − 1)degrees of freedom, where a1 =

s21n1

and a2 =s22n2

.

iii

Testing variances

v =(n− 1)s2

σ20

and H0 : σ2 = σ20 :

H1 Critical region P-probability

σ2 > σ20 v ≥ h2,α 1− F (v)

σ2 < σ20 v ≤ h1,α F (v)

σ2 6= σ20 v ≤ h1,α/2 or v ≥ h2,α/2 2 min

(F (v), 1− F (v)

)F is the cumulative χ2-distribution function with n− 1 degrees of freedom.

f =1

k

s21s22

and H0 : σ21 = kσ2

2 :

H1 Critical region P-probability

σ21 > kσ2

2 f ≥ f2,α 1−G(f)σ21 < kσ2

2 f ≤ f1,α G(f)σ21 6= kσ2

2 f ≤ f1,α/2 tai f ≥ f2,α/2 2 min(G(f), 1−G(f)

)G is the cumulative F-distribution function with n1 − 1 and n2 − 1 degrees of freedom.

Goodness-of-fit test

Ho : P(T1) = p1, . . . ,P(Tk) = pk

H =k∑i=1

(Fi − npi)2

npi≈ χ2 with k − 1 degrees of freedom

Independence test. Contingence tables

P(T1) = p1, . . . ,P(Tk) = pk and P(S1) = q1, . . . ,P(Sl) = ql

P(Ti ∩ Sj) = pi,j (i = 1, . . . , k and j = 1, . . . , l)

H0 : pi,j = piqj (i = 1, . . . , k and j = 1, . . . , l)

H =k∑i=1

l∑j=1

(Fi,j − FiGj/n)2

FiGj/n≈ χ2 with (k − 1)(l − 1) degrees of freedom

Contingence table:

S1 S2 · · · Sl ΣT1 f1,1 f1,2 · · · f1,l f1T2 f2,1 f2,2 · · · f2,l f2...

......

. . ....

...Tk fk,1 fk,2 · · · fk,l fkΣ g1 g2 · · · gl n

iv

Maximum likelihood estimation

L(θ1, . . . , θm;x1, . . . , xn) = f(x1; θ1, . . . , θm) · · · f(xn; θ1, . . . , θm)

l(θ1, . . . , θm;x1, . . . , xn) = ln f(x1; θ1, . . . , θm) + · · ·+ ln f(xn; θ1, . . . , θm)

(θ1, . . . , θm) = argmaxθ1,...,θm

L(θ1, . . . , θm;x1, . . . , xn) or

(θ1, . . . , θm) = argmaxθ1,...,θm

l(θ1, . . . , θm;x1, . . . , xn)

Linear regression

Model: y = β0 + β1x1 + · · ·+ βkxk + ε

Data:

x1 x2 · · · xk yx1,1 x1,2 · · · x1,k y1x2,1 x2,2 · · · x2,k y2

......

......

xn,1 xn,2 · · · xn,k yn

X =

1 x1,1 x1,2 · · · x1,k1 x2,1 x2,2 · · · x2,k...

......

. . ....

1 xn,1 xn,2 · · · xn,k

, y =

y1y2...yn

, ε =

ε1ε2...εn

, β =

β0β1...βk

Data model: y = Xβ + ε.

β = b = (XTX)−1XTy = β + (XTX)−1XTε

C = (cij) = (XTX)−1

H = XCXT (hat matrix)

P = In −H

E(bi) = βi , var(bi) = ciiσ2 and cov(bi, bj) = cijσ

2

y = Xb

e = y − y = Py = Pε

SSE = ‖e‖2 =n∑i=1

e2i =n∑i=1

(yi − yi)2

σ2 =SSE

n− k − 1= MSE ,

√MSE = RMSE

v

SST =n∑i=1

(yi − y)2 , SSR =n∑i=1

(yi − 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

σ2 = MSE

(MST)

F =MSR

MSE

H0 : β1 = · · · = βk = 0 : F =MSR

MSE∼ F with k and n− k − 1 degrees of freedom

H0 : βi = β0,i; Ti =bi − β0,i

RMSE√cii∼ t with n− k − 1 degrees of freedom

R2 =SSR

SST= 1− SSE

SST

R2adj = 1− MSE

MST= 1− n− 1

n− k − 1

SSE

SST,

Categorical regressors:

zi zi,1 zi,2 · · · zi,mi−1

Ai,1 1 0 · · · 0Ai,2 0 1 · · · 0...

......

...Ai,mi−1 0 0 · · · 1Ai,mi

0 0 · · · 0

y = β0 + β1x1 + · · ·+ βkxk +l∑

i=1

(βi,1zi,1 + · · ·+ βi,mi−1zi,mi−1) + ε

Logistical regression

P(y = A) =1

1 + e−β0−β1x1−···−βkxk

L(β0, . . . , βk) = L1(β0, . . . , βk) · · ·Ln(β0, . . . , βk) , where

Li(β0, . . . , βk) =

pi =

1

1 + e−β0−β1xi,1−···−βkxi,k, if yi = A

1− pi =e−β0−β1xi,1−···−βkxi,k

1 + e−β0−β1xi,1−···−βkxi,k, if yi = B

vi

(b0, b1, . . . , bk) = argmaxβ0,β1...,βk

L(β0, β1, . . . , βk)

p0 =1

1 + e−b0−b1x0,1−···−bkx0,k.

Kruskal–Wallis-test

H0 : µ1 = · · · = µk

H =12

n(n+ 1)

k∑j=1

W 2j

nj− 3(n+ 1) ≈ χ2 with k − 1 degrees of freedom

Spearman rank correlation coefficient

Samples: x1,1, . . . ,x1,n and x2,1, . . . ,x2,n

Sequence numbers: r1,1 , . . . ,r1,n and r2,1 , . . . ,r2,n

rS =

n∑i=1

(r1,i − r)(r2,i − r)√n∑i=1

(r1,i − r)2√

n∑i=1

(r2,i − r)2, where r =

n+ 1

2

rS = 1− 6

n(n2 − 1)

n∑i=1

d2i , where di = r1,i − r2,i

(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 0.53590.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.68790.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.72240.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.75490.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.78520.8 0.7881 0.7910 0.7939 0.7969 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.83891.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8513 0.8554 0.8577 0.8529 0.86211.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.91771.4 0.9192 0.9207 0.9222 0.9236 0.9215 0.9265 0.9279 0.9292 0.9306 0.93191.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9492 0.94411.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.97061.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.97672.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.98172.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.98902.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.99362.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.99522.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.99742.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.99812.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.99863.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.99903.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.99933.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.99953.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.99973.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

viii

χ2-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 7.882 0.0100 0.0201 0.0506 0.103 0.211 0.575 1.39 2.77 4.61 5.99 7.38 9.21 10.63 0.0717 0.115 0.216 0.352 0.584 1.21 2.37 4.11 6.25 7.81 9.35 11.3 12.84 0.207 0.297 0.484 0.711 1.06 1.92 3.36 5.39 7.78 9.49 11.1 13.3 14.95 0.412 0.554 0.831 1.15 1.61 2.67 4.35 6.63 9.24 11.1 12.8 15.1 16.76 0.676 0.872 1.24 1.64 2.20 3.45 5.35 7.84 10.6 12.6 14.4 16.8 18.57 0.989 1.24 1.69 2.17 2.83 4.25 6.35 9.04 12.0 14.1 16.0 18.5 20.38 1.34 1.65 2.18 2.73 3.49 5.07 7.34 10.2 13.4 15.5 17.5 20.1 22.09 1.73 2.09 2.70 3.33 4.17 5.9 8.34 11.4 14.7 16.9 19.0 21.7 23.610 2.16 2.56 3.25 3.94 4.87 6.74 9.34 12.5 16.0 18.3 20.5 23.2 25.211 2.60 3.05 3.82 4.57 5.58 7.58 10.3 13.7 17.3 19.7 21.9 24.7 26.812 3.07 3.57 4.40 5.23 6.30 8.44 11.3 14.8 18.5 21.0 23.3 26.2 28.313 3.57 4.11 5.01 5.89 7.04 9.3 12.3 16.0 19.8 22.4 24.7 27.7 29.814 4.07 4.66 5.63 6.57 7.79 10.2 13.3 17.1 21.1 23.7 26.1 29.1 31.315 4.60 5.23 6.26 7.26 8.55 11.0 14.3 18.2 22.3 25.0 27.5 30.6 32.816 5.14 5.81 6.91 7.96 9.31 11.9 15.3 19.4 23.5 26.3 28.8 32.0 34.317 5.70 6.41 7.56 8.67 10.1 12.8 16.3 20.5 24.8 27.6 30.2 33.4 35.718 6.26 7.01 8.23 9.39 10.9 13.7 17.3 21.6 26.0 28.9 31.5 34.8 37.219 6.84 7.63 8.91 10.1 11.7 14.6 18.3 22.7 27.2 30.1 32.9 36.2 38.620 7.43 8.26 9.59 10.9 12.4 15.5 19.3 23.8 28.4 31.4 34.2 37.6 40.021 8.03 8.90 10.3 11.6 13.2 16.3 20.3 24.9 29.6 32.7 35.5 38.9 41.422 8.64 9.54 11.0 12.3 14.0 17.2 21.3 26.0 30.8 33.9 36.8 40.3 42.823 9.26 10.2 11.7 13.1 14.8 18.1 22.3 27.1 32.0 35.2 38.1 41.6 44.224 9.89 10.9 12.4 13.8 15.7 19.0 23.3 28.2 33.2 36.4 39.4 43.0 45.625 10.5 11.5 13.1 14.6 16.5 19.9 24.3 29.3 34.4 37.7 40.6 44.3 46.926 11.2 12.2 13.8 15.4 17.3 20.8 25.3 30.4 35.6 38.9 41.9 45.6 48.327 11.8 12.9 14.6 16.2 18.1 21.7 26.3 31.5 36.7 40.1 43.2 47.0 49.628 12.5 13.6 15.3 16.9 18.9 22.7 27.3 32.6 37.9 41.3 44.5 48.3 51.029 13.1 14.3 16.0 17.7 19.8 23.6 28.3 33.7 39.1 42.6 45.7 49.6 52.330 13.8 15.0 16.8 18.5 20.6 24.5 29.3 34.8 40.3 43.8 47.0 50.9 53.731 14.5 15.7 17.5 19.3 21.4 25.4 30.3 35.9 41.4 45.0 48.2 52.2 55.032 15.1 16.4 18.3 20.1 22.3 26.3 31.3 37.0 42.6 46.2 49.5 53.5 56.333 15.8 17.1 19.0 20.9 23.1 27.2 32.3 38.1 43.7 47.4 50.7 54.8 57.634 16.5 17.8 19.8 21.7 24.0 28.1 33.3 39.1 44.9 48.6 52.0 56.1 59.035 17.2 18.5 20.6 22.5 24.8 29.1 34.3 40.2 46.1 49.8 53.2 57.3 60.336 17.9 19.2 21.3 23.3 25.6 30.0 35.3 41.3 47.2 51.0 54.4 58.6 61.637 18.6 20.0 22.1 24.1 26.5 30.9 36.3 42.4 48.4 52.2 55.7 59.9 62.938 19.3 20.7 22.9 24.9 27.3 31.8 37.3 43.5 49.5 53.4 56.9 61.2 64.239 20.0 21.4 23.7 25.7 28.2 32.7 38.3 44.5 50.7 54.6 58.1 62.4 65.540 20.7 22.2 24.4 26.5 29.1 33.7 39.3 45.6 51.8 55.8 59.3 63.7 66.841 21.4 22.9 25.2 27.3 29.9 34.6 40.3 46.7 52.9 56.9 60.6 65.0 68.142 22.1 23.7 26.0 28.1 30.8 35.5 41.3 47.8 54.1 58.1 61.8 66.2 69.343 22.9 24.4 26.8 29.0 31.6 36.4 42.3 48.8 55.2 59.3 63.0 67.5 70.644 23.6 25.1 27.6 29.8 32.5 37.4 43.3 49.9 56.4 60.5 64.2 68.7 71.945 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 f2, for F-distribution for = 0.05, 0.025, 0.01 and degrees of freedom v1 and v2. The inverses are the left tail quantiles f1, for the same :s, and degrees of freedom v2 and v1.

v1v2 v2

xi

Signed-rank test

Critical Values of the Wilcoxon Signed Ranks Test

Two-Tailed Test One-Tailed Test n ! = .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)

m n

!

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20 .05

--

--

6

7

7 8 8 9 9 10 10 11 11 12

12

13 13 14

3 .01

--

--

--

--

-- -- 6 6 6 7 7 7 8 8

8

8 9 9

.05

--

10

11

12

13 14 14 15 16 17 18 19 20 21

21

22 23 24 4

.01

--

--

--

10

10 11 11 12 12 13 13 14 15 15

16

16 17 18 .05

15

16

17

18

20 21 22 23 24 26 27 28 29 30

32

33 34 35

5 .01

--

--

15

16

16 17 18 19 20 21 22 22 23 24

25

26 27 28

.05

22

23

24

26

27 29 31 32 34 35 37 38 40 42

43

45 46 48 6

.01

--

21

22

23

24 25 26 27 28 30 31 32 33 34

36

37 38 39 .05

29

31

33

34

36 38 40 42 44 46 48 50 52 54

56

58 60 62

7 .01

--

28

29

31

32 34 35 37 38 40 41 43 44 46

47

49 50 52

.05

38

40

42

44

46 49 51 53 55 58 60 62 65 67

70

72 74 77 8

.01

--

37

38

40

42 43 45 47 49 51 53 54 56 58

60

62 64 66 .05

47

49

52

55

57 60 62 65 68 71 73 76 79 82

84

87 90 93

9 .01

45

46

48

50

52 54 56 58 61 63 65 67 69 72

74

76 78 81

.05

58

60

63

66

69 72 75 78 81 84 88 91 94 97

100

103 107 110 10

.01

55

57

59

61

64 66 68 71 73 76 79 81 84 86

89

92 94 97 .05

69

72

75

79

82

85

89

92

96

99

103

106

110

113

117

121

124

128

11 .01

66

68

71

73

76 79 82 84 87 90 93 96 99 102

105

108 111 114

.05

82

85

89

92

96 100 104 107 111 115 119 123 127 131

135

139 143 147 12

.01

79

81

84

87

90 93 96 99 102 105 109 112 115 119

122

125 129 132 .05

95

99

103

107

111 115 119 124 128 132 136 141 145 150

154

158 163 167

13 .01

92

94

98

101

104 108 111 115 118 122 125 129 133 136

140

144 147 151

.05

110

114

118

122

127 131 136 141 145 150 155 160 164 169

172

179 183 188 14

.01

106

109

112

116

120 123 127 131 135 139 143 147 151 155

159

163 168 172 .05

125

130

134

139

144 149 154 159 164 169 174 179 184 190

195

200 205 210

15 .01

122

125

128

132

136 140 144 149 153 157 162 166 171 175

180

184 189 193

.05

142

147

151

157

162 167 173 178 183 189 195 200 206 211

217

222 228 234 16

.01

138

141

145

149

154 158 163 167 172 177 181 186 191 196

201

206 210 215 .05

159

164

170

175

181 187 192 198 204 210 216 220 228 234

240

246 252 258

17 .01

155

159

163

168

172 177 182 187 192 197 202 207 213 218

223

228 234 239

.05

178

183

189

195

201 207 213 219 226 232 238 245 251 257

264

270 277 283 18

.01

173

177

182

187

192 197 202 208 213 218 224 229 235 241

246

252 258 263 .05

197

203

209

215

222 228 235 242 248 255 262 268 275 282

289

296 303 309

19 .01

193

197

202

207

212 218 223 229 235 241 246 253 259 264

271

277 283 289

.05

218

224

230

237

244 251 258 265 272 279 286 293 300 308

315

322 329 337 20

.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 = 0.01n 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 = 0.015 3.4993 4.1424 5.3868 4.2906 5.0767 6.5977 6.6563 7.8711 10.2226 3.1407 3.7225 4.8498 3.7325 4.4223 5.7581 5.3833 6.3656 8.29107 2.9129 3.4558 4.5087 3.3895 4.0196 5.2409 4.6570 5.5198 7.19078 2.7542 3.2699 4.2707 3.1560 3.7454 4.8892 4.1883 4.9694 6.48129 2.6367 3.1322 4.0945 2.9864 3.5459 4.6328 3.8596 4.5810 5.980310 2.5459 3.0257 3.9579 2.8563 3.3935 4.4370 3.6162 4.2952 5.610611 2.4734 2.9407 3.8488 2.7536 3.2727 4.2818 3.4286 4.0725 5.324312 2.4139 2.8706 3.7591 2.6701 3.1748 4.1555 3.2793 3.8954 5.095613 2.3643 2.8122 3.6841 2.6011 3.0932 4.0505 3.1557 3.7509 4.909114 2.3219 2.7624 3.6200 2.5424 3.0241 3.9616 3.0537 3.6310 4.753215 2.2855 2.7196 3.5648 2.4923 2.9648 3.8852 2.9669 3.5285 4.621216 2.2536 2.6822 3.5166 2.4485 2.9135 3.8189 2.8926 3.4406 4.507817 2.2257 2.6491 3.4740 2.4102 2.8685 3.7605 2.8277 3.3637 4.408418 2.2007 2.6197 3.4361 2.3762 2.8283 3.7088 2.7711 3.2966 4.321319 2.1784 2.5934 3.4022 2.3460 2.7925 3.6627 2.7202 3.2361 4.243320 2.1583 2.5697 3.3715 2.3188 2.7603 3.6210 2.6758 3.1838 4.174721 2.1401 2.5482 3.3437 2.2941 2.7312 3.5832 2.6346 3.1360 4.112522 2.1234 2.5285 3.3183 2.2718 2.7047 3.5490 2.5979 3.0924 4.056223 2.1083 2.5105 3.2951 2.2513 2.6805 3.5176 2.5641 3.0528 4.004424 2.0943 2.4940 3.2735 2.2325 2.6582 3.4888 2.5342 3.0169 3.958025 2.0813 2.4786 3.2538 2.2151 2.6378 3.4622 2.5060 2.9836 3.914726 2.0693 2.4644 3.2354 2.1990 2.6187 3.4375 2.4797 2.9533 3.875127 2.0581 2.4512 3.2182 2.1842 2.6012 3.4145 2.4560 2.9247 3.838528 2.0477 2.4389 3.2023 2.1703 2.5846 3.3933 2.4340 2.8983 3.804829 2.0380 2.4274 3.1873 2.1573 2.5693 3.3733 2.4133 2.8737 3.772130 2.0289 2.4166 3.1732 2.1450 2.5548 3.3546 2.3940 2.8509 3.742631 2.0203 2.4065 3.1601 2.1337 2.5414 3.3369 2.3758 2.8299 3.714832 2.0122 2.3969 3.1477 2.1230 2.5285 3.3205 2.3590 2.8095 3.688533 2.0045 2.3878 3.1360 2.1128 2.5167 3.3048 2.3430 2.7900 3.663834 1.9973 2.3793 3.1248 2.1033 2.5053 3.2901 2.3279 2.7727 3.640535 1.9905 2.3712 3.1143 2.0942 2.4945 3.2761 2.3139 2.7557 3.618536 1.9840 2.3635 3.1043 2.0857 2.4844 3.2628 2.3003 2.7396 3.597637 1.9779 2.3561 3.0948 2.0775 2.4748 3.2503 2.2875 2.7246 3.578238 1.9720 2.3492 3.0857 2.0697 2.4655 3.2382 2.2753 2.7105 3.559339 1.9664 2.3425 3.0771 2.0623 2.4568 3.2268 2.2638 2.6966 3.541440 1.9611 2.3362 3.0688 2.0552 2.4484 3.2158 2.2527 2.6839 3.524441 1.9560 2.3301 3.0609 2.0485 2.4404 3.2055 2.2424 2.6711 3.508542 1.9511 2.3244 3.0533 2.0421 2.4327 3.1955 2.2324 2.6593 3.492743 1.9464 2.3188 3.0461 2.0359 2.4254 3.1860 2.2228 2.6481 3.478044 1.9419 2.3134 3.0391 2.0300 2.4183 3.1768 2.2137 2.6371 3.463845 1.9376 2.3083 3.0324 2.0243 2.4117 3.1679 2.2049 2.6268 3.450246 1.9334 2.3034 3.0260 2.0188 2.4051 3.1595 2.1964 2.6167 3.437047 1.9294 2.2987 3.0199 2.0136 2.3989 3.1515 2.1884 2.6071 3.424548 1.9256 2.2941 3.0139 2.0086 2.3929 3.1435 2.1806 2.5979 3.412549 1.9218 2.2897 3.0081 2.0037 2.3871 3.1360 2.1734 2.5890 3.400850 1.9183 2.2855 3.0026 1.9990 2.3816 3.1287 2.1660 2.5805 3.389955 1.9022 2.2663 2.9776 1.9779 2.3564 3.0960 2.1338 2.5421 3.339560 1.8885 2.2500 2.9563 1.9599 2.3351 3.0680 2.1063 2.5094 3.296865 1.8766 2.2359 2.9378 1.9444 2.3166 3.0439 2.0827 2.4813 3.260470 1.8662 2.2235 2.9217 1.9308 2.3005 3.0228 2.0623 2.4571 3.228275 1.8570 2.2126 2.9074 1.9188 2.2862 3.0041 2.0442 2.4355 3.200280 1.8488 2.2029 2.8947 1.9082 2.2735 2.9875 2.0282 2.4165 3.175385 1.8415 2.1941 2.8832 1.8986 2.2621 2.9726 2.0139 2.3994 3.152990 1.8348 2.1862 2.8728 1.8899 2.2519 2.9591 2.0008 2.3839 3.132795 1.8287 2.1790 2.8634 1.8820 2.2425 2.9468 1.9891 2.3700 3.1143100 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 = 0.01n 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 = 0.015 2.7423 3.3998 4.6660 3.4066 4.2027 5.7411 5.3617 6.5783 8.93906 2.4937 3.0919 4.2425 3.0063 3.7077 5.0620 4.4111 5.4055 7.33467 2.3327 2.8938 3.9720 2.7554 3.3994 4.6417 3.8591 4.7279 6.41208 2.2186 2.7543 3.7826 2.5819 3.1873 4.3539 3.4972 4.2852 5.81189 2.1329 2.6499 3.6414 2.4538 3.0312 4.1430 3.2404 3.9723 5.388910 2.0656 2.5684 3.5316 2.3546 2.9110 3.9811 3.0479 3.7383 5.073711 2.0113 2.5026 3.4434 2.2753 2.8150 3.8523 2.8977 3.5562 4.829012 1.9662 2.4483 3.3707 2.2101 2.7364 3.7471 2.7767 3.4099 4.633013 1.9281 2.4024 3.3095 2.1554 2.6705 3.6592 2.6770 3.2896 4.472014 1.8954 2.3631 3.2572 2.1088 2.6144 3.5845 2.5931 3.1886 4.337215 1.8669 2.3289 3.2118 2.0684 2.5660 3.5201 2.5215 3.1024 4.222416 1.8418 2.2990 3.1720 2.0330 2.5237 3.4640 2.4594 3.0279 4.123317 1.8195 2.2724 3.1369 2.0017 2.4862 3.4144 2.4051 2.9627 4.036718 1.7995 2.2486 3.1054 1.9738 2.4530 3.3703 2.3570 2.9051 3.960419 1.7815 2.2272 3.0771 1.9487 2.4231 3.3308 2.3142 2.8539 3.892420 1.7652 2.2078 3.0515 1.9260 2.3960 3.2951 2.2757 2.8079 3.831621 1.7503 2.1901 3.0282 1.9053 2.3714 3.2628 2.2408 2.7663 3.776622 1.7366 2.1739 3.0069 1.8864 2.3490 3.2332 2.2091 2.7285 3.726823 1.7240 2.1589 2.9873 1.8690 2.3283 3.2061 2.1801 2.6940 3.681224 1.7124 2.1451 2.9691 1.8530 2.3093 3.1811 2.1535 2.6623 3.639525 1.7015 2.1323 2.9524 1.8381 2.2917 3.1579 2.1290 2.6331 3.601126 1.6914 2.1204 2.9367 1.8242 2.2753 3.1365 2.1063 2.6062 3.565627 1.6820 2.1092 2.9221 1.8114 2.2600 3.1165 2.0852 2.5811 3.532628 1.6732 2.0988 2.9085 1.7993 2.2458 3.0978 2.0655 2.5577 3.501929 1.6649 2.0890 2.8958 1.7880 2.2324 3.0804 2.0471 2.5359 3.473330 1.6571 2.0798 2.8837 1.7773 2.2198 3.0639 2.0298 2.5155 3.446531 1.6497 2.0711 2.8724 1.7673 2.2080 3.0484 2.0136 2.4963 3.421432 1.6427 2.0629 2.8617 1.7578 2.1968 3.0338 1.9984 2.4782 3.397733 1.6361 2.0551 2.8515 1.7489 2.1862 3.0200 1.9840 2.4612 3.375434 1.6299 2.0478 2.8419 1.7403 2.1762 3.0070 1.9703 2.4451 3.354335 1.6239 2.0407 2.8328 1.7323 2.1667 2.9946 1.9574 2.4298 3.334336 1.6182 2.0341 2.8241 1.7246 2.1577 2.9828 1.9452 2.4154 3.315537 1.6128 2.0277 2.8158 1.7173 2.1491 2.9716 1.9335 2.4016 3.297538 1.6076 2.0216 2.8080 1.7102 2.1408 2.9609 1.9224 2.3885 3.280439 1.6026 2.0158 2.8004 1.7036 2.1330 2.9507 1.9118 2.3760 3.264140 1.5979 2.0103 2.7932 1.6972 2.1255 2.9409 1.9017 2.3641 3.248641 1.5934 2.0050 2.7863 1.6911 2.1183 2.9316 1.8921 2.3528 3.233742 1.5890 1.9998 2.7796 1.6852 2.1114 2.9226 1.8828 2.3418 3.219543 1.5848 1.9949 2.7733 1.6795 2.1048 2.9141 1.8739 2.3314 3.205944 1.5808 1.9902 2.7672 1.6742 2.0985 2.9059 1.8654 2.3214 3.192945 1.5769 1.9857 2.7613 1.6689 2.0924 2.8979 1.8573 2.3118 3.180446 1.5732 1.9813 2.7556 1.6639 2.0865 2.8903 1.8495 2.3025 3.168447 1.5695 1.9771 2.7502 1.6591 2.0808 2.8830 1.8419 2.2937 3.156848 1.5661 1.9730 2.7449 1.6544 2.0753 2.8759 1.8346 2.2851 3.145749 1.5627 1.9691 2.7398 1.6499 2.0701 2.8690 1.8275 2.2768 3.134950 1.5595 1.9653 2.7349 1.6455 2.0650 2.8625 1.8208 2.2689 3.124655 1.5447 1.9481 2.7126 1.6258 2.0419 2.8326 1.7902 2.2330 3.078060 1.5320 1.9333 2.6935 1.6089 2.0222 2.8070 1.7641 2.2024 3.038265 1.5210 1.9204 2.6769 1.5942 2.0050 2.7849 1.7414 2.1759 3.003970 1.5112 1.9090 2.6623 1.5812 1.9898 2.7654 1.7216 2.1526 2.973975 1.5025 1.8990 2.6493 1.5697 1.9765 2.7481 1.7040 2.1321 2.947480 1.4947 1.8899 2.6377 1.5594 1.9644 2.7326 1.6883 2.1137 2.923785 1.4877 1.8817 2.6272 1.5501 1.9536 2.7187 1.6742 2.0973 2.902490 1.4813 1.8743 2.6176 1.5416 1.9438 2.7061 1.6613 2.0824 2.883295 1.4754 1.8675 2.6089 1.5338 1.9348 2.6945 1.6497 2.0688 2.8657100 1.4701 1.8612 2.6009 1.5268 1.9265 2.6839 1.6390 2.0563 2.8496

xv