Tài liệu MEI STRUCTURED MATHEMATICS EXAMINATION FORMULAE AND TABLES ppt

MEI STRUCTURED MATHEMATICS

EXAMINATION FORMULAE AND TABLES

1


Arithmetic series
General (kth) term,
last (nth) term, l =
Sum to n terms,
Geometric series
General (kth) term,
Sum to n terms,
Sum to infinity

Infinite series
f(x)

uk = a + (k – 1)d
un = a + (n – l)d
–
–
Sn = 1 n(a + l) = 1 n[2a + (n – 1)d]
2
2

x2
xr
= f(0) + xf'(0) + –– f"(0) + ... + –– f (r)(0) + ...
2!

r!

f(x)
f(a + x)

uk = a r k–1
a(1 – r n)
a(r n – 1)
Sn = –––––––– = ––––––––
1–r
r–1

(x – a)2
(x – a)rf(r)(a)
= f(a) + (x – a)f'(a) + –––––– f"(a) + ... + –––––––––– + ...
r!
2!
x2
xr
= f(a) + xf'(a) + –– f"(a) + ... + –– f(r)(a) + ...
2!
r!

x2
xr
ex = exp(x) = 1 + x + –– + ... + –– + ... , all x
2!
r!

a

S∞ = ––––– , – 1 < r < 1
1–r

x2
x3
xr
= x – –– + –– – ... + (–1)r+1 –– + ... , – 1 < x р 1
2
3
r

sin x

x3
x5
x 2r+1
= x – –– + –– – ... + (–1)r –––––––– + ... , all x
3!
5!
(2r + 1)!

cos x

x2
x4
x 2r
= 1 – –– + –– – ... + (–1)r –––– + ... , all x
2!
4!
(2r)!

arctan x

x3
x5
x 2r+1
= x – –– + –– – ... + (–1)r –––––– + ... , – 1 р x р 1
3
5
2r + 1

General case

sinh x

n(n – 1)
n(n – 1) ... (n – r + 1)
(1 + x)n = 1 + nx + ––––––– x2 + ... + ––––––––––––––––– xr + ... , |x| < 1,
2!
1.2 ... r
n∈‫ޒ‬

x3
x5
x 2r+1
= x + –– + –– + ... + –––––––– + ... , all x
3!
5!
(2r + 1)!

cosh x

x2
x4
x 2r
= 1 + –– + –– + ... + –––– + ... , all x
2!
4!
(2r)!

artanh x

x3
x5
x 2r+1
= x + –– + –– + ... + –––––––– + ... , – 1 < x < 1
3
5
(2r + 1)

Binomial expansions
When n is a positive integer
n
n
n
(a + b)n = an + 1 an –1 b + 2 an–2 b2 + ... + r an–r br + ... bn , n ∈ ‫ގ‬
where
n
n
n

n+1
n!
n
r = Cr = ––––––––
r + r+1 = r+1
r!(n – r)!

()

()

()

()

() ( ) (

)

2
Logarithms and exponentials
exln a = ax

logbx
loga x = –––––
logba

Numerical solution of equations
f(xn)
Newton-Raphson iterative formula for solving f(x) = 0, xn+1 = xn – ––––

f'(xn)
Complex Numbers
{r(cos θ + j sin θ)}n = r n(cos nθ + j sin nθ)
ejθ = cos θ + j sin θ
2πk
The roots of zn = 1 are given by z = exp( –––– j) for k = 0, 1, 2, ..., n – 1
n
Finite series
n
n
1
1
∑ r2 = – n(n + 1)(2n + 1)
∑ r3 = – n2(n + 1)2
4
6
r=1
r=1

ALGEBRA

ln(1 + x)

Hyperbolic functions
cosh2x – sinh2x = 1, sinh2x = 2sinhx coshx, cosh2x = cosh2x + sinh2x
arcosh x = ln(x +
x 2 + 1 ),
1+x
1
artanh x = – ln ––––– , |x| < 1

1–x
2
arsinh x = ln(x +

(

x 2 – 1 ), x у 1

)

Matrices
Anticlockwise rotation through angle θ, centre O:
Reflection in the line y = x tan θ :

( cosθθ
sin
( cos22θθ
sin

–sin θ
cos θ

)

sin 2θ
–cos 2θ

)

Cosine rule

A

b2 + c2 – a2
cos A = –––––––––– (etc.)
2bc

c

a2 = b2 + c2 –2bc cos A (etc.)
B

Trigonometry

Perpendicular distance of a point from a line and a plane
b

a

Line: (x1,y1) from ax + by + c = 0 :

a2 + b2
n1α + n2β + n3γ + d
Plane: (α,β,γ) from n1x + n2y + n3z + d = 0 : ––––––––––––––––––
√(n12 + n22 + n32)

C

sin (θ ± φ) = sin θ cos φ ± cos θ sin φ

cos (θ ± φ) = cos θ cos φ ϯ sin θ sin φ

Vector product

i a1 b1
a2b3 – a3b2
^ = j a b = a b –a b
a × b = |a| |b| sinθ n
2 2
3 1
1 3
k a3 b3
a1b2 – a2b1

tan θ ± tan φ
tan (θ ± φ) = –––––––––––– , [(θ ± φ) ≠ (k + W)π]
1 ϯ tan θ tan φ

|

a × (b × c) = (c . a) b – (a . b) c

–
–
sin θ – sin φ = 2 cos 1 (θ + φ) sin 1 (θ – φ)
2
2

Conics
Ellipse

Hyperbola

Rectangular
hyperbola

y2
x2
–– + –– = 1
2
b2
a

y2 = 4ax

y2
x2
–– – –– = 1
2
b2
a

x y = c2

Parametric form

(acosθ, bsinθ)

(at2, 2at)

(asecθ, btanθ)

c
(ct, –
–)
t

3

Parabola

Standard
form

1
– (θ – φ)
2
1
– (θ – φ)
2

Vectors and 3-D coordinate geometry
(The position vectors of points A, B, C are a, b, c.)
The position vector of the point dividing AB in the ratio λ:µ
µa + b
is
( + à)
Line:

)

a1 b1 c1
a. (b ì c) = a2 b2 c2 = b. (c × a) = c. (a × b)
a3 b3 c3

(1 + t )

–
–
sin θ + sin φ = 2 sin 1 (θ + φ) cos 1 (θ – φ)
2
2

–
cos θ + cos φ = 2 cos 1 (θ + φ) cos
2
–
cos θ – cos φ = –2 sin 1 (θ + φ) sin
2

| |(
|

Cartesian equation of line through A in direction u is
x – a1
y – a2
z – a3
–––––– = –––––– = –––––– = t
u1
u2

u3

( )

e<1
= a2 (1 – e2)

e=1

Foci

(± ae, 0)

(a, 0)

(± ae, 0)

(±c√2, ±c√2)

Directrices

a
x=±–
e

x = –a

a
x=±–
e

x + y = ±c√2

Asymptotes

none

none

y
x
–
a=±–
b

x = 0, y = 0

Eccentricity

b2

a.u
The resolved part of a in the direction u is –––––
|u|

Any of these conics can be expressed in polar
coordinates (with the focus as the origin) as:
where l is the length of the semi-latus rectum.

Plane: Cartesian equation of plane through A with normal n is

n1 x + n2y + n3z + d = 0 where d = –a . n

Mensuration

The plane through non-collinear points A, B and C has vector equation
r = a + s(b – a) + t(c – a) = (1 – s – t) a + sb + tc
The plane through A parallel to u and v has equation
r = a + su + tv

Cone :

b2

Sphere : Surface area = 4π r2
Curved surface area = π r × slant height

e>1
= a2 (e2 – 1)

e = √2

l
– = 1 + e cos θ
r

TRIGONOMETRY, VECTORS AND GEOMETRY

(1 – t2)
2t
–

For t = tan 1 θ : sin θ = –––––– , cos θ = ––––––
2
2
2
(1 + t )

ax1 + by1 + c


Differentiation f(x)
tan kx
sec x
cot x
cosec x
arcsin x

f'(x)
ksec2 kx
sec x tan x
–cosec2 x
–cosec x cot x
1
–––––––
√(1 – x2)
–1
–––––––
√(1 – x2)
1
–––––
1 + x2

cosh x
sinh x
sech2 x
1
–––––––
√(1 + x2)

arccos x
arctan x
sinh x
cosh x
tanh x
arsinh x

artanh x
4
du
dv
v ––– – u –––
dx
dx
u dy
Quotient rule y = – , ––– =
2
v
v dx
Trapezium rule

b–a
–

∫a ydx ≈ 1 h{(y0 + yn) + 2(y1 + y2 + ... + yn–1)}, where h = –––––
2
n
b

Integration by parts
Area of a sector

Arc length

dv
du
∫ u ––– dx = uv – ∫ v ––– dx
dx
dx
–
A = 1 ∫ r dθ (polar coordinates)
2
–
y
˙
A = 1 ∫ (x˙ – yx) dt (parametric form)
2
˙
˙
s = ∫ √ (x + y ) dt (parametric form)
dy
s = ∫ √ (1 + [ –––] ) dx (cartesian coordinates)
dx
dr

s = ∫ √ (r + [ –––] ) dθ (polar coordinates)
dθ
2

2

2

2

2

2

sec x
1
––––––
2 – a2
x
1
–––––––
√(a2 – x2)
1
––––––
a2 + x2
1
––––––
a2 – x2
sinh x
cosh x

tanh x
1
–––––––
√(a2 + x2)
1
–––––––
√(x2 – a2)
Surface area of revolution

∫f(x) dx (+ a constant)
(l/k) tan kx
ln |sec x|
ln |sin x|
x
–ln |cosec x + cot x| = ln |tan – |
2
x π
ln |sec x + tan x| = ln tan – + –
2 4
1
x–
–– ln ––– a
x +––
a
2a

| (

| |
x

arcsin ( – ) , |x| < a
a
1 arctan –
x
–
( a)
a
1
x
a +––
x
–– ln | ––– x | = 1 artanh ( – ) , |x| < a
–
a
a
a–
2a

cosh x
sinh x
ln cosh x
x
arsinh – or ln (x + x 2 + a 2 ),
a

()
x
arcosh ( – ) or ln (x + x
a
S = 2π∫y ds = 2π∫y√(x

˙
S = 2π∫x ds = 2π∫x√(x
˙

2

– a 2 ), x > a , a > 0

2

+ y 2) dt
˙

2

+ y 2) dt


x
y

Curvature

)|

d2y

d
x x y

ă
dx2
= = = –––––––––––––
2 + y2)3/2
ds
dy 2 3/2
(˙ ˙
x
1 + ––
dx

( [ ])

1
Radius of curvature ρ = –– ,
κ

^
Centre of curvature c = r + ρ n

L'Hôpital’s rule
If f(a) = g(a) = 0 and g'(a) ≠ 0 then

f(x)
f'(a)
Lim –––– = ––––
x ➝a g(x)
g'(a)

Multi-variable calculus

∂g/∂x
∂w
∂w
∂w
For w = g(x, y, z), δw = ––– δx + ––– δy + ––– δz
grad g = ∂g/∂y
∂x
∂y
∂z
∂g/∂z

( )

CALCULUS

1
–––––––
√(x2 – 1)
1
–––––––
(1 – x2)

arcosh x

Integration f(x)
sec2 kx
tan x
cot x
cosec x

Centre of mass (uniform bodies)
Triangular lamina:
Solid hemisphere of radius r:
Hemispherical shell of radius r:
Solid cone or pyramid of height h:

Moments of inertia (uniform bodies, mass M)
2 along median from vertex
–
3
3 r from centre
–
8
1 r from centre
–
2
1 h above the base on the
–
4

line from centre of
base to vertex

Sector of circle, radius r, angle 2θ:

2r sin θ
––––––– from centre
3θ

r sin θ
Arc of circle, radius r, angle 2θ at centre: ––––––– from centre
θ
1 h above the base on the
–
Conical shell, height h:
3 line from the centre of

Thin rod, length 2l, about perpendicular axis through centre:
Rectangular lamina about axis in plane bisecting edges of length 2l:
Thin rod, length 2l, about perpendicular axis through end:
Rectangular lamina about edge perpendicular to edges of length 2l:

Rectangular lamina, sides 2a and 2b, about perpendicular
1
– M(a2 + b2)
axis through centre:
3
Hoop or cylindrical shell of radius r about perpendicular
axis through centre:
Hoop of radius r about a diameter:
Disc or solid cylinder of radius r about axis:

base to the vertex
Solid sphere of radius r about a diameter:
5

Motion in polar coordinates
Spherical shell of radius r about a diameter:

˙
Transverse velocity:
v = rθ
v2
˙
Radial acceleration:
–rθ 2 =
r
ă

Transverse acceleration: v = r
General motion
Radial velocity:
Transverse velocity:
Radial acceleration:
Transverse acceleration:

r


r

r r 2
ă
d
ă + 2r = 1 –– (r2θ )
˙
–
˙˙
rθ

r dt

Moments as vectors
The moment about O of F acting at r is r × F

Mr2
1
– Mr2
2
1
– Mr2
2
1
– Mr2
4
2
– Mr2
5
2
– Mr2
3

Parallel axes theorem:

IA = IG + M(AG)2

Perpendicular axes theorem:

Iz = Ix + Iy (for a lamina in the (x, y) plane)

MECHANICS

Disc of radius r about a diameter:

Motion in a circle

1
– Ml2
3
1
– Ml2
3
4
– Ml2
3
4
– Ml2
3


Probability

Product-moment correlation: Pearson’s coefficient
∑ xi yi
–xy
Sxy
Σ( xi – x )( yi – y )
n
r=
=

=
2
2
Sxx Syy
 ∑ xi 2
  ∑ yi 2

Σ( xi – x ) Σ( yi – y )
− x2
− y2

 n

 n

P(A∪B) = P(A) + P(B) – P(A∩B)
P(A∩B) = P(A) . P(B|A)
P(B|A)P(A)
P(A|B) = ––––––––––––––––––––––
P(B|A)P(A) + P(B|A')P(A')

[

P(Aj)P(B|Aj)
Bayes’ Theorem: P(A j |B) = ––––––––––––
∑P(Ai)P(B|Ai)

Rank correlation: Spearman’s coefficient

Populations

6∑di2
rs = 1 – ––––––––
n(n2 – 1)

Discrete distributions
X is a random variable taking values xi in a discrete distribution with
P(X = xi) = pi
Expectation:
µ = E(X) = ∑xi pi
Variance:
σ2 = Var(X) = ∑(xi – µ)2 pi = ∑xi2pi – µ2
For a function g(X): E[g(X)] = ∑g(xi)pi

Regression

6

X is a continuous variable with probability density function (p.d.f.) f(x)
Expectation:
µ = E(X) = ∫ x f(x)dx
Variance:
σ2 = Var (X)
= ∫(x – µ)2 f(x)dx = ∫x2 f(x)dx – µ2
For a function g(X):
E[g(X)] = ∫g(x)f(x)dx
Cumulative
x
distribution function F(x) = P(X р x) = ∫–∞f(t)dt

Sxx = ∑(xi – x )2 = ∑xi2 –

n

, Syy = ∑(yi – y )2 = ∑yi2 –

(∑xi)(∑yi)
Sxy = ∑(xi – x )(yi – y ) = ∑xi yi – –––––––––
n
Covariance

Sxy
–––– = ∑( xi – x )( yi – y ) = ∑ xi yi – x y
n
n
n

1
S2 for population variance σ 2 where S2 = –––– ∑(xi – x )2fi
n–1
Probability generating functions
For a discrete distribution

For a sample of n pairs of observations (xi, yi)
(∑xi)2
–––––

Least squares regression line of y on x: y – y = b(x – x )
∑ xi yi
–xy

Sxy
∑(xi – x) (yi – y )
n
b = ––– = ––––––––––––––– =
Sxx
∑ xi 2
∑(xi – x )2
– x2
n
Estimates
Unbiased estimates from a single sample
σ2
X for population mean µ; Var X = ––
n

(∑yi)2
–––––
n

,

G(t) = E(tX)
E(X) = G'(1); Var(X) = G"(1) + µ – µ2
GX + Y (t) = GX (t) GY (t) for independent X, Y
Moment generating functions:
MX(θ) = E(eθX)
E(X) = M'(0) = µ;

E(Xn) = M(n)(0)

Var(X) = M"(0) – {M'(0)}2
MX + Y (θ) = MX (θ) MY (θ) for independent X, Y

STATISTICS

Continuous distributions

Correlation and regression

]


Markov Chains
pn + 1 = pnP
Long run proportion p = pP
Bivariate distributions
Covariance
Cov(X, Y) = E[(X – µX)(Y – µY)] = E(XY) – µXµY
Cov(X, Y)
Product-moment correlation coefficient
ρ = ––––––––
σX σY
Sum and difference
Var(aX ± bY) = a2Var(X) + b2Var(Y) ± 2ab Cov (X,Y)
If X, Y are independent: Var(aX ± bY) = a2Var(X) + b2Var(Y)
E(XY) = E(X) E(Y)
Coding
X = aX' + b
⇒ Cov(X, Y) = ac Cov(X', Y')
Y = cY' + d

}

One-factor model: xij = µ + αi + εij, where εij ~ N(0,σ2)
2
Ti2
––
SSB = ∑ni ( x i – x )2 = ∑ ––– – T
ni
n
i
i
2
––
SST = ∑ ∑ (xij – x )2 = ∑ ∑ xij2 – T
n
i j
i j

Yi
α + βxi + εi

RSS
∑(yi – a – bxi)2

α + βf(xi) + εi
α + βxi + γzi + εi

∑(yi – a –

εi ~ N(0, σ2)

No. of parameters, p
2

bf(xi))2

∑(yi – a – bxi –

2

czi)2

3

a, b, c are estimates for α, β, γ.

For the model Yi = α + βxi + εi,
Sxy
σ2
b = ––– , b ~ N β, ––– ,
Sxx
Sxx

(

b−β
~ tn –2
ˆ
σ 2 / Sxx

)

σ 2∑xi2
–––––––
a = y – b x , a ~ N α, n S
xx

(

RSS
^
––––
σ2 = n – p

)

(x0 – x )2
a + bx0 ~ N(α + βx0, σ2 1 + –––––––
–
n
Sxx
2
(Sxy)
RSS = Syy – ––––– = Syy (1 – r2)
Sxx

{

}

Randomised response technique
y
– – (1 – θ)
n
^
E(p) = ––––––––––
(2θ – 1)

[(2θ – 1) p + (1 – θ)][θ – (2θ – 1)p]

^
Var(p) = ––––––––––––––––––––––––––––––
2
n(2θ – 1)

Factorial design
Interaction between 1st and 2nd of 3 treatments
(–)

{

(Abc – abc) + (AbC – abC)
(ABc – aBc) + (ABC – aBC)
––––––––––––––––––––– – ––––––––––––––––––––––
2
2

}

Exponential smoothing
^
yn+1 = α yn + α(1 – α)yn–1 + α(1 – α)2 yn–2 + ... + α(1 – α)n–1 y1
+ (1 – α)ny0
^ = y + α(y – y )
^
^
yn+1
n
n
n
^
^
yn+1 = α yn + (1 – α) yn

STATISTICS

7

Analysis of variance

Regression


Description
Pearson’s product
moment
correlation test

r=

Distribution

∑ xi yi
–xy
n
 ∑ xi 2
  ∑ yi 2

– x2
– y2

 n
 n



t-test for the
difference in the
means of
2 samples

6∑di2
rs = 1 – –––––––
n(n2 – 1)

8

Normal test for
a mean

x–µ
σ/ n

N(0, 1)

t-test for a mean

x–µ
s/ n

tn – 1

χ2

test

t-test for
paired sample

Normal test for the
difference in the
means of 2 samples
with different
variances

Description

∑

(f

o

– fe )

( x1 – x2 ) – µ
s/ n

( x – y ) – ( µ1 – µ2 )
1 1
+
s
n1 n2

See tables

Wilcoxon Rank-sum
(or Mann-Whitney)
2-Sample test

Samples size m, n: m р n
Wilcoxon
W = sum of ranks of
sample size m
Mann-Whitney
–
T = W – 1 m(m + 1)
2

See tables

p –θ

Normal test on
binomial proportion

N(0, 1)

+ n2 – 2

1

(n1 – 1)s12 + (n2 – 1)s22
where s2 = –––––––––––––––––––––––
n1 + n2 – 2

 θ (1 – θ ) 
 n 

χ2 test for variance

(n – 1)s 2
σ2

( x – y ) – ( µ1 – µ2 )

σ 12 σ 2 2
+
n1

n2

tn

A statistic T is calculated
from the ranked data.

χ 2v

t with (n – 1)
degrees of
freedom

Distribution

Wilcoxon single
sample test

2

fe

Test statistic

F-test on ratio of
two variances

s12 /σ12
–––––––
s22 /σ22

, s12 > s22

N(0, 1)

χ2n – 1
Fn

1

–1, n2 –1

STATISTICS: HYPOTHESIS TESTS

Spearman rank
correlation test

Test statistic


Name

Mean

Function

Variance

p.g.f. G(t) (discrete)
m.g.f. M(θ) (continuous)

P(X = r) = nCr qn–rpr ,
for r = 0, 1, ... ,n , 0 < p < 1, q = 1 – p

np

npq

G(t) = (q + pt)n

Poisson (λ)
Discrete

λr
P(X = r) = e–λ ––– ,
r!
for r = 0, 1, ... , λ > 0

λ

λ

G(t) = eλ(t – 1)

µ

σ2

M(θ) = exp(µθ + Wσ 2θ 2)

1

x–µ
exp – W –––––
σ

( (

Normal N(µ, σ2)
Continuous

f(x) =

Uniform (Rectangular) on
[a, b] Continuous

1
f(x) = –––––
b–a

Exponential
Continuous

f(x) = λe–λx

Geometric
Discrete

P(X = r) = q r – 1p ,

σ 2π

) ),
2

–∞ < x < ∞

9
Negative binomial
Discrete

, aрxрb

,

0
x у 0, λ > 0
r = 1, 2, ... ,

, q=1–p

P(X = r) = r – 1Cn – 1 qr – n pn ,
r = n, n + 1, ... ,
0
q=1–p

a+b
–––––

2

1
––
12

(b – a)2

ebθ – eaθ
M(θ) = –––––––––
(b – a)θ

1
––
λ

1
––
λ2

λ
M(θ) = –––––
λ–θ

1
––
p

q
––

p2

pt
G(t) = –––––
1 – qt

n
––
p

nq
––
p2

pt
G(t) = –––––
1 – qt

(

n

)

STATISTICS: DISTRIBUTIONS

Binomial B(n, p)
Discrete

Numerical Solution of Equations

f(xn)
The Newton-Raphson iteration for solving f(x) = 0 : xn + 1 = xn – ––––
f'(xn)
Numerical integration
The trapezium rule
b
1
b–a
ydx ≈ – h{(y0 + yn) + 2(y1 + y2 + ... + yn–1)}, where h = –––––
2
n
a

∫

b

1
–
2

f(x)

(x – a)2
= f(a) + (x – a)f '(a) + –––––– f"(a) + error
2!
(x – a)2
= f(a) + (x – a)f '(a) + –––––– f"(η) , a < η < x

2!

b–a
+ y1 1 + ... + yn– 1 1 + yn– 1 ), where h = –––––
–
–
–
n
2
2
2

–
∫a ydx ≈ 1 h{(y0 + yn) + 4(y1 + y3 + ... + yn–1) + 2(y2 + y4 + ... + yn–2)},
3
b

b–a
where h = –––––
n
The Gaussian 2-point integration rule
10

  −h   h  
f ( x )dx ≈ h f   + f   
–h
  3  3
Interpolation/finite differences
h

Numerical solution of differential equations
dy
For –– = f(x, y):
dx
Euler’s method : yr + 1 = yr + hf(xr, yr); xr+1 = xr + h
Runge-Kutta method (order 2) (modified Euler method)
–
yr + 1 = yr + 1 (k1 + k2)
2

where k1 = h f(xr, yr), k2 = h f(xr + h, yr + k1)
Runge-Kutta method, order 4:

n
x – xi
Lagrange’s polynomial : Pn(x) = ∑ Lr(x)f(x) where Lr(x) = ∏ ––––––
xr – xi
i=0
i≠r

Newton’s forward difference interpolation formula
(x – x0)(x – x1)
(x – x0)
f(x) = f(x0) + –––––– ∆f(x0) + –––––––––––– ∆2f(x0) + ...
2!h2
h
Newton’s divided difference interpolation formula

–
yr+1 = yr + 1 (k1 + 2k2 + 2k3 + k4),

6

where k1 = hf(xr, yr)
–
–
k3 = hf(xr + 1 h, yr + 1 k2)
2
2

–
–
k2 = hf(xr + 1 h, yr + 1 k1)
2
2

k4 = hf(xr + h, yr + k3).

Logic gates

f(x) = f[x0] + (x – x0]f[x0, x1] + (x – x0) (x – x1)f[x0, x1, x2] + ...
Numerical differentiation
f(x + h) – 2f(x) + f(x – h)
f"(x) ≈ –––––––––––––––––––––
h2

NOT

AND

OR

NAND

NUMERICAL ANALYSIS
DECISION & DISCRETE MATHEMATICS

Simpson’s rule
for n even

∫

h2
f(a + h) = f(a) + hf '(a) + ––– f"(a + ξ), 0 < ξ < h
2!

f(x)

The mid-ordinate rule

∫a ydx ≈ h(y

Taylor polynomials
h2
f(a + h) = f(a) + hf '(a) + ––– f"(a) + error
2!


Statistical Tables

12 – 17

18 – 20
21
22
23
23
24 – 25
26 – 27
28 – 29
30
30
31
31 – 32

Cumulative binomial probability
Cumulative Poisson probability
Critical values for correlation coefficients
The Normal distribution and its inverse
Percentage points of the χ2 distribution
Percentage points of the t-distribution
Critical values for the F-test
Critical values for the Mann-Whitney test
Critical values for the Wilcoxon Rank Sum 2-sample test
Critical values for the Wilcoxon Single sample and Paired sample tests
Shewhart Chart: Action and Warning lines
Estimation of standard deviation from range
Random permutations

11

The Binomial distribution: cumulative probabilites
x

P (X р x) = ∑ nCr (1 – p) n–r pr
r=0

n

p

0.250 0.300

1/3

0.350 0.400 0.450

0.500 0.550

0.600 0.650 2/3

2

0
1
2

0.9025 0.8100 0.7225 0.6944 0.6400 0.5625 0.4900 0.4444 0.4225 0.3600 0.3025 0.2500 0.2025 0.1600 0.1225 0.1111 0.0900 0.0625 0.0400 0.0278 0.0225 0.0100 0.0025
0.9975 0.9900 0.9775 0.9722 0.9600 0.9375 0.9100 0.8889 0.8775 0.8400 0.7975 0.7500 0.6975 0.6400 0.5775 0.5556 0.5100 0.4375 0.3600 0.3056 0.2775 0.1900 0.0975
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

3

0
1
2
3

0.8574
0.9928
0.9999
1.0000

0.7290
0.9720
0.9990
1.0000

0.6141
0.9392
0.9966
1.0000

0.5787
0.9259
0.9954
1.0000

0.5120
0.8960
0.9920

1.0000

0.4219
0.8437
0.9844
1.0000

0.3430
0.7840
0.9730
1.0000

0.2963
0.7407
0.9630
1.0000

0.2746
0.7183
0.9571
1.0000

0.2160
0.6480
0.9360
1.0000

0.1664
0.5748
0.9089

1.0000

0.1250
0.5000
0.8750
1.0000

0.0911
0.4252
0.8336
1.0000

0.0640
0.3520
0.7840
1.0000

0.0429
0.2818
0.7254
1.0000

0.0370
0.2593
0.7037
1.0000

0.0270
0.2160
0.6570

1.0000

0.0156
0.1563
0.5781
1.0000

0.0080
0.1040
0.4880
1.0000

0.0046
0.0741
0.4213
1.0000

0.0034
0.0608
0.3859
1.0000

0.0010
0.0280
0.2710
1.0000

0.0001
0.0073
0.1426

1.0000

4

0
1
2
3
4

0.8145
0.9860
0.9995
1.0000

0.6561
0.9477
0.9963
0.9999
1.0000

0.5220
0.8905
0.9880
0.9995
1.0000

0.4823
0.8681
0.9838

0.9992
1.0000

0.4096
0.8192
0.9728
0.9984
1.0000

0.3164
0.7383
0.9492
0.9961
1.0000

0.2401
0.6517
0.9163
0.9919
1.0000

0.1975
0.5926
0.8889
0.9877
1.0000

0.1785
0.5630
0.8735

0.9850
1.0000

0.1296
0.4752
0.8208
0.9744
1.0000

0.0915
0.3910
0.7585
0.9590
1.0000

0.0625
0.3125
0.6875
0.9375
1.0000

0.0410
0.2415
0.6090
0.9085
1.0000

0.0256
0.1792
0.5248

0.8704
1.0000

0.0150
0.1265
0.4370
0.8215
1.0000

0.0123
0.1111
0.4074
0.8025
1.0000

0.0081
0.0837
0.3483
0.7599
1.0000

0.0039
0.0508
0.2617
0.6836
1.0000

0.0016
0.0272
0.1808

0.5904
1.0000

0.0008
0.0162
0.1319
0.5177
1.0000

0.0005
0.0120
0.1095
0.4780
1.0000

0.0001
0.0037
0.0523
0.3439
1.0000

0.0000
0.0005
0.0140
0.1855
1.0000

5

0

1
2
3
4
5

0.7738
0.9774
0.9988
1.0000

0.5905
0.9185
0.9914
0.9995
1.0000

0.4437
0.8352
0.9734
0.9978
0.9999
1.0000

0.4019
0.8038
0.9645
0.9967
0.9999
1.0000

0.3277
0.7373
0.9421
0.9933
0.9997
1.0000

0.2373
0.6328
0.8965
0.9844
0.9990
1.0000

0.1681
0.5282
0.8369
0.9692
0.9976
1.0000

0.1317
0.4609
0.7901
0.9547
0.9959
1.0000

0.1160

0.4284
0.7648
0.9460
0.9947
1.0000

0.0778
0.3370
0.6826
0.9130
0.9898
1.0000

0.0503
0.2562
0.5931
0.8688
0.9815
1.0000

0.0313
0.1875
0.5000
0.8125
0.9688
1.0000

0.0185
0.1312
0.4069

0.7438
0.9497
1.0000

0.0102
0.0870
0.3174
0.6630
0.9222
1.0000

0.0053
0.0540
0.2352
0.5716
0.8840
1.0000

0.0041
0.0453
0.2099
0.5391
0.8683
1.0000

0.0024
0.0308
0.1631
0.4718
0.8319

1.0000

0.0010
0.0156
0.1035
0.3672
0.7627
1.0000

0.0003
0.0067
0.0579
0.2627
0.6723
1.0000

0.0001
0.0033
0.0355
0.1962
0.5981
1.0000

0.0001
0.0022
0.0266
0.1648
0.5563
1.0000

0.0000
0.0005
0.0086
0.0815
0.4095
1.0000

0.0000
0.0012
0.0226
0.2262
1.0000

0
1
2
3
4
5
6

0.7351
0.9672
0.9978
0.9999
1.0000

0.5314
0.8857
0.9841

0.9987
0.9999
1.0000

0.3771
0.7765
0.9527
0.9941
0.9996
1.0000

0.3349
0.7368
0.9377
0.9913
0.9993
1.0000

0.2621
0.6554
0.9011
0.9830
0.9984
0.9999
1.0000

0.1780
0.5339
0.8306
0.9624

0.9954
0.9998
1.0000

0.1176
0.4202
0.7443
0.9295
0.9891
0.9993
1.0000

0.0878
0.3512
0.6804
0.8999
0.9822
0.9986
1.0000

0.0754
0.3191
0.6471
0.8826
0.9777
0.9982
1.0000

0.0467
0.2333

0.5443
0.8208
0.9590
0.9959
1.0000

0.0277
0.1636
0.4415
0.7447
0.9308
0.9917
1.0000

0.0156
0.1094
0.3438
0.6563
0.8906
0.9844
1.0000

0.0083
0.0692
0.2553
0.5585
0.8364
0.9723
1.0000

0.0041
0.0410
0.1792
0.4557
0.7667
0.9533
1.0000

0.0018
0.0223
0.1174
0.3529
0.6809
0.9246
1.0000

0.0014
0.0178
0.1001
0.3196
0.6488
0.9122
1.0000

0.0007
0.0109
0.0705
0.2557
0.5798
0.8824

1.0000

0.0002
0.0046
0.0376
0.1694
0.4661
0.8220
1.0000

0.0001
0.0016
0.0170
0.0989
0.3446
0.7379
1.0000

0.0000
0.0007
0.0087
0.0623
0.2632
0.6651
1.0000

0.0000
0.0004
0.0059
0.0473

0.2235
0.6229
1.0000

0.0000
0.0001
0.0013
0.0159
0.1143
0.4686
1.0000

0.0000
0.0001
0.0022
0.0328
0.2649
1.0000

0
1
2
3
4
5
6
7

0.6983
0.9556

0.9962
0.9998
1.0000

0.4783
0.8503
0.9743
0.9973
0.9998
1.0000

0.3206
0.7166
0.9262
0.9879
0.9988
0.9999
1.0000

0.2791
0.6698
0.9042
0.9824
0.9980
0.9999
1.0000
1.0000

0.2097
0.5767

0.8520
0.9667
0.9953
0.9996
1.0000
1.0000

0.1335
0.4449
0.7564
0.9294
0.9871
0.9987
0.9999
1.0000

0.0824
0.3294
0.6471
0.8740
0.9712
0.9962
0.9998
1.0000

0.0585
0.2634
0.5706
0.8267
0.9547

0.9931
0.9995
1.0000

0.0490
0.2338
0.5323
0.8002
0.9444
0.9910
0.9994
1.0000

0.0280
0.1586
0.4199
0.7102
0.9037
0.9812
0.9984
1.0000

0.0152
0.1024
0.3164
0.6083
0.8471
0.9643
0.9963
1.0000

0.0078
0.0625
0.2266
0.5000
0.7734
0.9375
0.9922
1.0000

0.0037
0.0357
0.1529
0.3917
0.6836
0.8976
0.9848
1.0000

0.0016
0.0188
0.0963
0.2898
0.5801
0.8414
0.9720
1.0000

0.0006
0.0090

0.0556
0.1998
0.4677
0.7662
0.9510
1.0000

0.0005
0.0069
0.0453
0.1733
0.4294
0.7366
0.9415
1.0000

0.0002
0.0038
0.0288
0.1260
0.3529
0.6706
0.9176
1.0000

0.0001
0.0013
0.0129
0.0706
0.2436

0.5551
0.8665
1.0000

0.0000
0.0004
0.0047
0.0333
0.1480
0.4233
0.7903
1.0000

0.0000
0.0001
0.0020
0.0176
0.0958
0.3302
0.7209
1.0000

0.0000
0.0001
0.0012
0.0121
0.0738
0.2834
0.6794
1.0000

0.0000
0.0002
0.0027
0.0257
0.1497
0.5217
1.0000

0.0000
0.0002
0.0038
0.0444
0.3017
1.0000

CUMULATIVE BINOMIAL PROBABILITY

0.9500 0.9000 0.8500 0.8333 0.8000 0.7500 0.7000 0.6667 0.6500 0.6000 0.5500 0.5000 0.4500 0.4000 0.3500 0.3333 0.3000 0.2500 0.2000 0.1667 0.1500 0.1000 0.0500
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

12

0
1

7

0.200

0.850 0.900 0.950

0.050 0.100

1

6

0.150 1/6

0.700 0.750 0.800 5/6

x


n
8

9

13
11

0.050 0.100

0.150 1/6

0.200

0.250 0.300

1/3

0.350 0.400 0.450

0.500 0.550

0.600 0.650 2/3

0.700 0.750 0.800

5/6

0.850 0.900 0.950

0
1
2
3
4
5
6
7
8

0.6634
0.9428
0.9942
0.9996
1.0000

0.4305
0.8131
0.9619
0.9950
0.9996
1.0000

0.2725
0.6572
0.8948
0.9786
0.9971
0.9998
1.0000

0.2326
0.6047
0.8652
0.9693
0.9954
0.9996
1.0000

0.1678
0.5033
0.7969
0.9437
0.9896
0.9988

0.9999
1.0000

0.1001
0.3671
0.6785
0.8862
0.9727
0.9958
0.9996
1.0000

0.0576
0.2553
0.5518
0.8059
0.9420
0.9887
0.9987
0.9999
1.0000

0.0390
0.1951
0.4682
0.7414
0.9121
0.9803
0.9974
0.9998

1.0000

0.0319
0.1691
0.4278
0.7064
0.8939
0.9747
0.9964
0.9998
1.0000

0.0168
0.1064
0.3154
0.5941
0.8263
0.9502
0.9915
0.9993
1.0000

0.0084
0.0632
0.2201
0.4770
0.7396
0.9115
0.9819
0.9983

1.0000

0.0039
0.0352
0.1445
0.3633
0.6367
0.8555
0.9648
0.9961
1.0000

0.0017
0.0181
0.0885
0.2604
0.5230
0.7799
0.9368
0.9916
1.0000

0.0007
0.0085
0.0498
0.1737
0.4059
0.6846
0.8936
0.9832

1.0000

0.0002
0.0036
0.0253
0.1061
0.2936
0.5722
0.8309
0.9681
1.0000

0.0002
0.0026
0.0197
0.0879
0.2587
0.5318
0.8049
0.9610
1.0000

0.0001
0.0013
0.0113
0.0580
0.1941
0.4482
0.7447
0.9424

1.0000

0.0000
0.0004
0.0042
0.0273
0.1138
0.3215
0.6329
0.8999
1.0000

0.0000
0.0001
0.0012
0.0104
0.0563
0.2031
0.4967
0.8322
1.0000

0.0000
0.0004
0.0046
0.0307
0.1348
0.3953
0.7674
1.0000

0.0000
0.0002
0.0029
0.0214
0.1052
0.3428
0.7275
1.0000

0.0000
0.0004
0.0050
0.0381
0.1869
0.5695
1.0000

0.0000
0.0004
0.0058
0.0572
0.3366
1.0000

0
1
2
3
4

5
6
7
8
9

0.6302
0.9288
0.9916
0.9994
1.0000

0.3874
0.7748
0.9470
0.9917
0.9991
0.9999
1.0000

0.2316
0.5995
0.8591
0.9661
0.9944
0.9994
1.0000

0.1938
0.5427

0.8217
0.9520
0.9911
0.9989
0.9999
1.0000

0.1342
0.4362
0.7382
0.9144
0.9804
0.9969
0.9997
1.0000

0.0751
0.3003
0.6007
0.8343
0.9511
0.9900
0.9987
0.9999
1.0000

0.0404
0.1960
0.4628
0.7297

0.9012
0.9747
0.9957
0.9996
1.0000

0.0260
0.1431
0.3772
0.6503
0.8552
0.9576
0.9917
0.9990
0.9999
1.0000

0.0207
0.1211
0.3373
0.6089
0.8283
0.9464
0.9888
0.9986
0.9999
1.0000

0.0101
0.0705

0.2318
0.4826
0.7334
0.9006
0.9750
0.9962
0.9997
1.0000

0.0046
0.0385
0.1495
0.3614
0.6214
0.8342
0.9502
0.9909
0.9992
1.0000

0.0020
0.0195
0.0898
0.2539
0.5000
0.7461
0.9102
0.9805
0.9980
1.0000

0.0008
0.0091
0.0498
0.1658
0.3786
0.6386
0.8505
0.9615
09954
1.0000

0.0003
0.0038
0.0250
0.0994
0.2666
0.5174
0.7682
0.9295
0.9899
1.0000

0.0001
0.0014
0.0112
0.0536
0.1717
0.3911
0.6627

0.8789
0.9793
1.0000

0.0001
0.0010
0.0083
0.0424
0.1448
0.3497
0.6228
0.8569
0.9740
1.0000

0.0000
0.0004
0.0043
0.0253
0.0988
0.2703
0.5372
0.8040
0.9596
1.0000

00000
0.0001
0.0013
0.0100

0.0489
0.1657
0.3993
0.6997
0.9249
1.0000

0.0000
0.0003
0.0031
0.0196
0.0856
0.2618
0.5638
0.8658
1.0000

0.0000
0.0001
0.0011
0.0090
0.0480
0.1783
0.4573
0.8062
1.0000

0.0000
0.0006
0.0056

0.0339
0.1409
0.4005
0.7684
1.0000

0.0000
0.0001
0.0009
0.0083
0.0530
0.2252
0.6126
1.0000

0.0000
0.0006
0.0084
0.0712
0.3698
1.0000

0
1
2
3
4
5
6
7

8
9
10

0.5987
0.9139
0.9885
0.9990
0.9999
1.0000

0.3487
0.7361
0.9298
0.9872
0.9984
0.9999
1.0000

0.1969
0.5443
0.8202
0.9500
0.9901
0.9986
0.9999
1.0000

0.1615
0.4845

0.7752
0.9303
0.9845
0.9976
0.9997
1.0000

0.1074
0.3758
0.6778
0.8791
0.9672
0.9936
0.9991
0.9999
1.0000

0.0563
0.2440
0.5256
0.7759
0.9219
0.9803
0.9965
0.9996
1.0000

0.0282
0.1493
0.3828

0.6496
0.8497
0.9527
0.9894
0.9984
0.9999
1.0000

0.0173
0.1040
0.2991
0.5593
0.7869
0.9234
0.9803
0.9966
0.9996
1.0000

0.0135
0.0860
0.2616
0.5138
0.7515
0.9051
0.9740
0.9952
0.9995
1.0000

0.0060
0.0464
0.1673
0.3823
0.6331
0.8338
0.9452
0.9877
0.9983
0.9999
1.0000

0.0025
0.0233
0.0996
0.2660
0.5044
0.7384
0.8980
0.9726
0.9955
0.9997
1.0000

0.0010
0.0107
0.0547
0.1719
0.3770
0.6230

0.8281
0.9453
0.9893
0.9990
1.0000

0.0003
0.0045
0.0274
0.1020
0.2616
0.4956
0.7340
0.9004
0.9767
0.9975
1.0000

0.0001
0.0017
0.0123
0.0548
0.1662
0.3669
0.6177
0.8327
0.9536
0.9940
1.0000

0.0000
0.0005
0.0048
0.0260
0.0949
0.2485
0.4862
0.7384
0.9140
0.9865
1.0000

0.0000
0.0004
0.0034
0.0197
0.0766
0.2131
0.4407
0.7009
0.8960
0.9827
1.0000

0.0000
0.0001
0.0016
0.0106
0.0473
0.1503

0.3504
0.6172
0.8507
0.9718
1.0000

0.0000
0.0004
0.0035
0.0197
0.0781
0.2241
0.4744
0.7560
0.9437
1.0000

0.0000
0.0001
0.0009
0.0064
0.0328
0.1209
0.3222
0.6242
0.8926
1.0000

0.0000
0.0003

0.0024
0.0155
0.0697
0.2248
0.5155
0.8385
1.0000

0.0000
0.0001
0.0014
0.0099
0.0500
0.1798
0.4557
0.8031
1.0000

0.0000
0.0001
0.0016
0.0128
0.0702
0.2639
0.6513
1.0000

0.0000
0.0001
0.0010

0.0115
0.0861
0.4013
1.0000

0
1
2
3
4
5
6
7
8
9
10
11

0.5688
0.8981
0.9848
0.9984
0.9999
1.0000

0.3138
0.6974
0.9104
0.9815
0.9972

0.9997
1.0000

0.1673
0.4922
0.7788
0.9306
0.9841
0.9973
0.9997
1.0000

0.1346
0.4307
0.7268
0.9044
0.9755
0.9954
0.9994
0.9999
1.0000

0.0859
0.3221
0.6174
0.8389
0.9496
0.9883
0.9980
0.9998

1.0000

0.0422
0.1971
0.4552
0.7133
0.8854
0.9657
0.9924
0.9988
0.9999
1.0000

0.0198
0.1130
0.3127
0.5696
0.7897
0.9218
0.9784
0.9957
0.9994
1.0000

0.0116
0.0751
0.2341
0.4726
0.7110
0.8779

0.9614
0.9912
0.9986
0.9999
1.0000

0.0088
0.0606
0.2001
0.4256
0.6683
0.8513
0.9499
0.9878
0.9980
0.9998
1.0000

0.0036
0.0302
0.1189
0.2963
0.5328
0.7535
0.9006
0.9707
0.9941
0.9993
1.0000

0.0014
0.0139
0.0652
0.1911
0.3971
0.6331
0.8262
0.9390
0.9852
0.9978
0.9998
1.0000

0.0005
0.0059
0.0327
0.1133
0.2744
0.5000
0.7256
0.8867
0.9673
0.9941
0.9995
1.0000

0.0002
0.0022
0.0148
0.0610

0.1738
0.3669
0.6029
0.8089
0.9348
0.9861
0.9986
1.0000

0.0000
0.0007
0.0059
0.0293
0.0994
0.2465
0.4672
0.7037
0.8811
0.9698
0.9964
1.0000

0.0000
0.0002
0.0020
0.0122
0.0501
0.1487
0.3317
0.5744

0.7999
0.9394
0.9912
1.0000

.00000
0.0001
0.0014
0.0088
0.0386
0.1221
0.2890
0.5274
0.7659
0.9249
0.9884
1.0000

0.0000
0.0006
0.0043
0.0216
0.0782
0.2103
0.4304
0.6873
0.8870
0.9802
1.0000

0.0000
0.0001
0.0012
0.0076
0.0343
0.1146
0.2867
0.5448
0.8029
0.9578
1.0000

0.0000
0.0002
0.0020
0.0117
0.0504
0.1611
0.3826
0.6779
0.9141
1.0000

0.0000
0.0001
0.0006
0.0046
0.0245
0.0956
0.2732

0.5693
0.8654
1.0000

0.0000
0.0003
0.0027
0.0159
0.0694
0.2212
0.5078
0.8327
1.0000

0.0000
0.0003
0.0028
0.0185
0.0896
0.3026
0.6862
1.0000

0.0000
0.0001
0.0016
0.0152
0.1019
0.4312
1.0000

CUMULATIVE BINOMIAL PROBABILITY

10

p
x


n
12

14
14

0.050 0.100

0.150 1/6

0.200

0.250 0.300

1/3

0.350 0.400 0.450

0.500 0.550

0.600 0.650 2/3

0.700 0.750 0.800

5/6

0.850 0.900 0.950

0
1
2
3
4
5
6
7
8
9
10
11
12

0.5404
0.8816
0.9804
0.9978
0.9998
1.0000

0.2824
0.6590

0.8891
0.9744
0.9957
0.9995
0.9999
1.0000

0.1422
0.4435
0.7358
0.9078
0.9761
0.9954
0.9993
0.9999
1.0000

0.1122
0.3813
0.6774
0.8748
0.9637
0.9921
0.9987
0.9998
1.0000

0.0687
0.2749
0.5583

0.7946
0.9274
0.9806
0.9961
0.9994
0.9999
1.0000

0.0317
0.1584
0.3907
0.6488
0.8424
0.9456
0.9857
0.9972
0.9996
1.0000

0.0138
0.0850
0.2528
0.4925
0.7237
0.8822
0.9614
0.9905
0.9983
0.9998
1.0000

0.0077
0.0540
0.1811
0.3931
0.6315
0.8223
0.9336
0.9812
0.9961
0.9995
1.0000

0.0057
0.0424
0.1513
0.3467
0.5833
0.7873
0.9154
0.9745
0.9944
0.9992
0.9999
1.0000

0.0022
0.0196
0.0834
0.2253

0.4382
0.6652
0.8418
0.9427
0.9847
0.9972
0.9997
1.0000

0.0008
0.0083
0.0421
0.1345
0.3044
0.5269
0.7393
0.8883
0.9644
0.9921
0.9989
0.9999
1.0000

0.0002
0.0032
0.0193
0.0730
0.1938
0.3872
0.6128

0.8062
0.9270
0.9807
0.9968
0.9998
1.0000

0.0001
0.0011
0.0079
0.0356
0.1117
0.2607
0.4731
0.6956
0.8655
0.9579
0.9917
0.9992
1.0000

0.0000
0.0003
0.0028
0.0153
0.0573
0.1582
0.3348
0.5618
0.7747

0.9166
0.9804
0.9978
1.0000

0.0000
0.0001
0.0008
0.0056
0.0255
0.0846
0.2127
0.4167
0.6533
0.8487
0.9576
0.9943
1.0000

0.0000
0.0005
0.0039
0.0188
0.0664
0.1777
0.3685
0.6069
0.8189
0.9460
0.9923

1.0000

0.0000
0.0002
0.0017
0.0095
0.0386
0.1178
0.2763
0.5075
0.7472
0.9150
0.9862
1.0000

0.0000
0.0004
0.0028
0.0143
0.0544
0.1576
0.3512
0.6093
0.8416
0.9683
1.0000

0.0000
0.0001
0.0006

0.0039
0.0194
0.0726
0.2054
0.4417
0.7251
0.9313
1.0000

0.0000
0.0002
0.0013
0.0079
0.0364
0.1252
0.3226
0.6187
0.8878
1.0000

0.0000
0.0001
0.0007
0.0046
0.0239
0.0922
0.2642
0.5565
0.8578
1.0000

0.0000
0.0001
0.0005
0.0043
0.0256
0.1109
0.3410
0.7176
1.0000

0.0000
0.0002
0.0022
0.0196
0.1184
0.4596
1.0000

0
1
2
3
4
5
6
7
8
9
10

11
12
13

0.5133
0.8646
0.9755
0.9969
0.9997
1.0000

0.2542
0.6213
0.8661
0.9658
0.9935
0.9991
0.9999
1.0000

01209
0.3983
0.6920
0.8820
0.9658
0.9925
0.9987
0.9998
1.0000

0.0935
0.3365
0.6281
0.8419
0.9488
0.9873
0.9976
0.9997
1.0000

0.0550
0.2336
0.5017
0.7473
0.9009
0.9700
0.9930
0.9988
0.9998
1.0000

0.0238
0.1267
0.3326
0.5843
0.7940
0.9198
0.9757
0.9944
0.9990

0.9999
1.0000

0.0097
0.0637
0.2025
0.4206
0.6543
0.8346
0.9376
0.9818
0.9960
0.9993
0.9999
1.0000

0.0051
0.0385
0.1387
0.3224
0.5520
0.7587
0.8965
0.9653
0.9912
0.9984
0.9998
1.0000

0.0037

0.0296
0.1132
0.2783
0.5005
0.7159
0.8705
0.9538
0.9874
0.9975
0.9997
1.0000

0.0013
0.0126
0.0579
0.1686
0.3530
0.5744
0.7712
0.9023
0.9679
0.9922
0.9987
0.9999
1.0000

0.0004
0.0049
0.0269
0.0929

0.2279
0.4268
0.6437
0.8212
0.9302
0.9797
0.9959
0.9995
1.0000

0.0001
0.0017
0.0112
0.0461
0.1334
0.2905
0.5000
0.7095
0.8666
0.9539
0.9888
0.9983
0.9999
1.0000

0.0000
0.0005
0.0041
0.0203
0.0698

0.1788
0.3563
0.5732
0.7721
0.9071
0.9731
0.9951
0.9996
1.0000

0.0000
0.0001
0.0013
0.0078
0.0321
0.0977
0.2288
0.4256
0.6470
0.8314
0.9421
0.9874
0.9987
1.0000

0.0000
0.0003
0.0025
0.0126
0.0462

0.1295
0.2841
0.4995
0.7217
0.8868
0.9704
0.9963
1.0000

0.0000
0.0002
0.0016
0.0088
0.0347
0.1035
0.2413
0.4480
0.6776
0.8613
0.9615
0.9949
1.0000

0.0000
0.0001
0.0007
0.0040
0.0182
0.0624
0.1654

0.3457
0.5794
0.7975
0.9363
0.9903
1.0000

0.0000
0.0001
0.0010
0.0056
0.0243
0.0802
0.2060
0.4157
0.6674
0.8733
0.9762
1.0000

0.0000
0.0002
0.0012
0.0070
0.0300
0.0991
0.2527
0.4983
0.7664
0.9450

1.0000

0.0000
0.0003
0.0024
0.0127
0.0512
0.1581
0.3719
0.6635
0.9065
1.0000

0.0000
0.0002
0.0013
0.0075
0.0342
0.1180
0.3080
0.6017
0.8791
1.0000

0.0000
0.0001
0.0009
0.0065
0.0342
0.1339

0.3787
0.7458
1.0000

0.0000
0.0003
0.0031
0.0245
0.1354
0.4867
1.0000

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14

0.4877
0.8470

0.9699
0.9958
0.9996
1.0000

0.2288
0.5846
0.8416
0.9559
0.9908
0.9985
0.9998
1.0000

0.1028
0.3567
0.6479
0.8535
0.9533
0.9885
0.9978
0.9997
1.0000

0.0779
0.2960
0.5795
0.8063
0.9310
0.9809

0.9959
0.9993
0.9999
1.0000

0.0440
0.1979
0.4481
0.6982
0.8702
0.9561
0.9884
0.9976
0.9996
1.0000

0.0178
0.1010
0.2811
0.5213
0.7415
0.8883
0.9617
0.9897
0.9978
0.9997
1.0000

0.0068
0.0475

0.1608
0.3552
0.5842
0.7805
0.9067
0.9685
0.9917
0.9983
0.9998
1.0000

0.0034
0.0274
0.1053
0.2612
0.4755
0.6898
0.8505
0.9424
0.9826
0.9960
0.9993
0.9999
1.0000

0.0024
0.0205
0.0839
0.2205
0.4227

0.6405
0.8164
0.9247
0.9757
0.9940
0.9989
0.9999
1.0000

0.0008
0.0081
0.0398
0.1243
0.2793
0.4859
0.6925
0.8499
0.9417
0.9825
0.9961
0.9994
0.9999
1.0000

0.0002
0.0029
0.0170
0.0632
0.1672
0.3373

0.5461
0.7414
0.8811
0.9574
0.9886
0.9978
0.9997
1.0000

0.0001
0.0009
0.0065
0.0287
0.0898
0.2120
0.3953
0.6047
0.7880
0.9102
0.9713
0.9935
0.9991
0.9999
1.0000

0.0000
0.0003
0.0022
0.0114
0.0426

0.1189
0.2586
0.4539
0.6627
0.8328
0.9368
0.9830
0.9971
0.9998
1.0000

0.0000
0.0001
0.0006
0.0039
0.0175
0.0583
0.1501
0.3075
0.5141
0.7207
0.8757
0.9602
0.9919
0.9992
1.0000

0.0000
0.0001
0.0011

0.0060
0.0243
0.0753
0.1836
0.3595
0.5773
0.7795
0.9161
0.9795
0.9976
1.0000

0.0000
0.0001
0.0007
0.0040
0.0174
0.0576
0.1495
0.3102
0.5245
0.7388
0.8947
0.9726
0.9966
1.0000

0.0000
0.0002
0.0017

0.0083
0.0315
0.0933
0.2195
0.4158
0.6448
0.8392
0.9525
0.9932
1.0000

0.0000
0.0003
0.0022
0.0103
0.0383
0.1117
0.2585
0.4787
0.7189
0.8990
0.9822
1.0000

0.0000
0.0004
0.0024
0.0116
0.0439
0.1298

0.3018
0.5519
0.8021
0.9560
1.0000

0.0000
0.0001
0.0007
0.0041
0.0191
0.0690
0.1937
0.4205
0.7040
0.9221
1.0000

0.0000
0.0003
0.0022
0.0115
0.0467
0.1465
0.3521
0.6433
0.8972
1.0000

0.0000

0.0002
0.0015
0.0092
0.0441
0.1584
0.4154
0.7712
1.0000

0.0000
0.0004
0.0042
0.0301
0.1530
0.5123
1.0000

CUMULATIVE BINOMIAL PROBABILITY

13

p
x


n
15

p
x

0.050
0.4633
0.8290
0.9638
0.9945
0.9994
0.9999
1.0000

0.4401
0.8108
0.9571
0.9930
0.9991
0.9999
1.0000

0.4181
0.7922
0.9497
0.9912
0.9988
0.9999
1.0000

0.100
0.2059
0.5490
0.8159

0.9444
0.9873
0.9978
0.9997
1.0000

0.1853
0.5147
0.7892
0.9316
0.9830
0.9967
0.9995
0.9999
1.0000

0.1668
0.4818
0.7618
0.9174
0.9779
0.9953
0.9992
0.9999
1.0000

0.150
0.0874
0.3186
0.6042

0.8227
0.9383
0.9832
0.9964
0.9994
0.9999
1.0000

1/6
0.0649
0.2596
0.5322
0.7685
0.9102
0.9726
0.9934
0.9987
0.9998
1.0000

0.200
0.0352
0.1671
0.3980
0.6482
0.8358
0.9389
0.9819
0.9958
0.9992

0.9999
1.0000

0.250
0.0134
0.0802
0.2361
0.4613
0.6865
0.8516
0.9434
0.9827
0.9958
0.9992
0.9999
1.0000

0.300
0.0047
0.0353
0.1268
0.2969
0.5155
0.7216
0.8689
0.9500
0.9848
0.9963
0.9993
0.9999

1.0000

1/3
0.0023
0.0194
0.0794
0.2092
0.4041
0.6184
0.7970
0.9118
0.9692
0.9915
0.9982
0.9997
1.0000

0.350
0.0016
0.0142
0.0617
0.1727
0.3519
0.5643
0.7548
0.8868
0.9578
0.9876
0.9972
0.9995

0.9999
1.0000

0.400
0.0005
0.0052
0.0271
0.0905
0.2173
0.4032
0.6098
0.7869
0.9050
0.9662
0.9907
0.9981
0.9997
1.0000

0.450
0.0001
0.0017
0.0107
0.0424
0.1204
0.2608
0.4522
0.6535
0.8182
0.9231

0.9745
0.9937
0.9989
0.9999
1.0000

0.500
0.0000
0.0005
0.0037
0.0176
0.0592
0.1509
0.3036
0.5000
0.6964
0.8491
0.9408
0.9824
0.9963
0.9995
1.0000

0.0743
0.2839
0.5614
0.7899
0.9209
0.9765
0.9944

0.9989
0.9998
1.0000

0.0541
0.2272
0.4868
0.7291
0.8866
0.9622
0.9899
0.9979
0.9996
1.0000

0.0281
0.1407
0.3518
0.5981
0.7982
0.9183
0.9733
0.9930
0.9985
0.9998
1.0000

0.0100
0.0635
0.1971

0.4050
0.6302
0.8103
0.9204
0.9729
0.9925
0.9984
0.9997
1.0000

0.0033
0.0261
0.0994
0.2459
0.4499
0.6598
0.8247
0.9256
0.9743
0.9929
0.9984
0.9997
1.0000

0.0015
0.0137
0.0594
0.1659
0.3391
0.5469

0.7374
0.8735
0.9500
0.9841
0.9960
0.9992
0.9999
1.0000

0.0010
0.0098
0.0451
0.1339
0.2892
0.4900
0.6881
0.8406
0.9329
0.9771
0.9938
0.9987
0.9998
1.0000

0.0003
0.0033
0.0183
0.0651
0.1666
0.3288

0.5272
0.7161
0.8577
0.9417
0.9809
0.9951
0.9991
0.9999
1.0000

0.0001
0.0010
0.0066
0.0281
0.0853
0.1976
0.3660
0.5629
0.7441
0.8759
0.9514
0.9851
0.9965
0.9994
0.9999
1.0000

0.0000
0.0003
0.0021

0.0106
0.0384
0.1051
0.2272
0.4018
0.5982
0.7728
0.8949
0.9616
0.9894
0.9979
0.9997
1.0000

0.0631
0.2525
0.5198
0.7556
0.9013
0.9681
0.9917
0.9983
0.9997
1.0000

0.0451
0.1983
0.4435
0.6887
0.8604

0.9496
0.9853
0.9965
0.9993
0.9999
1.0000

0.0225
0.1182
0.3096
0.5489
0.7582
0.8943
0.9623
0.9891
0.9974
0.9995
0.9999
1.0000

0.0075
0.0501
0.1637
0.3530
0.5739
0.7653
0.8929
0.9598
0.9876
0.9969

0.9994
0.9999
1.0000

0.0023
0.0193
0.0774
0.2019
0.3887
0.5968
0.7752
0.8954
0.9597
0.9873
0.9968
0.9993
0.9999
1.0000

0.0010
0.0096
0.0442
0.1304
0.2814
0.4777
0.6739
0.8281
0.9245
0.9727
0.9920

0.9981
0.9997
1.0000

0.0007
0.0067
0.0327
0.1028
0.2348
0.4197
0.6188
0.7872
0.9006
0.9617
0.9880
0.9970
0.9994
0.9999
1.0000

0.0002
0.0021
0.0123
0.0464
0.1260
0.2639
0.4478
0.6405
0.8011
0.9081

0.9652
0.9894
0.9975
0.9995
0.9999
1.0000

0.0000
0.0006
0.0041
0.0184
0.0596
0.1471
0.2902
0.4743
0.6626
0.8166
0.9174
0.9699
0.9914
0.9981
0.9997
1.0000

0.0000
0.0001
0.0012
0.0064
0.0245
0.0717

0.1662
0.3145
0.5000
0.6855
0.8338
0.9283
0.9755
0.9936
0.9988
0.9999
1.0000

0.550
0.0000
0.0001
0.0011
0.0063
0.0255
0.0769
0.1818
0.3465
0.5478
0.7392
0.8796
0.9576
0.9893
0.9983
0.9999
1.0000
0.0000

0.0001
0.0006
0.0035
0.0149
0.0486
0.1241
0.2559
0.4371
0.6340
0.8024
0.9147
0.9719
0.9934
0.9990
0.9999
1.0000
0.0000
0.0003
0.0019
0.0086
0.0301
0.0826
0.1834
0.3374
0.5257
0.7098
0.8529
0.9404
0.9816
0.9959

0.9994
1.0000

0.600 0.650 2/3

0.700 0.750 0.800

5/6

0.850 0.900 0.950

0.0000
0.0003
0.0019
0.0093
0.0338
0.0950
0.2131
0.3902
0.5968
0.7827
0.9095
0.9729
0.9948
0.9995
1.0000

0.0000
0.0001
0.0005

0.0028
0.0124
0.0422
0.1132
0.2452
0.4357
0.6481
0.8273
0.9383
0.9858
0.9984
1.0000

0.0000
0.0003
0.0018
0.0085
0.0308
0.0882
0.2030
0.3816
0.5959
0.7908
0.9206
0.9806
0.9977
1.0000

0.0000
0.0001

0.0007
0.0037
0.0152
0.0500
0.1311
0.2784
0.4845
0.7031
0.8732
0.9647
0.9953
1.0000

0.0000
0.0001
0.0008
0.0042
0.0173
0.0566
0.1484
0.3135
0.5387
0.7639
0.9198
0.9866
1.0000

0.0000
0.0001
0.0008

0.0042
0.0181
0.0611
0.1642
0.3518
0.6020
0.8329
0.9648
1.0000

0.0000
0.0002
0.0013
0.0066
0.0274
0.0898
0.2315
0.4678
0.7404
0.9351
1.0000

0.0000
0.0001
0.0006
0.0036
0.0168
0.0617
0.1773
0.3958

0.6814
0.9126
1.0000

0.0000
0.0003
0.0022
0.0127
0.0556
0.1841
0.4510
0.7941
1.0000

0.0000
0.0001
0.0006
0.0055
0.0362
0.1710
0.5367
1.0000

0.0000
0.0001
0.0009
0.0049
0.0191
0.0583
0.1423

0.2839
0.4728
0.6712
0.8334
0.9349
0.9817
0.9967
0.9997
1.0000

0.0000
0.0002
0.0013
0.0062
0.0229
0.0671
0.1594
0.3119
0.5100
0.7108
0.8661
0.9549
0.9902
0.9990
1.0000

0.0000
0.0001
0.0008
0.0040

0.0159
0.0500
0.1265
0.2626
0.4531
0.6609
0.8341
0.9406
0.9863
0.9985
1.0000

0.0000
0.0003
0.0016
0.0071
0.0257
0.0744
0.1753
0.3402
0.5501
0.7541
0.9006
0.9739
0.9967
1.0000

0.0000
0.0003
0.0016

0.0075
0.0271
0.0796
0.1897
0.3698
0.5950
0.8029
0.9365
0.9900
1.0000

0.0000
0.0002
0.0015
0.0070
0.0267
0.0817
0.2018
0.4019
0.6482
0.8593
0.9719
1.0000

0.0000
0.0004
0.0021
0.0101
0.0378
0.1134

0.2709
0.5132
0.7728
0.9459
1.0000

0.0000
0.0002
0.0011
0.0056
0.0235
0.0791
0.2101
0.4386
0.7161
0.9257
1.0000

0.0000
0.0001
0.0005
0.0033
0.0170
0.0684
0.2108
0.4853
0.8147
1.0000

0.0000

0.0001
0.0009
0.0070
0.0429
0.1892
0.5599
1.0000

0.0000
0.0001
0.0005
0.0025
0.0106
0.0348
0.0919
0.1989
0.3595
0.5522
0.7361
0.8740
0.9536
0.9877
0.9979
0.9998
1.0000

0.0000
0.0001
0.0006
0.0030

0.0120
0.0383
0.0994
0.2128
0.3812
0.5803
0.7652
0.8972
0.9673
0.9933
0.9993
1.0000

0.0000
0.0003
0.0019
0.0080
0.0273
0.0755
0.1719
0.3261
0.5223
0.7186
0.8696
0.9558
0.9904
0.9990
1.0000

0.0000

0.0001
0.0007
0.0032
0.0127
0.0403
0.1046
0.2248
0.4032
0.6113
0.7981
0.9226
0.9807
0.9977
1.0000

0.0000
0.0001
0.0006
0.0031
0.0124
0.0402
0.1071
0.2347
0.4261
0.6470
0.8363
0.9499
0.9925
1.0000

0.0000
0.0001
0.0005
0.0026
0.0109
0.0377
0.1057
0.2418
0.4511
0.6904
0.8818
0.9775
1.0000

0.0000
0.0001
0.0007
0.0035
0.0147
0.0504
0.1396
0.3113
0.5565
0.8017
0.9549
1.0000

0.0000
0.0003
0.0017

0.0083
0.0319
0.0987
0.2444
0.4802
0.7475
0.9369
1.0000

0.0000
0.0001
0.0008
0.0047
0.0221
0.0826
0.2382
0.5182
0.8332
1.0000

0.0000
0.0001
0.0012
0.0088
0.0503
0.2078
0.5819
1.0000

CUMULATIVE BINOMIAL PROBABILITY

15

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16 0
1
2
3
4
5
6
7
8
9
10

11
12
13
14
15
16
17 0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17


n
18

16

0.050 0.100

0.150 1/6

0.200

0.250 0.300

1/3

0.350 0.400 0.450

0.500 0.550

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14

15
16
17
18

0.3972
0.7735
0.9419
0.9891
0.9985
0.9998
1.0000

0.0536
0.2241
0.4794
0.7202
0.8794
0.9581
0.9882
0.9973
0.9995
0.9999
1.0000

0.0180
0.0991
0.2713
0.5010
0.7164

0.8671
0.9487
0.9837
0.9957
0.9991
0.9998
1.0000

0.0056
0.0395
0.1353
0.3057
0.5187
0.7175
0.8610
0.9431
0.9807
0.9946
0.9988
0.9998
1.0000

0.0016
0.0142
0.0600
0.1646
0.3327
0.5344
0.7217
0.8593

0.9404
0.9790
0.9939
0.9986
0.9997
1.0000

0.0007
0.0068
0.0326
0.1017
0.2311
0.4122
0.6085
0.7767
0.8924
0.9567
0.9856
0.9961
0.9991
0.9999
1.0000

0.0004
0.0046
0.0236
0.0783
0.1886
0.3550
0.5491

0.7283
0.8609
0.9403
0.9788
0.9938
0.9986
0.9997
1.0000

0.0001
0.0013
0.0082
0.0328
0.0942
0.2088
0.3743
0.5634
0.7368
0.8653
0.9424
0.9797
0.9942
0.9987
0.9998
1.0000

0.0000
0.0003
0.0025
0.0120

0.0411
0.1077
0.2258
0.3915
0.5778
0.7473
0.8720
0.9463
0.9817
0.9951
0.9990
0.9999
1.0000

0.0000
0.0001
0.0007
0.0038
0.0154
0.0481
0.1189
0.2403
0.4073
0.5927
0.7597
0.8811
0.9519
0.9846
0.9962
0.9993

0.9999
1.0000

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

0.3774
0.7547
0.9335
0.9868
0.9980
0.9998

1.0000

0.0144
0.0829
0.2369
0.4551
0.6733
0.8369
0.9324
0.9767
0.9933
0.9984
0.9997
1.0000

0.0042
0.0310
0.1113
0.2631
0.4654
0.6678
0.8251
0.9225
0.9713
0.9911
0.9977
0.9995
0.9999
1.0000

0.0011
0.0104
0.0462
0.1332
0.2822
0.4739
0.6655
0.8180
0.9161
0.9674
0.9895
0.9972
0.9994
0.9999
1.0000

0.0005
0.0047
0.0240
0.0787
0.1879
0.3519
0.5431
0.7207
0.8538
0.9352
0.9759
0.9926
0.9981
0.9996

0.9999
1.0000

0.0003
0.0031
0.0170
0.0591
0.1500
0.2968
0.4812
0.6656
0.8145
0.9125
0.9653
0.9886
0.9969
0.9993
0.9999
1.0000

0.0001
0.0008
0.0055
0.0230
0.0696
0.1629
0.3081
0.4878
0.6675
0.8139

0.9115
0.9648
0.9884
0.9969
0.9994
0.9999
1.0000

0.0000
0.0002
0.0015
0.0077
0.0280
0.0777
0.1727
0.3169
0.4940
0.6710
0.8159
0.9129
0.9658
0.9891
0.9972
0.9995
0.9999
1.0000

0.1501
0.4503
0.7338

0.9018
0.9718
0.9936
0.9988
0.9998
1.0000

0.1351
0.4203
0.7054
0.8850
0.9648
0.9914
0.9983
0.9997
1.0000

0.0456
0.1985
0.4413
0.6841
0.8556
0.9463
0.9837
0.9959
0.9992
0.9999
1.0000

0.0376

0.1728
0.4027
0.6479
0.8318
0.9347
0.9794
0.9947
0.9989
0.9998
1.0000

0.0313
0.1502
0.3643
0.6070
0.8011
0.9176
0.9719
0.9921
0.9982
0.9996
0.9999
1.0000

0.0000
0.0004
0.0022
0.0096
0.0318
0.0835

0.1796
0.3238
0.5000
0.6762
0.8204
0.9165
0.9682
0.9904
0.9978
0.9996
1.0000

0.0000
0.0001
0.0010
0.0049
0.0183
0.0537
0.1280
0.2527
0.4222
0.6085
0.7742
0.8923
0.9589
0.9880
0.9975
0.9997
1.0000

0.0000
0.0001
0.0005
0.0028
0.0109
0.0342
0.0871
0.1841
0.3290
0.5060
0.6831
0.8273
0.9223
0.9720
0.9923
0.9985
0.9998
1.0000

0.600 0.650 2/3

0.700 0.750 0.800

5/6

0.850 0.900 0.950

0.0000
0.0002
0.0013

0.0058
0.0203
0.0576
0.1347
0.2632
0.4366
0.6257
0.7912
0.9058
0.9672
0.9918
0.9987
0.9999
1.0000

0.0000
0.0003
0.0014
0.0062
0.0212
0.0597
0.1391
0.2717
0.4509
0.6450
0.8114
0.9217
0.9764
0.9954
0.9996

1.0000

0.0000
0.0001
0.0009
0.0039
0.0144
0.0433
0.1076
0.2233
0.3915
0.5878
0.7689
0.8983
0.9674
0.9932
0.9993
1.0000

0.0000
0.0003
0.0014
0.0061
0.0210
0.0596
0.1407
0.2783
0.4656
0.6673
0.8354

0.9400
0.9858
0.9984
1.0000

0.0000
0.0002
0.0012
0.0054
0.0193
0.0569
0.1390
0.2825
0.4813
0.6943
0.8647
0.9605
0.9944
1.0000

0.0000
0.0002
0.0009
0.0043
0.0163
0.0513
0.1329
0.2836
0.4990
0.7287

0.9009
0.9820
1.0000

0.0000
0.0002
0.0011
0.0053
0.0206
0.0653
0.1682
0.3521
0.5973
0.8272
0.9624
1.0000

0.0000
0.0001
0.0005
0.0027
0.0118
0.0419
0.1206
0.2798
0.5203
0.7759
0.9464
1.0000

0.0000
0.0002
0.0012
0.0064
0.0282
0.0982
0.2662
0.5497
0.8499
1.0000

0.0000
0.0002
0.0015
0.0109
0.0581
0.2265
0.6028
1.0000

0.0000
0.0001
0.0006
0.0031
0.0116
0.0352
0.0885
0.1861
0.3325
0.5122

0.6919
0.8371
0.9304
0.9770
0.9945
0.9992
0.9999
1.0000

0.0000
0.0001
0.0007
0.0031
0.0114
0.0347
0.0875
0.1855
0.3344
0.5188
0.7032
0.8500
0.9409
0.9830
0.9969
0.9997
1.0000

0.0000
0.0001
0.0004

0.0019
0.0074
0.0241
0.0648
0.1462
0.2793
0.4569
0.6481
0.8121
0.9213
0.9760
0.9953
0.9995
1.0000

0.0000
0.0001
0.0006
0.0028
0.0105
0.0326
0.0839
0.1820
0.3345
0.5261
0.7178
0.8668
0.9538
0.9896
0.9989

1.0000

0.0000
0.0001
0.0005
0.0023
0.0089
0.0287
0.0775
0.1749
0.3322
0.5346
0.7369
0.8887
0.9690
0.9958
1.0000

0.0000
0.0003
0.0016
0.0067
0.0233
0.0676
0.1631
0.3267
0.5449
0.7631
0.9171
0.9856

1.0000

0.0000
0.0001
0.0004
0.0018
0.0079
0.0281
0.0824
0.1989
0.3930
0.6357
0.8498
0.9687
1.0000

0.0000
0.0001
0.0008
0.0041
0.0163
0.0537
0.1444
0.3159
0.5587
0.8015
0.9544
1.0000

0.0000

0.0003
0.0017
0.0086
0.0352
0.1150
0.2946
0.5797
0.8649
1.0000

0.0002
0.0020
0.0132
0.0665
0.2453
0.6226
1.0000

CUMULATIVE BINOMIAL PROBABILITY

19

p
x


n
20

p

x

0.150 1/6

0.200

0.250 0.300

1/3

0.350 0.400 0.450

0.3585
0.7358
0.9245
0.9841
0.9974
0.9997
1.0000

0.0388
0.1756
0.4049
0.6477
0.8298
0.9327
0.9781
0.9941
0.9987
0.9998

1.0000

0.0115
0.0692
0.2061
0.4114
0.6296
0.8042
0.9133
0.9679
0.9900
0.9974
0.9994
0.9999
1.0000

0.0032
0.0243
0.0913
0.2252
0.4148
0.6172
0.7858
0.8982
0.9591
0.9861
0.9961
0.9991
0.9998
1.0000

0.0003
0.0033
0.0176
0.0604
0.1515
0.2972
0.4793
0.6615
0.8095
0.9081
0.9624
0.9870
0.9963
0.9991
0.9998
1.0000

0.0002
0.0021
0.0121
0.0444
0.1182
0.2454
0.4166
0.6010
0.7624
0.8782
0.9468
0.9804

0.9940
0.9985
0.9998
1.0000

0.1216
0.3917
0.6769
0.8670
0.9568
0.9887
0.9976
0.9996
0.9999
1.0000

0.0261
0.1304
0.3287
0.5665
0.7687
0.8982
0.9629
0.9887
0.9972
0.9994
0.9999
1.0000

0.0008

0.0076
0.0355
0.1071
0.2375
0.4164
0.6080
0.7723
0.8867
0.9520
0.9829
0.9949
0.9987
0.9997
1.0000

0.0000
0.0005
0.0036
0.0160
0.0510
0.1256
0.2500
0.4159
0.5956
0.7553
0.8725
0.9435
0.9790
0.9935
0.9984

0.9997
1.0000

0.0000
0.0001
0.0009
0.0049
0.0189
0.0553
0.1299
0.2520
0.4143
0.5914
0.7507
0.8692
0.9420
0.9786
0.9936
0.9985
0.9997
1.0000

0.500 0.550
0.0000
0.0002
0.0013
0.0059
0.0207
0.0577
0.1316

0.2517
0.4119
0.5881
0.7483
0.8684
0.9423
0.9793
0.9941
0.9987
0.9998
1.0000

0.0000
0.0003
0.0015
0.0064
0.0214
0.0580
0.1308
0.2493
0.4086
0.5857
0.7480
0.8701
0.9447
0.9811
0.9951
0.9991
0.9999
1.0000

0.600 0.650 2/3

0.0000
0.0003
0.0016
0.0065
0.0210
0.0565
0.1275
0.2447
0.4044
0.5841
0.7500
0.8744
0.9490
0.9840
0.9964
0.9995
1.0000

0.0000
0.0003
0.0015
0.0060
0.0196
0.0532
0.1218
0.2376
0.3990

0.5834
0.7546
0.8818
0.9556
0.9879
0.9979
0.9998
1.0000

0.0000
0.0002
0.0009
0.0037
0.0130
0.0376
0.0919
0.1905
0.3385
0.5207
0.7028
0.8485
0.9396
0.9824
0.9967
0.9997
1.0000

0.700 0.750 0.800

0.0000

0.0003
0.0013
0.0051
0.0171
0.0480
0.1133
0.2277
0.3920
0.5836
0.7625
0.8929
0.9645
0.9924
0.9992
1.0000

0.0000
0.0002
0.0009
0.0039
0.0139
0.0409
0.1018
0.2142
0.3828
0.5852
0.7748
0.9087
0.9757
0.9968

1.0000

0.0000
0.0001
0.0006
0.0026
0.0100
0.0321
0.0867
0.1958
0.3704
0.5886
0.7939
0.9308
0.9885
1.0000

5/6

0.0000
0.0001
0.0006
0.0028
0.0113
0.0371
0.1018
0.2313
0.4335
0.6713
0.8696

0.9739
1.0000

0.850 0.900 0.950

0.0000
0.0002
0.0013
0.0059
0.0219
0.0673
0.1702
0.3523
0.5951
0.8244
0.9612
1.0000

0.0000
0.0001
0.0004
0.0024
0.0113
0.0432
0.1330
0.3231
0.6083
0.8784
1.0000

0.0000
0.0003
0.0026
0.0159
0.0755
0.2642
0.6415
1.0000

17

CUMULATIVE BINOMIAL PROBABILITY

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

17
18
19
20

0.050 0.100


The Poisson distribution: cumulative probabilities
x

λr
P (X р x) = ∑ e–λ ––
r!
r=0
xλ

0.01

0
1
2
3

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09
0.9139
0.9962
0.9999
1.0000

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.9048
0.9953
0.9998
1.0000
.....
.....
.....
.....

0.8187
0.9825
0.9989
0.9999
1.0000
.....
.....
.....

0.7408
0.9631
0.9964
0.9997
1.0000
.....
.....
.....

0.6703

0.9384
0.9921
0.9992
0.9999
1.0000
.....
.....

0.6065
0.9098
0.9856
0.9982
0.9998
1.0000
.....
.....

0.5488
0.8781
0.9769
0.9966
0.9996
1.0000
.....
.....

0.4966
0.8442
0.9659
0.9942

0.9992
0.9999
1.0000
.....

0.4493
0.8088
0.9526
0.9909
0.9986
0.9998
1.0000
.....

0.4066
0.7725
0.9371
0.9865
0.9977
0.9997
1.0000
.....

1.10

1.20

1.30

1.40

1.50

1.60

1.70

1.80

1.90

0.3329
0.6990
0.9004
0.9743
0.9946
0.9990
0.9999
1.0000
.....
.....

0.3012
0.6626
0.8795
0.9662
0.9923
0.9985
0.9997
1.0000

.....
.....

0.2725
0.6268
0.8571
0.9569
0.9893
0.9978
0.9996
0.9999
1.0000
.....

0.2466
0.5918
0.8335
0.9463
0.9857
0.9968
0.9994
0.9999
1.0000
.....

0.2231
0.5578
0.8088
0.9344
0.9814

0.9955
0.9991
0.9998
1.0000
.....

0.2019
0.5249
0.7834
0.9212
0.9763
0.9940
0.9987
0.9997
1.0000
.....

0.1827
0.4932
0.7572
0.9068
0.9704
0.9920
0.9981
0.9996
0.9999
1.0000

0.1653
0.4628

0.7306
0.8913
0.9636
0.9896
0.9974
0.9994
0.9999
1.0000

0.1496
0.4337
0.7037
0.8747
0.9559
0.9868
0.9966
0.9992
0.9998
1.0000

x λ 2.00

2.10

2.20

2.30

2.40

2.50

2.60

2.70

2.80

2.90

0
1
2
3
4
5
6
7
8
9
10
11
12

0.1225
0.3796
0.6496
0.8386
0.9379
0.9796

0.9941
0.9985
0.9997
0.9999
1.0000
.....
.....

0.1108
0.3546
0.6227
0.8194
0.9275
0.9751
0.9925
0.9980
0.9995
0.9999
1.0000
.....
.....

0.1003
0.3309
0.5960
0.7993
0.9162
0.9700
0.9906
0.9974

0.9994
0.9999
1.0000
.....
.....

0.0907
0.3084
0.5697
0.7787
0.9041
0.9643
0.9884
0.9967
0.9991
0.9998
1.0000
.....
.....

0.0821
0.2873
0.5438
0.7576
0.8912
0.9580
0.9858
0.9958
0.9989
0.9997

0.9999
1.0000
.....

0.0743
0.2674
0.5184
0.7360
0.8774
0.9510
0.9828
0.9947
0.9985
0.9996
0.9999
1.0000
.....

0.0672
0.2487
0.4936
0.7141
0.8629
0.9433
0.9794
0.9934
0.9981
0.9995
0.9999
1.0000

.....

0.0608
0.2311
0.4695
0.6919
0.8477
0.9349
0.9756
0.9919
0.9976
0.9993
0.9998
1.0000
.....

0.0550
0.2146
0.4460
0.6696
0.8318
0.9258
0.9713
0.9901
0.9969
0.9991
0.9998
0.9999
1.0000

xλ
0
1
2
3
4
5
6
7

x λ 1.00
18

0
1
2
3
4
5
6
7
8
9

0.3679
0.7358
0.9197
0.9810
0.9963
0.9994

0.9999
1.0000
.....
.....

0.1353
0.4060
0.6767
0.8571
0.9473
0.9834
0.9955
0.9989
0.9998
1.0000
.....
.....
.....

3.10

3.20

3.30

3.40

3.50

3.60

3.70

3.80

3.90

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14

0.0450
0.1847
0.4012
0.6248
0.7982
0.9057
0.9612

0.9858
0.9953
0.9986
0.9996
0.9999
1.0000
.....
.....

0.0408
0.1712
0.3799
0.6025
0.7806
0.8946
0.9554
0.9832
0.9943
0.9982
0.9995
0.9999
1.0000
.....
.....

0.0369
0.1586
0.3594
0.5803
0.7626

0.8829
0.9490
0.9802
0.9931
0.9978
0.9994
0.9998
1.0000
.....
.....

0.0334
0.1468
0.3397
0.5584
0.7442
0.8705
0.9421
0.9769
0.9917
0.9973
0.9992
0.9998
0.9999
1.0000
.....

0.0302
0.1359
0.3208

0.5366
0.7254
0.8576
0.9347
0.9733
0.9901
0.9967
0.9990
0.9997
0.9999
1.0000
.....

0.0273
0.1257
0.3027
0.5152
0.7064
0.8441
0.9267
0.9692
0.9883
0.9960
0.9987
0.9996
0.9999
1.0000
.....

0.0247

0.1162
0.2854
0.4942
0.6872
0.8301
0.9182
0.9648
0.9863
0.9952
0.9984
0.9995
0.9999
1.0000
.....

0.0224
0.1074
0.2689
0.4735
0.6678
0.8156
0.9091
0.9599
0.9840
0.9942
0.9981
0.9994
0.9998
1.0000
.....

0.0202
0.0992
0.2531
0.4532
0.6484
0.8006
0.8995
0.9546
0.9815
0.9931
0.9977
0.9993
0.9998
0.9999
1.0000

x λ 4.00

4.10

4.20

4.30

4.40

4.50

4.60

4.70

4.80

4.90

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

0.0166
0.0845
0.2238
0.4142
0.6093

0.7693
0.8786
0.9427
0.9755
0.9905
0.9966
0.9989
0.9997
0.9999
1.0000
.....
.....

0.0150
0.0780
0.2102
0.3954
0.5898
0.7531
0.8675
0.9361
0.9721
0.9889
0.9959
0.9986
0.9996
0.9999
1.0000
.....
.....

0.0136
0.0719
0.1974
0.3772
0.5704
0.7367
0.8558
0.9290
0.9683
0.9871
0.9952
0.9983
0.9995
0.9998
1.0000
.....
.....

0.0123
0.0663
0.1851
0.3594
0.5512
0.7199
0.8436
0.9214
0.9642
0.9851
0.9943

0.9980
0.9993
0.9998
0.9999
1.0000
.....

0.0111
0.0611
0.1736
0.3423
0.5321
0.7029
0.8311
0.9134
0.9597
0.9829
0.9933
0.9976
0.9992
0.9997
0.9999
1.0000
.....

0.0101
0.0563
0.1626
0.3257
0.5132

0.6858
0.8180
0.9049
0.9549
0.9805
0.9922
0.9971
0.9990
0.9997
0.9999
1.0000
.....

0.0091
0.0518
0.1523
0.3097
0.4946
0.6684
0.8046
0.8960
0.9497
0.9778
0.9910
0.9966
0.9988
0.9996
0.9999
1.0000
.....

0.0082
0.0477
0.1425
0.2942
0.4763
0.6510
0.7908
0.8867
0.9442
0.9749
0.9896
0.9960
0.9986
0.9995
0.9999
1.0000
.....

0.0074
0.0439
0.1333
0.2793
0.4582
0.6335
0.7767
0.8769
0.9382
0.9717
0.9880

0.9953
0.9983
0.9994
0.9998
0.9999
1.0000

0.0498
0.1991
0.4232
0.6472
0.8153
0.9161
0.9665
0.9881
0.9962
0.9989
0.9997
0.9999
1.0000
.....
.....

0.0183
0.0916
0.2381
0.4335
0.6288
0.7851
0.8893

0.9489
0.9786
0.9919
0.9972
0.9991
0.9997
0.9999
1.0000
.....
.....

CUMULATIVE POISSON PROBABILITY

0.9900 0.9802 0.9704 0.9608 0.9512 0.9418 0.9324 0.9231
1.0000 0.9998 0.9996 0.9992 0.9988 0.9983 0.9977 0.9970
. . . . . 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.0000 1.0000

x λ 3.00


5.20

5.30

5.40

5.50

5.60

5.70

5.80

5.90

x λ 7.00

7.10

7.20

7.30

7.40

7.50

7.60

7.70

7.80

7.90

0.0067
0.0404
0.1247

0.2650
0.4405
0.6160
0.7622
0.8666
0.9319
0.9682
0.9863
0.9945
0.9980
0.9993
0.9998
0.9999
1.0000
.....

0.0061
0.0372
0.1165
0.2513
0.4231
0.5984
0.7474
0.8560
0.9252
0.9644
0.9844
0.9937
0.9976
0.9992

0.9997
0.9999
1.0000
.....

0.0055
0.0342
0.1088
0.2381
0.4061
0.5809
0.7324
0.8449
0.9181
0.9603
0.9823
0.9927
0.9972
0.9990
0.9997
0.9999
1.0000
.....

0.0050
0.0314
0.1016
0.2254
0.3895
0.5635

0.7171
0.8335
0.9106
0.9559
0.9800
0.9916
0.9967
0.9988
0.9996
0.9999
1.0000
.....

0.0045
0.0289
0.0948
0.2133
0.3733
0.5461
0.7017
0.8217
0.9027
0.9512
0.9775
0.9904
0.9962
0.9986
0.9995
0.9998
0.9999

1.0000

0.0041
0.0266
0.0884
0.2017
0.3575
0.5289
0.6860
0.8095
0.8944
0.9462
0.9747
0.9890
0.9955
0.9983
0.9994
0.9998
0.9999
1.0000

0.0037
0.0244
0.0824
0.1906
0.3422
0.5119
0.6703
0.7970
0.8857

0.9409
0.9718
0.9875
0.9949
0.9980
0.9993
0.9998
0.9999
1.0000

0.0033
0.0224
0.0768
0.1800
0.3272
0.4950
0.6544
0.7841
0.8766
0.9352
0.9686
0.9859
0.9941
0.9977
0.9991
.09997
0.9999
1.0000

0.0030

0.0206
0.0715
0.1700
0.3127
0.4783
0.6384
0.7710
0.8672
0.9292
0.9651
0.9841
0.9932
0.9973
0.9990
0.9996
0.9999
1.0000

0.0027
0.0189
0.0666
0.1604
0.2987
0.4619
0.6224
0.7576
0.8574
0.9228
0.9614
0.9821

0.9922
0.9969
0.9988
0.9996
0.9999
1.0000

xλ

6.00

6.10

6.20

6.30

6.40

6.50

6.60

6.70

6.80

6.90

0

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21

0.0009
0.0073
0.0296
0.0818
0.1730
0.3007
0.4497
0.5987

0.7291
0.8305
0.9015
0.9467
0.9730
0.9872
0.9943
0.9976
0.9990
0.9996
0.9999
1.0000
.....
.....

0.0008
0.0067
0.0275
0.0767
0.1641
0.2881
0.4349
0.5838
0.7160
0.8202
0.8942
0.9420
0.9703
0.9857
0.9935

0.9972
0.9989
0.9996
0.9998
0.9999
1.0000
.....

0.0007
0.0061
0.0255
0.0719
0.1555
0.2759
0.4204
0.5689
0.7027
0.8096
0.8867
0.9371
0.9673
0.9841
0.9927
0.9969
0.9987
0.9995
0.9998
0.9999
1.0000
.....

0.0007
0.0056
0.0236
0.0674
0.1473
0.2640
0.4060
0.5541
0.6892
0.7988
0.8788
0.9319
0.9642
0.9824
0.9918
0.9964
0.9985
0.9994
0.9998
0.9999
1.0000
.....

00006
0.0051
0.0219
0.0632
0.1395
0.2526

0.3920
0.5393
0.6757
0.7877
0.8707
0.9265
0.9609
0.9805
0.9908
0.9959
0.9983
0.9993
0.9997
0.9999
1.0000
.....

0.0006
0.0047
0.0203
0.0591
0.1321
0.2414
0.3782
0.5246
0.6620
0.7764
0.8622
0.9208
0.9573

0.9784
0.9897
0.9954
0.9980
0.9992
0.9997
0.9999
1.0000
.....

0.0005
0.0043
0.0188
0.0554
0.1249
0.2307
0.3646
0.5100
0.6482
0.7649
0.8535
0.9148
0.9536
0.9762
0.9886
0.9948
09978
0.9991
0.9996
0.9999

1.0000
.....

0.0005
0.0039
0.0174
0.0518
0.1181
0.2203
0.3514
0.4956
0.6343
0.7531
0.8445
0.9085
0.9496
0.9739
0.9873
0.9941
0.9974
0.9989
0.9996
0.9998
0.9999
1.0000

0.0004
0.0036
0.0161
0.0485

0.1117
0.2103
0.3384
0.4812
0.6204
0.7411
0.8352
0.9020
0.9454
0.9714
0.9859
0.9934
0.9971
0.9988
0.9995
0.9998
0.9999
1.0000

0.0004
0.0033
0.0149
0.0453
0.1055
0.2006
0.3257
0.4670
0.6065
0.7290
0.8257

0.8952
0.9409
0.9687
0.9844
0.9926
09967
0.9986
0.9994
0.9998
0.9999
1.0000

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17

18
19

0.0025
0.0174
0.0620
0.1512
0.2851
0.4457
0.6063
0.7440
0.8472
0.9161
0.9574
0.9799
0.9912
0.9964
0.9986
0.9995
0.9998
0.9999
1..0000
.....

0.0022
0.0159
0.0577
0.1425
0.2719
0.4298

0.5902
0.7301
0.8367
0.9090
0.9531
0.9776
0.9900
0.9958
0.9984
0.9994
0.9998
0.9999
1.0000
.....

0.0020
0.0146
0.0536
0.1342
0.2592
0.4141
0.5742
0.7160
0.8259
0.9016
0.9486
0.9750
0.9887
0.9952
0.9981

0.9993
0.9997
0.9999
1.0000
.....

0.0018
0.0134
0.0498
0.1264
0.2469
0.3988
0.5582
0.7017
0.8148
0.8939
0.9437
0.9723
0.9873
0.9945
0.9978
0.9992
0.9997
0.9999
1.0000
.....

0.0017
0.0123
0.0463

0.1189
0.2351
0.3837
0.5423
0.6873
0.8033
0.8858
0.9386
0.9693
0.9857
0.9937
0.9974
0.9990
0.9996
0.9999
1.0000
.....

0.0015
0.0113
0.0430
0.1118
0.2237
0.3690
0.5265
0.6728
0.7916
0.8774
0.9332
0.9661

0.9840
0.9929
0.9970
0.9988
0.9996
0.9998
0.9999
1.0000

0.0014
0.0103
0.0400
0.1052
0.2127
0.3547
0.5108
0.6581
0.7796
0.8686
0.9274
0.9627
0.9821
0.9920
0.9966
0.9986
0.9995
0.9998
0.9999
1.0000

0.0012
0.0095
0.0371
0.0988
0.2022
0.3406
0.4953
0.6433
0.7673
0.8596
0.9214
0.9591
0.9801
0.9909
0.9961
0.9984
0.9994
0.9998
0.9999
1.0000

0.0011
0.0087
0.0344
0.0928
0.1920
0.3270
0.4799
0.6285
0.7548

0.8502
0.9151
0.9552
0.9779
0.9898
0.9956
0.9982
0.9993
0.9997
0.9999
1.0000

0.0010
0.0080
0.0320
0.0871
0.1823
0.3137
0.4647
0.6136
0.7420
0.8405
0.9084
0.9510
0.9755
0.9885
0.9950
0.9979
0.9992
0.9997

0.9999
1.0000

x λ

8.00

8.10

8.20

8.30

8.40

8.50

8.60

8.70

8.80

8.90

0
1
2
3
4

5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23

0.0003
0.0030
0.0138
0.0424
0.0996
0.1912
0.3134
0.4530
0.5925
0.7166

0.8159
0.8881
0.9362
0.9658
0.9827
0.9918
0.9963
0.9984
0.9993
0.9997
0.9999
1.0000
.....
.....

0.0003
0.0028
0.0127
0.0396
0.0940
0.1822
0.3013
0.4391
0.5786
0.7041
0.8058
0.8807
0.9313
0.9628
0.9810

0.9908
0.9958
0.9982
0.9992
0.9997
0.9999
1.0000
.....
.....

0.0003
0.0025
0.0118
0.0370
0.0887
0.1736
0.2896
0.4254
0.5647
0.6915
0.7955
0.8731
0.9261
0.9595
0.9791
0.9898
0.9953
0.9979
0.9991
0.9997

0.9999
1.0000
.....
.....

0.0002
0.0023
0.0109
0.0346
0.0837
0.1653
0.2781
0.4119
0.5507
0.6788
0.7850
0.8652
0.9207
0.9561
0.9771
0.9887
0.9947
0.9977
0.9990
0.9996
0.9998
0.9999
1.0000
.....

0.0002
0.0021
0.0100
0.0323
0.0789
0.1573
0.2670
0.3987
0.5369
0.6659
0.7743
0.8571
0.9150
0.9524
0.9749
0.9875
0.9941
0.9973
0.9989
0.9995
0.9998
0.9999
1.0000
.....

0.0002
0.0019
0.0093
0.0301
0.0744

0.1496
0.2562
0.3856
0.5231
0.6530
0.7634
0.8487
0.9091
0.9486
0.9726
0.9862
0.9934
0.9970
0.9987
0.9995
0.9998
0.9999
1.0000
.....

0.0002
0.0018
0.0086
0.0281
0.0701
0.1422
0.2457
0.3728
0.5094
0.6400

0.7522
0.8400
0.9029
0.9445
0.9701
0.9848
0.9926
0.9966
0.9985
0.9994
0.9998
0.9999
1.0000
.....

0.0002
0.0016
0.0079
0.0262
0.0660
0.1352
0.2355
0.3602
0.4958
0.6269
0.7409
0.8311
0.8965
0.9403
0.9675

0.9832
0.9918
0.9962
0.9983
0.9993
0.9997
0.9999
1.0000
.....

0.0002
0.0015
0.0073
0.0244
0.0621
0.1284
0.2256
0.3478
0.4823
0.6137
0.7294
0.8220
0.8898
0.9358
0.9647
0.9816
0.9909
0.9957
0.9981
0.9992

0.9997
0.9999
1.0000
.....

0.0001
0.0014
0.0068
0.0228
0.0584
0.1219
0.2160
0.3357
0.4689
0.6006
0.7178
0.8126
0.8829
0.9311
0.9617
0.9798
0.9899
0.9952
0.9978
0.9991
0.9996
0.9998
0.9999
1.0000

CUMULATIVE POISSON PROBABILITY

5.10

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17

19

x λ 5.00


9.00

9.10

9.20

9.30

9.40

9.50

9.60

9.70

9.80

9.90

0
1
2
3
4
5
6
7
8
9
10
11

12
13
14
15
16
17
18
19
20
21
22
23
24

0.0001
0.0012
0.0062
0.0212
0.0550
0.1157
0.2068
0.3239
0.4557
0.5874
0.7060
0.8030
0.8758
0.9261
0.9585
0.9780

0.9889
0.9947
0.9976
0.9989
0.9996
0.9998
0.9999
1.0000
.....

0.0001
0.0011
0.0058
0.0198
0.0517
0.1098
0.1978
0.3123
0.4426
0.5742
0.6941
0.7932
0.8684
0.9210
0.9552
0.9760
0.9878
0.9941
0.9973
0.9988

0.9995
0.9998
0.9999
1.0000
.....

0.0001
0.0010
0.0053
0.0184
0.0486
0.1041
0.1892
0.3010
0.4296
0.5611
0.6820
0.7832
0.8607
0.9156
0.9517
0.9738
0.9865
0.9934
0.9969
0.9986
0.9994
0.9998
0.9999
1.0000

.....

0.0001
0.0009
0.0049
0.0172
0.0456
0.0986
0.1808
0.2900
0.4168
0.5479
0.6699
0.7730
0.8529
0.9100
0.9480
0.9715
0.9852
0.9927
0.9966
0.9985
0.9993
0.9997
0.9999
1.0000
.....

0.0001
0.0009

0.0045
0.0160
0.0429
0.0935
0.1727
0.2792
0.4042
0.5349
0.6576
0.7626
0.8448
0.9042
0.9441
0.9691
0.9838
0.9919
0.9962
0.9983
0.9992
0.9997
0.9999
1.0000
.....

0.0001
0.0008
0.0042
0.0149
0.0403
0.0885

0.1649
0.2687
0.3918
0.5218
0.6453
0.7520
0.8364
0.8981
0.9400
0.9665
0.9823
0.9911
0.9957
0.9980
0.9991
0.9996
0.9999
0.9999
1.0000

0.0001
0.0007
0.0038
0.0138
0.0378
0.0838
0.1574
0.2584
0.3796
0.5089

0.6329
0.7412
0.8279
0.8919
0.9357
0.9638
0.9806
0.9902
0.9952
0.9978
0.9990
0.9996
0.9998
0.9999
1.0000

0.0001
0.0007
0.0035
0.0129
0.0355
0.0793
0.1502
0.2485
0.3676
0.4960
0.6205
0.7303
0.8191
0.8853

0.9312
0.9609
0.9789
0.9892
0.9947
0.9975
0.9989
0.9995
0.9998
0.9999
1.0000

0.0001
0.0006
0.0033
0.0120
0.0333
0.0750
0.1433
0.2388
0.3558
0.4832
0.6080
0.7193
0.8101
0.8786
0.9265
0.9579
0.9770
0.9881

0.9941
0.9972
0.9987
0.9995
0.9998
0.9999
1.0000

0.0001
0.0005
0.0030
0.0111
0.0312
0.0710
0.1366
0.2294
0.3442
0.4705
0.5955
0.7081
0.8009
0.8716
0.9216
0.9546
0.9751
0.9870
0.9935
0.9969
0.9986
0.9994

0.9997
0.9999
1.0000

x λ

10.00

10.10

10.20

10.30

10.40

10.50

10.60

10.70

10.80

10.90

0
1
2
3

4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26

0.0000
0.0005
0.0028
0.0103
0.0293
0.0671

0.1301
0.2202
0.3328
0.4579
0.5830
0.6968
0.7916
0.8645
0.9165
0.9513
0.9730
0.9857
0.9928
0.9965
0.9984
0.9993
0.9997
0.9999
1.0000
.....
.....

0.0000
0.0005
0.0026
0.0096
0.0274
0.0634
0.1240
0.2113

0.3217
0.4455
0.5705
0.6853
0.7820
0.8571
0.9112
0.9477
0.9707
0.9844
0.9921
0.9962
0.9982
0.9992
0.9997
0.9999
0.9999
1.0000
.....

0.0000
0.0004
0.0023
0.0089
0.0257
0.0599
0.1180
0.2027
0.3108
0.4332

0.5580
0.6738
0.7722
0.8494
0.9057
0.9440
0.9684
0.9830
0.9913
0.9957
0.9980
0.9991
0.9996
0.9998
0.9999
1.0000
.....

0.0000
0.0004
0.0022
0.0083
0.0241
0.0566
0.1123
0.1944
0.3001
0.4210
0.5456
0.6622

0.7623
0.8416
0.9000
0.9400
0.9658
0.9815
0.9904
0.9953
0.9978
0.9990
0.9996
0.9998
0.9999
1.0000
.....

0.0000
0.0003
0.0020
0.0077
0.0225
0.0534
0.1069
0.1863
0.2896
0.4090
0.5331
0.6505
0.7522
0.8336

0.8940
0.9359
0.9632
0.9799
0.9895
0.9948
0.9975
0.9989
0.9995
0.9998
0.9999
1.0000
.....

0.0000
0.0003
0.0018
0.0071
0.0211
0.0504
0.1016
0.1785
0.2794
0.3971
0.5207
0.6387
0.7420
0.8253
0.8879
0.9317

0.9604
0.9781
0.9885
0.9942
0.9972
0.9987
0.9994
0.9998
0.9999
1.0000
.....

0.0000
0.0003
0.0017
0.0066
0.0197
0.0475
0.0966
0.1710
0.2694
0.3854
0.5084
0.6269
0.7316
0.8169
0.8815
0.9272
0.9574
0.9763

0.9874
0.9936
0.9969
0.9986
0.9994
0.9997
0.9999
1.0000
.....

0.0000
0.0003
0.0016
0.0062
0.0185
0.0448
0.0918
0.1636
0.2597
0.3739
0.4961
0.6150
0.7210
0.8083
0.8750
0.9225
0.9543
0.9744
0.9863
0.9930

0.9966
0.9984
0.9993
0.9997
0.9999
0.9999
1.0000

0.0000
0.0002
0.0014
0.0057
0.0173
0.0423
0.0872
0.1566
0.2502
0.3626
0.4840
0.6031
0.7104
0.7995
0.8682
0.9177
0.9511
0.9723
0.9850
0.9923
0.9962
0.9982

0.9992
0.9996
0.9998
0.9999
1.0000

0.0000
0.0002
0.0013
0.0053
0.0162
0.0398
0.0828
0.1498
0.2410
0.3515
0.4719
0.5912
0.6996
0.7905
0.8612
0.9126
0.9477
0.9701
0.9837
0.9915
0.9958
0.9980
0.9991
0.9996

0.9998
0.9999
1.0000

CUMULATIVE POISSON PROBABILITY

20

x λ


Critical values for the product moment correlation coefficient, r

n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

17
18
19
20
21
22
23
24
25
26
27
28
29
30

21/2%
5%

1%
2%

–
–
0.9877
0.9000
0.8054
0.7293
0.6694
0.6215
0.5822

0.5494
0.5214
0.4973
0.4762
0.4575
0.4409
0.4259
0.4124
0.4000
0.3887
0.3783
0.3687
0.3598
0.3515
0.3438
0.3365
0.3297
0.3233
0.3172
0.3115
0.3061

–
–
0.9969
0.9500
0.8783
0.8114
0.7545
0.7067

0.6664
0.6319
0.6021
0.5760
0.5529
0.5324
0.5140
0.4973
0.4821
0.4683
0.4555
0.4438
0.4329
0.4227
0.4132
0.4044
0.3961
0.3882
0.3809
0.3739
0.3673
0.3610

–
–
0.9995
0.9800
0.9343
0.8822
0.8329

0.7887
0.7498
0.7155
0.6851
0.6581
0.6339
0.6120
0.5923
0.5742
0.5577
0.5425
0.5285
0.5155
0.5034
0.4921
0.4815
0.4716
0.4622
0.4534
0.4451
0.4372
0.4297
0.4226

1

/2%
1%

–

–
0.9999
0.9900
0.9587
0.9172
0.8745
0.8343
0.7977
0.7646
0.7348
0.7079
0.6835
0.6614
0.6411
0.6226
0.6055
0.5897
0.5751
0.5614
0.5487
0.5368
0.5256
0.5151
0.5052
0.4958
0.4869
0.4785
0.4705
0.4629

1-Tail Test
2-Tail Test

n
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56

57
58
59
60

5%
10%

21/2%
5%

1%
2%

0.3009
0.2960
0.2913
0.2869
0.2826
0.2785
0.2746
0.2709
0.2673
0.2638
0.2605
0.2573
0.2542
0.2512
0.2483
0.2455

0.2429
0.2403
0.2377
0.2353
0.2329
0.2306
0.2284
0.2262
0.2241
0.2221
0.2201
0.2181
0.2162
0.2144

0.3550
0.3494
0.3440
0.3388
0.3388
0.3291
0.3246
0.3202
0.3160
0.3120
0.3081
0.3044
0.3008
0.2973
0.2940

0.2907
0.2876
0.2845
0.2816
0.2787
0.2759
0.2732
0.2706
0.2681
0.2656
0.2632
0.2609
0.2586
0.2564
0.2542

0.4158
0.4093
0.4032
0.3972
0.3916
0.3862
0.3810
0.3760
0.3712
0.3665
0.3621
0.3578
0.3536
0.3496

0.3457
0.3420
0.3384
0.3348
0.3314
0.3281
0.3249
0.3218
0.3188
0.3158
0.3129
0.3102
0.3074
0.3048
0.3022
0.2997

1

5%
10%

/2%
1%

0.4556
0.4487
0.4421
0.4357
0.4926

0.4238
0.4182
0.4128
0.4076
0.4026
0.3978
0.3932
0.3887
0.3843
0.3801
0.3761
0.3721
0.3683
0.3646
0.3610
0.3575
0.3542
0.3509
0.3477
0.3445
0.3415
0.3385
0.3357
0.3328
0.3301

n
1
2
3

4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30

21/2%
5%

1%
2%

–
–
–
1.0000
0.9000
0.8286
0.7143
0.6429
0.6000
0.5636
0.5364
0.5035
0.4835
0.4637
0.4464
0.4294
0.4142
0.4014
0.3912
0.3805
0.3701
0.3608
0.3528
0.3443
0.3369
0.3306

0.3242
0.3180
0.3118
0.3063

–
–
–
–
1.0000
0.8857
0.7857
0.7381
0.7000
0.6485
0.6182
0.5874
0.5604
0.5385
0.5214
0.5029
0.4877
0.4716
0.4596
0.4466
0.4364
0.4252
0.4160
0.4070
0.3977

0.3901
0.3828
0.3755
0.3685
0.3624

–
–
–
–
1.0000
0.9429
0.8929
0.8333
0.7833
0.7455
0.7091
0.6783
0.6484
0.6264
0.6036
0.5824
0.5662
0.5501
0.5351
0.5218
0.5091
0.4975
0.4862
0.4757

0.4662
0.4571
0.4487
0.4401
0.4325
0.4251

1

/2%
1%

–
–
–
–
–
1.0000
0.9286
0.8810
0.8333
0.7939
0.7545
0.7273
0.7033
0.6791
0.6536
0.6353
0.6176
0.5996

0.5842
0.5699
0.5558
0.5438
0.5316
0.5209
0.5108
0.5009
0.4915
0.4828
0.4749
0.4670

1-Tail Test
2-Tail Test

n
31
32
33
34
35
36
37
38
39
40
41
42
43

44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60

5%
10%

21/2%
5%

1%
2%

0.3012
0.2962
0.2914

0.2871
0.2829
0.2788
0.2748
0.2710
0.2674
0.2640
0.2606
0.2574
0.2543
0.2513
0.2484
0.2456
0.2429
0.2403
0.2378
0.2353
0.2329
0.2307
0.2284
0.2262
0.2242
0.2221
0.2201
0.2181
0.2162
0.2144

0.3560
0.3504

0.3449
0.3396
0.3347
0.3300
0.3253
0.3209
0.3168
0.3128
0.3087
0.3051
0.3014
0.2978
0.2945
0.2913
0.2880
0.2850
0.2820
0.2791
0.2764
0.2736
0.2710
0.2685
0.2659
0.2636
0.2612
0.2589
0.2567
0.2545

0.4185

0.4117
0.4054
0.3995
0.3936
0.3882
0.3829
0.3778
0.3729
0.3681
0.3636
0.3594
0.3550
0.3511
0.3470
0.3433
0.3396
0.3361
0.3326
0.3293
0.3260
0.3228
0.3198
0.3168
0.3139
0.3111
0.3083
0.3057
0.3030
0.3005

1

/2%
1%

0.4593
0.4523
0.4455
0.4390
0.4328
0.4268
0.4211
0.4155
0.4103
0.4051
0.4002
0.3955
0.3908
0.3865
0.3882
0.3781
0.3741
0.3702
0.3664
0.3628
0.3592
0.3558
0.3524
0.3492
0.3460

0.3429
0.3400
0.3370
0.3342
0.3314

CRITICAL VALUES FOR CORRELATION COEFFICIENTS

21

5%
10%

Critical values for Spearman’s rank correlation coefficient, rs


The Normal distribution: values of Φ(z) = p

The Inverse Normal function: values of Φ–1(p) = z

The table gives the probability, p, of a
random variable distributed as N(0, 1)
being less than z.

N(0, 1)

p
z

(add)

.01

.02

.03

.04

.05

.06

.07

.08

.09

1 2

.5000
.5398
.5793
.6179
.6554

5040
5438
5832
6217

6591

5080
5478
5871
6255
6628

5120
5517
5910
6293
6664

5160
5557
5948
6331
6700

5199
5596
5987
6368
6736

5239
5636
6026
6406

6772

5279
5675
6064
6443
6808

5319
5714
6103
6480
6844

5359
5753
6141
6517
6879

4
4
4
4
4

8
8
8
8

7

3

4

5

6

7

8

9

12
12
12
11
11

16
16
15
15
14

20
20

19
19
18

24
24
23
23
22

28
28
27
26
25

32
32
31
30
29

36
35
35
34
32

0.5
0.6

0.7
0.8
0.9

.6915
.7257
.7580
.7881
.8159

6950
7291
7611
7910
8186

6985
7324
7642
7939
8212

7019
7357
7673
7967
8238

7054
7389

7704
7995
8264

7088
7422
7734
8023
8289

7123
7454
7764
8051
8315

7157
7486
7794
8078
8340

7190
7517
7823
8106
8365

7224
7549

7852
8133
8389

3
3
3
3
3

7 10 14 17 21 24
6 10 13 16 19 23
6 9 12 15 18 21
6 8 11 14 17 19
5 8 10 13 15 18

27
26
24
22
20

31
29
27
25
23

1.0
1.1

1.2
1.3
1.4

.8413
.8643
.8849
.9032
.9192

8438
8665
8869
9049
9207

8461
8686
8888
9066
9222

8485
8708
8907
9082
9236

8508
8729

8925
9099
9251

8531
8749
8944
9115
9265

8554
8770
8962
9131
9279

8577
8790
8980
9147
9292

8599
8810
8997
9162
9306

8621
8830

9015
9177
9319

2
2
2
2
1

5
4
4
3
3

7
6
6
5
4

9 12 14 16 18 21
8 10 12 14 16 19
7 9 11 13 15 16
6 8 10 11 13 14
6 7 8 10 11 13

1.5
1.6

1.7
1.8
1.9

.9332
.9452
.9554
.9641
.9713

9345
9463
9564
9649
9719

9357
9474
9573
9656
9726

9370
9484
9582
9664
9732

9382
9495

9591
9671
9738

9394
9505
9599
9678
9744

9406
9515
9608
9686
9750

9418
9525
9616
9693
9756

9429
9535
9625
9699
9761

9441
9545

9633
9706
9767

1
1
1
1
1

2
2
2
1
1

4
3
3
2
2

5
4
3
3
2

6
5

4
4
3

7
6
5
4
4

8 10 11
7 8 9
6 7 8
5 6 6
4 5 5

2.0
2.1
2.2
2.3
2.4

.9772
.9821
.9861
.9893
.9918

9778
9826

9864
9896
9920

9783
9830
9868
9898
9922

9788
9834
9871
9901
9925

9793
9838
9875
9904
9927

9798
9842
9878
9906
9929

9803
9846

9881
9909
9931

9808
9850
9884
9911
9932

9812
9854
9887
9913
9934

9817
9857
9890
9916
9936

0
0
0
0
0

1
1

1
1
0

1
1
1
1
1

2
2
1
1
1

2
2
2
1
1

3
2
2
2
1

3
3

2
2
1

2.5
2.6
2.7
2.8
2.9

.9938
.9953
.9965
.9974
.9981

9940
9955
9966
9975
9982

9941
9956
9967
9976
9982

9943
9957

9968
9977
9983

9945
9959
9969
9977
9984

9946
9960
9970
9978
9984

9948
9961
9971
9979
9985

9949
9962
9972
9979
9985

9951
9963

9973
9980
9986

9952
9964
9974
9981
9986

3.0
3.1
3.2
3.3
3.4

.9987
.9990
.9993
.9995
.9997

9987
9991
9993
9995
9997

9987
9991

9994
9996
9997

9988
9991
9994
9996
9997

9988
9992
9994
9996
9997

9989
9992
9994
9996
9997

9989
9992
9994
9996
9997

9989
9992

9995
9996
9997

9990
9993
9995
9996
9997

9990
9993
9995
9997
9998

differences
untrustworthy

4
3
3
2
2

4
4
3
2
2

.000
.0000
.0251
.0502
.0753
.1004
.1257
.1510
.1764
.2019
.2275
.2533
.2793
.3055
.3319
.3585
.3853
.4125
.4399
.4677
.4959
.5244
.5534
.5828
.6128
.6433
.6745
.7063
.7388

.7722
.8064
.8416
.8779
.9154
.9542
.9945
1.036
1.080
1.126
1.175
1.227
1.282
1.341
1.405
1.476
1.555
1.645
1.751
1.881
2.054
2.326

.001
.0025
.0276
.0527
.0778
.1030
.1282

.1535
.1789
.2045
.2301
.2559
.2819
.3081
.3345
.3611
.3880
.4152
.4427
.4705
.4987
.5273
.5563
.5858
.6158
.6464
.6776
.7095
.7421
.7756
.8099
.8452
.8816
.9192
.9581
.9986
1.041

1.085
1.131
1.180
1.232
1.287
1.347
1.412
1.483
1.563
1.655
1.762
1.896
2.075
2.366

.002
.0050
.0301
.0552
.0803
.1055
.1307
.1560
.1815
.2070
.2327
.2585
.2845
.3107
.3372

.3638
.3907
.4179
.4454
.4733
.5015
.5302
.5592
.5888
.6189
.6495
.6808
.7128
.7454
.7790
.8134
.8488
.8853
.9230
.9621
1.003
1.045
1.089
1.136
1.185
1.237
1.293
1.353
1.419
1.491

1.572
1.665
1.774
1.911
2.097
2.409

.003
.0075
.0326
.0577
.0828
.1080
.1332
.1586
.1840
.2096
.2353
.2611
.2871
.3134
.3398
.3665
.3934
.4207
.4482
.4761
.5044
.5330
.5622

.5918
.6219
.6526
.6840
.7160
.7488
.7824
.8169
.8524
.8890
.9269
.9661
1.007
1.049
1.094
1.141
1.190
1.243
1.299
1.360
1.426
1.499
1.581
1.675
1.787
1.927
2.120
2.457

.004

.0100
.0351
.0602
.0853
.1105
.1358
.1611
.1866
.2121
.2378
.2637
.2898
.3160
.3425
.3692
.3961
.4234
.4510
.4789
.5072
.5359
.5651
.5948
.6250
.6557
.6871
.7192
.7521
.7858
.8204

.8560
.8927
.9307
.9701
1.011
1.054
1.099
1.146
1.195
1.248
1.305
1.366
1.433
1.506
l.589
1.685
1.799
1.943
2.144
2.512

.005
.0125
.0376
.0627
.0878
.1130
.1383
.1637
.1891

.2147
.2404
.2663
.2924
.3186
.3451
.3719
.3989
.4261
.4538
.4817
.5101
.5388
.5681
.5978
.6280
.6588
.6903
.7225
.7554
.7892
.8239
.8596
.8965
.9346
.9741
1.015
1.058
1.103
1.150

1.200
l.254
1.311
1.372
1.440
1.514
1.598
1.695
1.812
1.960
2.170
2.576

.006
.0150
.0401
.0652
.0904
.1156
.1408
.1662
.1917
.2173
.2430
.2689
.2950
.3213
.3478
.3745
.4016

.4289
.4565
.4845
.5129
.5417
.5710
.6008
.6311
.6620
.6935
.7257
.7588
.7926
.8274
.8633
.9002
.9385
.9782
1.019
1.063
1.108
1.155
1.206
1.259
1.317
1.379
1.447
1.522
1.607
1.706

1.825
1.977
2.197
2.652

.007
.0175
.0426
.0677
.0929
.1181
.1434
.1687
.1942
.2198
.2456
.2715
.2976
.3239
.3505
.3772
.4043
.4316
.4593
.4874
.5158
.5446
.5740
.6038
.6341

.6651
.6967
.7290
.7621
.7961
.8310
.8669
.9040
.9424
.9822
1.024
1.067
1.112
1.160
1.211
1.265
1.323
1.385
1.454
1.530
1.616
1.717
1.838
1.995
2.226
2.748

.008
.0201
.0451

.0702
.0954
.1206
.1459
.1713
.1968
.2224
.2482
.2741
.3002
.3266
.3531
.3799
.4070
.4344
.4621
.4902
.5187
.5476
.5769
.6068
.6372
.6682
.6999
.7323
.7655
.7995
.8345
.8705
.9078

.9463
.9863
1.028
1.071
1.117
1.165
1.216
1.270
1.329
1.392
1.461
1.538
1.626
1.728
1.852
2.014
2.257
2.878

.009
.0226
.0476
.0728
.0979
.1231
.1484
.1738
.1993
.2250
.2508

.2767
.3029
.3292
.3558
.3826
.4097
.4372
.4649
.4930
.5215
.5505
.5799
.6098
.6403
.6713
.7031
.7356
.7688
.8030
.838l
.8742
.9116
.9502
.9904
1.032
1.076
1.122
1.170
1.221
1.276

1.335
1.398
1.468
1.546
1.635
1.739
1.866
2.034
2.290
3.090

THE NORMAL DISTRIBUTION AND ITS INVERSE

z

22

.00

0.0
0.1
0.2
0.3
0.4

p
.50
.51
.52
.53

.54
.55
.56
.57
.58
.59
.60
.61
.62
.63
.64
.65
.66
.67
.68
.69
.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
.80
.81
.82
.83

.84
.85
.86
.87
.88
.89
.90
.91
.92
.93
.94
.95
.96
.97
.98
.99


Percentage points of the χ2 (chi-squared) distribution

p%

p%
χ2

χ2
p%

97.5

95

90

10

5.0

2.5

1.0

0.5

.0001
.0201
0.115
0.297
0.554
0.872
1.239
1.646
2.088
2.558
3.053
3.571
4.107
4.660
5.229
5.812

6.408
7.015
7.633
8.260
8.897
9.542
10.20
10.86
11.52
12.20
12.88
13.56
14.26
14.95
18.51
22.16
29.71
70.06

.0010
.0506
0.216
0.484
0.831
1.237
1.690
2.180
2.700
3.247
3.816

4.404
5.009
5.629
6.262
6.908
7.564
8.231
8.907
9.591
10.28
10.98
11.69
12.40
13.12
13.84
14.57
15.31
16.05
16.79
20.57
24.43
32.36
74.22

.0039
0.103
0.352
0.711
1.145
1.635

2.167
2.733
3.325
3.940
4.575
5.226
5.892
6.571
7.261
7.962
8.672
9.390
10.12
10.85
11.59
12.34
13.09
13.85
14.61
15.38
16.15
16.93
17.71
18.49
22.47
26.51
34.76
77.93

.0158

0.211
0.584
1.064
1.610
2.204
2.833
3.490
4.168
4.865
5.578
6.304
7.042
7.790
8.547
9.312
10.09
10.86
11.65
12.44
13.24
14.04
14.85
15.66
16.47
17.29
18.11
18.94
19.77
20.60
24.80

29.05
37.69
82.36

2.706
4.605
6.251
7.779
9.236
10.64
12.02
13.36
14.68
15.99
17.28
18.55
19.81
21.06
22.31
23.54
24.77
25.99
27.20
28.41
29.62
30.81
32.01
33.20
34.38
35.56

36.74
37.92
39.09
40.26
46.06
51.81
63.17
118.5

3.841
5.991
7.815
9.488
11.07
12.59
14.07
15.51
16.92
18.31
19.68
21.03
22.36
23.68
25.00
26.30
27.59
28.87
30.14
31.41
32.67

33.92
35.17
36.42
37.65
38.89
40.11
41.34
42.56
43.77
49.80
55.76
67.50
124.3

5.024
7.378
9.348
11.14
12.83
14.45
16.01
17.53
19.02
20.48
21.92
23.34
24.74
26.12
27.49
28.85

30.19
31.53
32.85
34.17
35.48
36.78
38.08
39.36
40.65
41.92
43.19
44.46
45.72
46.98
53.20
59.34
71.42
129.6

6.635
9.210
11.34
13.28
15.09
16.81
18.48
20.09
21.67
23.21
24.72

26.22
27.69
29.14
30.58
32.00
33.41
34.81
36.19
37.57
38.93
40.29
41.64
42.98
44.31
45.64
46.96
48.28
49.59
50.89
57.34
63.69
76.15
135.8

7.879
10.60
12.84
14.86
16.75
18.55

20.28
21.95
23.59
25.19
26.76
28.30
29.82
31.32
32.80
34.27
35.72
37.16
38.58
40.00
41.40
42.80
44.18
45.56
46.93
48.29
49.64
50.99
52.34
53.67
60.27
66.77
79.49
140.2

1/2

p%

1/2

p%

t
p%
v=1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
20
30
50
100
∞

10

5

2

1

6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.725
1.697
1.676
1.660
1.645

12.71

4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.086
2.042
2.009
1.984
1.960

31.82
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764

2.718
2.681
2.650
2.624
2.602
2.528
2.457
2.403
2.364
2.326

63.66
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.845
2.750
2.678
2.626

2.576

2

99

χ
PERCENTAGE POINTS OF χ AND t – DISTRIBUTIONS

23

v=1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19

20
21
22
23
24
25
26
27
28
29
30
35
40
50
100

Percentage points of the t-distribution

= Percentage points of the
Normal distribution N(0, 1)


21/2% points of the F-distribution

5% points of the F-distribution

12

24

∞

v1

1

2

3

4

5

6

7

8

10

12

24

∞

238.9
19.4

8.85
6.04

241.9
19.4
8.79
5.96

243.9
19.4
8.74
5.91

249.0
19.5
8.64
5.77

254.3
19.5
8.53
5.63

1
2
3
4

648
38.5

17.4
12.22

800
39.0
16.0
10.65

864
39.2
15.4
9.98

900
39.2
15.1
9.60

922
39.3
14.9
9.36

937
39.3
14.7
9.20

948
39.4

14.6
9.07

957
39.4
14.5
8.98

969
39.4
14.4
8.84

977
39.4
14.3
8.75

997
39.5
14.1
8.51

1018
39.5
13.9
8.26

4.88
4.21

3.79
3.50
3.29

4.82
4.15
3.73
3.44
3.23

4.74
4.06
3.64
3.35
3.14

4.68
4.00
3.57
3.28
3.07

4.53
3.84
3.41
3.12
2.90

4.36
3.67

3.23
2.93
2.71

5
6
7
8
9

10.01
8.81
8.07
7.57
7.21

8.43
7.26
6.54
6.06
5.71

7.76
6.60
5.89
5.42
5.08

7.39
6.23

5.52
5.05
4.72

7.15
5.99
5.29
4.82
4.48

6.98
5.82
5.12
4.65
4.32

6.85
5.70
4.99
4.53
4.20

6.76
5.60
4.90
4.43
4.10

6.62
5.46

4.76
4.30
3.96

6.52
5.37
4.67
4.20
3.87

6.28
5.12
4.42
3.95
3.61

6.02
4.85
4.14
3.67
3.33

3.22
3.09
3.00
2.92
2.85

3.14
3.01

2.91
2.83
2.76

3.07
2.95
2.85
2.77
2.70

2.98
2.85
2.75
2.67
2.60

2.91
2.79
2.69
2.60
2.53

2.74
2.61
2.51
2.42
2.35

2.54
2.40

2.30
2.21
2.13

10
11
12
13
14

6.94
6.72
6.55
6.41
6.30

5.46
5.26
5.10
4.97
4.86

4.83
4.63
4.47
4.35
4.24

4.47
4.28

4.12
4.00
3.89

4.24
4.04
3.89
3.77
3.66

4.07
3.88
3.73
3.60
3.50

3.95
3.76
3.61
3.48
3.38

3.85
3.66
3.51
3.39
3.29

3.72
3.53

3.37
3.25
3.15

3.62
3.43
3.28
3.15
3.05

3.37
3.17
3.02
2.89
2.79

3.08
2.88
2.72
2.60
2.49

2.90
2.85
2.81
2.77
2.74

2.79
2.74

2.70
2.66
2.63

2.71
2.66
2.61
2.58
2.54

2.64
2.59
2.55
2.51
2.48

2.54
2.49
2.45
2.41
2.38

2.48
2.42
2.38
2.34
2.31

2.29
2.24

2.19
2.15
2.11

2.07
2.01
1.96
1.92
1.88

15
16
17
18
19

6.20
6.12
6.04
5.98
5.92

4.76
4.69
4.62
4.56
4.51

4.15
4.08

4.01
3.95
3.90

3.80
3.73
3.66
3.61
3.56

3.58
3.50
3.44
3.38
3.33

3.41
3.34
3.28
3.22
3.17

3.29
3.22
3.16
3.10
3.05

3.20
3.12

3.06
3.01
2.96

3.06
2.99
2.92
2.87
2.82

2.96
2.89
2.82
2.77
2.72

2.70
2.63
2.56
2.50
2.45

2.40
2.32
2.25
2.19
2.13

2.87
2.84

2.82
2.80
2.78

2.71
2.68
2.66
2.64
2.62

2.60
2.57
2.55
2.53
2.51

2.51
2.49
2.46
2.44
2.42

2.45
2.42
2.40
2.37
2.36

2.35
2.32

2.30
2.27
2.25

2.28
2.25
2.23
2.20
2.18

2.08
2.05
2.03
2.00
1.98

1.84
1.81
1.78
1.76
1.73

20
21
22
23
24

5.87
5.83

5.79
5.75
5.72

4.46
4.42
4.38
4.35
4.32

3.86
3.82
3.78
3.75
3.72

3.51
3.48
3.44
3.41
3.38

3.29
3.25
3.22
3.18
3.15

3.13
3.09

3.05
3.02
2.99

3.01
2.97
2.93
2.90
2.87

2.91
2.87
2.84
2.81
2.78

2.77
2.73
2.70
2.67
2.64

2.68
2.64
2.60
2.57
2.54

2.41
2.37

2.33
2.30
2.27

2.09
2.04
2.00
1.97
1.94

2.99
2.98
2.96
2.95
2.93

2.76
2.74
2.73
2.71
2.70

2.60
2.59
2.57
2.56
2.55

2.49
2.47

2.46
2.45
2.43

2.40
2.39
2.37
2.36
2.35

2.34
2.32
2.31
2.29
2.28

2.24
2.22
2.20
2.19
2.18

2.16
2.15
2.13
2.12
2.10

1.96
1.95

1.93
1.91
1.90

1.71
1.69
1.67
1.65
1.64

25
26
27
28
29

5.69
5.66
5.63
5.61
5.59

4.29
4.27
4.24
4.22
4.20

3.69
3.67

3.65
3.63
3.61

3.35
3.33
3.31
3.29
3.27

3.13
3.10
3.08
3.06
3.04

2.97
2.94
2.92
2.90
2.88

2.85
2.82
2.80
2.78
2.76

2.75
2.73

2.71
2.69
2.67

2.61
2.59
2.57
2.55
2.53

2.51
2.49
2.47
2.45
2.43

2.24
2.22
2.19
2.17
2.15

1.91
1.88
1.85
1.83
1.81

3.32
3.29

3.28
3.26
3.24

2.92
2.90
2.88
2.87
2.85

2.69
2.67
2.65
2.63
2.62

2.53
2.51
2.49
2.48
2.46

2.42
2.40
2.38
2.36
2.35

2.33
2.31

2.29
2.28
2.26

2.27
2.24
2.23
2.21
2.19

2.16
2.14
2.12
2.11
2.09

2.09
2.07
2.05
2.03
2.02

1.89
1.86
1.84
1.82
1.81

1.62
1.59

1.57
1.55
1.53

30
32
34
36
38

5.57
5.53
5.50
5.47
5.45

4.18
4.15
4.12
4.09
4.07

3.59
3.56
3.53
3.51
3.48

3.25
3.22

3.19
3.17
3.15

3.03
3.00
2.97
2.94
2.92

2.87
2.84
2.81
2.79
2.76

2.75
2.72
2.69
2.66
2.64

2.65
2.62
2.59
2.57
2.55

2.51
2.48

2.45
2.43
2.41

2.41
2.38
2.35
2.33
2.31

2.14
2.10
2.08
2.05
2.03

1.79
1.75
1.72
1.69
1.66

3.23
3.15
3.07
3.00

2.84
2.76
2.68

2.60

2.61
2.53
2.45
2.37

2.45
2.37
2.29
2.21

2.34
2.25
2.18
2.10

2.25
2.17
2.09
2.01

2.18
2.10
2.02
1.94

2.08
1.99
1.91

1.83

2.00
1.92
1.83
1.75

1.79
1.70
1.61
1.52

1.51
1.39
1.25
1.00

40
60
120
∞

5.42
5.29
5.15
5.02

4.05
3.93
3.80

3.69

3.46
3.34
3.23
3.12

3.13
3.01
2.89
2.79

2.90
2.79
2.67
2.57

2.74
2.63
2.52
2.41

2.62
2.51
2.39
2.29

2.53
2.41
2.30

2.19

2.39
2.27
2.16
2.05

2.29
2.17
2.05
1.94

2.01
1.88
1.76
1.64

1.64
1.48
1.31
1.00

2

3

4

5

6

7

8

161.4
18.5
10.13
7.71

199.5
19.0
9.55
6.94

215.7
19.2
9.28
6.59

224.6
19.2
9.12
6.39

230.2
19.3
9.01
6.26

234.0
19.3
8.94
6.16

236.8
19.4
8.89
6.09

5
6
7
8
9

6.61
5.99
5.59
5.32
5.12

5.79
5.14
4.74
4.46
4.26

5.41

4.76
4.35
4.07
3.86

5.19
4.53
4.12
3.84
3.63

5.05
4.39
3.97
3.69
3.48

4.95
4.28
3.87
3.58
3.37

10
11
12
13
14

24

1

4.96
4.84
4.75
4.67
4.60

4.10
3.98
3.89
3.81
3.74

3.71
3.59
3.49
3.41
3.34

3.48
3.36
3.26
3.18
3.11

3.33
3.20
3.11

3.03
2.96

15
16
17
18
19

4.54
4.49
4.45
4.41
4.38

3.68
3.63
3.59
3.55
3.52

3.29
3.24
3.20
3.16
3.13

3.06
3.01
2.96

2.93
2.90

20
21
22
23
24

4.35
4.32
4.30
4.28
4.26

3.49
3.47
3.44
3.42
3.40

3.10
3.07
3.05
3.03
3.01

25
26
27

28
29

4.24
4.23
4.21
4.20
4.18

3.39
3.37
3.35
3.34
3.33

30
32
34
36
38

4.17
4.15
4.13
4.11
4.10

40
60
120

∞

4.08
4.00
3.92
3.84

v2

CRITICAL VALUES FOR F – TEST

10

v1
1
2
3
4

v2


1% points of the F-distribution

0.1% points of the F-distribution

2

3

4

5

6

7

8

10

12

24

∞

v1

1

2

3

4

5

6

7

8

10

12

24

∞

4052
98.5
34.1
21.2

5000
99.0
30.8
18.0

5403
99.2
29.5
16.7

5625

99.2
28.7
16.0

5764
99.3
28.2
15.5

5859
99.3
27.9
15.2

5928
99.4
27.7
15.0

5981
99.4
27.5
14.8

6056
99.4
27.2
14.5

6106

99.4
27.1
14.4

6235
99.5
26.6
13.9

6366
99.5
26.1
13.5

1
2
3
4

4053
998.5.
167.0
74.14

5000
999.0
148.5
61.25

5404

999.2
141.1
56.18

5625
999.2
137.1
53.44

5764
999.3
134.6
51.71

5859
999.3
132.8
50.53

5929
999.4
131.5
49.66

5981
999.4
130.6
49.00

6056

999.4
129.2
48.05

6107
999.4
128.3
47.41

6235
999.5
125.9
45.77

6366
999 5
123.5
44.05

16.26
13.74
12.25
11.26
10.56

13.27
10.92
9.55
8.65
8.02

12.06
9.78
8.45
7.59
6.99

11.39
9.15
7.85
7.01
6.42

10.97
8.75
7.46
6.63
6.06

10.67
8.47
7.19
6.37
5.80

10.46
8.26
6.99
6.18
5.61

10.29
8.10
6.84
6.03
5.47

10.05
7.87
6.62
5.81
5.26

9.89
7.72
6.47
5.67
5.11

9.47
7.31
6.07
5.28
4.73

9.02
6.88
5.65
4.86
4.31

5
6
7
8
9

47.18
35.51
29.25
25.42
22.86

37.12
27.00
21.69
18.49
16.39

33.20
23.70
18.77
15.83
13.90

31.09
21.92
17.20
14.39
12.56

29.75
20.80
16.21
13.48
11.71

28.83
20.03
15.52
12.86
11.13

28.16
19.46
15.02
12.40
10.69

27.65
19.03
14.63
12.05
10.37

26.92
18.41
14.08
11.54
9.87

26.42
17.99
13.71
11.19
9.57

25.14
16.90
12.73
10.30
8.72

23.79
15.75
11.70
9.34
7.81

10
11
12
13
14

10.04
9.65
9.33
9.07
8.86

7.56
7.21
6.93
6.70
6.51

6.55
6.22
5.95
5.74
5.56

5.99
5.67
5.41
5.21
5.04

5.64
5.32
5.06
4.86
4.70

5.39
5.07
4.82
4.62
4.46

5.20
4.89
4.64
4.44
4.28

5.06
4.74
4.50
4.30
4.14

4.85
4.54
4.30
4.10
3.94

4.71
4.40
4.16
3.96
3.80

4.33
4.02
3.78
3.59
3.43

3.91
3.60
3.36
3.17
3.00

10
11
12
13
14

21.04
19.69
18.64
17.82
17.14

14.91
13.81
12.97
12.31
11.78

12.55
11.56
10.80
10.21
9.73

11.28
10.35
9.63
9.07
8.62

10.48
9.58
8.89
8.35
7.92

9.93
9.05
8.38
7.86
7.44

9.52
8.66
8.00
7.49
7.08

9.20
8.35
7.71
7.21
6.80

8.74
7.92
7.29
6.80
6.40

8.44
7.63
7.00
6.52
6.13

7.64
6.85
6.25
5.78
5.41

6.76
6.00
5.42
4.97
4.60

15
16
17
18
19

8.68
8.53
8.40
8.29
8.18

6.36
6.23
6.11
6.01
5.93

5.42
5.29
5.18
5.09
5.01

4.89
4.77
4.67
4.58
4.50

4.56
4.44
4.34
4.25
4.17

4.32
4.20
4.10
4.01
3.94

4.14
4.03
3.93
3.84
3.77

4.00
3.89
3.79
3.71
3.63

3.80
3.69
3.59
3.51
3.43

3.67
3.55
3.46
3.37
3.30

3.29
3.18
3.08
3.00
2.92

2.87
2.75
2.65
2.57
2.49

15
16
17
18
19

16.59
16.12
15.72
15.38
15.08

11.34
10.97
10.66
10.39
10.16

9.34
9.01
8.73
8.49
8.28

8.25
7.94
7.68
7.46
7.27

7.57
7.27
7.02
6.81
6.62

7.09
6.80
6.56
6.35
6.18

6.74
6.46
6.22
6.02
5.85

6.47
6.19
5.96
5.76
5.59

6.08
5.81
5.58
5.39
5.22

5.81
5.55
5.32
5.13
4.97

5.10
4.85
4.63
4.45
4.29

4.31
4.06
3.85
3.67
3.51

20
21
22
23
24

8.10
8.02
7.95
7.88
7.82

5.85
5.78
5.72
5.66
5.61

4.94
4.87
4.82
4.76
4.72

4.43
4.37
4.31
4.26
4.22

4.10
4.04
3.99
3.94
3.90

3.87
3.81
3.76
3.71
3.67

3.70
3.64
3.59
3.54
3.50

3.56
3.51
3.45
3.41
3.36

3.37
3.31
3.26
3.21
3.17

3.23
3.17
3.12
3.07
3.03

2.86
2.80
2.75
2.70
2.66

2.42
2.36
2.31
2.26
2.21

20
21
22
23
24

14.82
14.59
14.38
14.19
14.03

9.95
9.77
9.61
9.47
9.34

8.10
7.94
7.80
7.67
7.55

7.10
6.95
6.81
6.70
6.59

6.46
6.32
6.19
6.08
5.98

6.02
5.88
5.76
5.65
5.55

5.69
5.56
5.44
5.33
5.23

5.44
5.31
5.19
5.09
4.99

5.08
4.95
4.83
4.73
4.64

4.82
4.70
4.58
4.48
4.39

4.15
4.03
3.92
3.82
3.74

3.38
3.26
3.15
3.05
2.97

25
26
27
28
29

7.77
7.72
7.68
7.64
7.60

5.57
5.53
5.49
5.45
5.42

4.68
4.64
4.60
4.57
4.54

4.18
4.14
4.11
4.07
4.04

3.86
3.82
3.78
3.75
3.73

3.63
3.59
3.56
3.53
3.50

3.46
3.42
3.39
3.36
3.33

3.32
3.29
3.26
3.23
3.20

3.13
3.09
3.06
3.03
3.00

2.99
2.96
2.93
2.90
2.87

2.62
2.58
2.55
2.52
2.49

2.17
2.13
2.10
2.06
2.03

25
26
27
28
29

13.88
13.74
13.61
13.50
13.39

9.22
9.12
9.02
8.93
8.85

7.45
7.36
7.27
7.19
7.12

6.49
6.41
6.33
6.25
6.19

5.89
5.80
5.73
5.66
5.59

5.46
5.38
5.31
5.24
5.18

5.15
5.07
5.00
4.93
4.87

4.91
4.83
4.76
4.69
4.64

4.56
4.48
4.41
4.35
4.29

4.31
4.24
4.17
4.11
4.05

3.66
3.59
3.52
3.46
3.41

2.89
2.82
2.75
2.69
2.64

30
32
34
36
38

7.56
7.50
7.45
7.40
7.35

5.39
5.34
5.29
5.25
5.21

4.51
4.46
4.42
4.38
4.34

4.02
3.97
3.93
3.89
3.86

3.70
3.65
3.61
3.58
3.54

3.47
3.43
3.39
3.35
3.32

3.30
3.26
3.22
3.18
3.15

3.17
3.13
3.09
3.05
3.02

2.98
2.93
2.90
2.86
2.83

2.84
2.80
2.76
2.72
2.69

2.47
2.42
2.38
2.35
2.32

2.01
1.96
1.91
1.87
1.84

30
32
34
36
38

13.29
13.12
12.97
12.83
12.71

8.77
8.64
8.52
8.42
8.33

7.05
6.94
6.83
6.74
6.66

6.12
6.01
5.92
5.84
5.76

5.53
5.43
5.34
5.26
5.19

5.12
5.02
4.93
4.86
4.79

4.82
4.72
4.63
4.56
4.49

4.58
4.48
4.40
4.33
4.26

4.24
4.14
4.06
3.99
3.93

4.00
3.91
3.83
3.76
3.70

3.36
3.27
3.19
3.12
3.06

2.59
2.50
2.42
2.35
2.29

40
60
120
∞

7.31
7.08
6.85
6.63

5.18

4.98
4.79
4.61

4.31
4.13
3.95
3.78

3.83
3.65
3.48
3.32

3.51
3.34
3.17
3.02

3.29
3.12
2.96
2.80

3.12
2.95
2.79
2.64

2.99

2.82
2.66
2.51

2.80
2.63
2.47
2.32

2.66
2.50
2.34
2.18

2.29
2.12
1.95
1.79

1.80
1.60
1.38
1.00

40
60
120
∞

12.61

11.97
11.38
10.83

8.25
7.77
7.32
6.91

6.59
6.17
5.78
5.42

5.70
5.31
4.95
4.62

5.13
4.76
4.42
4.10

4.73
4.37
4.04
3.74

4.44

4.09
3.77
3.47

4.21
3.86
3.55
3.27

3.87
3.54
3.24
2.96

3.64
3.32
3.02
2.74

3.01
2.69
2.40
2.13

2.23
1.89
1.54
1.00

v2

CRITICAL VALUES FOR F – TEST

1

5
6
7
8
9

25

v1
1
2
3
4

v2