ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION · PDF fileADVANCED GENERAL CERTIFICATE...

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ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION ADVANCED GENERAL CERTIFICATE OF EDUCATION MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES (List MF1) MF1 CST252 January 2007

Transcript of ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION · PDF fileADVANCED GENERAL CERTIFICATE...

Page 1: ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION · PDF fileADVANCED GENERAL CERTIFICATE OF EDUCATION MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES (List MF1) MF1 CST252

ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION

ADVANCED GENERAL CERTIFICATE OF EDUCATION

MATHEMATICS

LIST OF FORMULAE

AND

STATISTICAL TABLES

(List MF1)

MF1CST252

January 2007

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Pure Mathematics

Mensuration

Surface area of sphere = 4πr2

Area of curved surface of cone = πr × slant height

Trigonometry

a2 = b2 + c2 − 2bc cos A

Arithmetic Series

un = a + (n − 1)dSn = 1

2n(a + l) = 1

2n{2a + (n − 1)d}

Geometric Series

un = arn−1

Sn = a(1 − rn)1 − r

S∞ = a1 − r

for |r | < 1

Summations

n

∑r=1

r2 = 16n(n + 1)(2n + 1)

n

∑r=1

r3 = 14n2(n + 1)2

Binomial Series

(nr) + ( n

r + 1) = (n + 1

r + 1)

(a + b)n = an + (n1)an−1b + (n

2)an−2b2 + . . . + (n

r)an−rbr + . . . + bn (n ∈ ),

where (nr) = nCr = n!

r!(n − r)!(1 + x)n = 1 + nx + n(n − 1)

1.2x2 + . . . + n(n − 1) . . . (n − r + 1)

1.2.3 . . . rxr + . . . (|x| < 1, n ∈ �)

Logarithms and exponentials

ex ln a = ax

Complex Numbers

{r(cos θ + i sin θ)}n = rn(cos nθ + i sin nθ)eiθ = cos θ + i sin θ

The roots of �n = 1 are given by � = e2πki

n , for k = 0, 1, 2, . . . , n − 1

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Maclaurin’s Series

f(x) = f(0) + xf ′(0) + x2

2!f ′′(0) + . . . + xr

r!f (r)(0) + . . .

ex = exp(x) = 1 + x + x2

2!+ . . . + xr

r!+ . . . for all x

ln(1 + x) = x − x2

2+ x3

3− . . . + (−1)r+1 xr

r+ . . . (−1 < x ≤ 1)

sin x = x − x3

3!+ x5

5!− . . . + (−1)r x2r+1

(2r + 1)! + . . . for all x

cos x = 1 − x2

2!+ x4

4!− . . . + (−1)r x2r

(2r)! + . . . for all x

tan−1 x = x − x3

3+ x5

5− . . . + (−1)r x2r+1

2r + 1+ . . . (−1 ≤ x ≤ 1)

sinh x = x + x3

3!+ x5

5!+ . . . + x2r+1

(2r + 1)! + . . . for all x

cosh x = 1 + x2

2!+ x4

4!+ . . . + x2r

(2r)! + . . . for all x

tanh−1 x = x + x3

3+ x5

5+ . . . + x2r+1

2r + 1+ . . . (−1 < x < 1)

Hyperbolic Functions

cosh2 x − sinh2 x = 1

sinh 2x = 2 sinh x cosh x

cosh 2x = cosh2 x + sinh2 x

cosh−1 x = ln{x + √(x2 − 1)} (x ≥ 1)sinh−1 x = ln{x + √(x2 + 1)}tanh−1 x = 1

2ln(1 + x

1 − x) (|x| < 1)

Coordinate Geometry

The perpendicular distance from (h, k) to ax + by + c = 0 is|ah + bk + c|√(a2 + b2)

The acute angle between lines with gradients m1 and m2 is tan−1 ∣∣∣∣ m1 − m2

1 +m1m2

∣∣∣∣Trigonometric Identities

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

tan(A ± B) = tan A ± tan B1 ∓ tan A tan B

(A ± B ≠ (k + 12)π)

For t = tan 12A: sin A = 2t

1 + t2, cos A = 1 − t2

1 + t2

sin A + sin B = 2 sinA + B

2cos

A − B2

sin A − sin B = 2 cosA + B

2sin

A − B2

cos A + cos B = 2 cosA + B

2cos

A − B2

cos A − cos B = −2 sinA + B

2sin

A − B2

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Vectors

The resolved part of a in the direction of b isa.b|b|

The point dividing AB in the ratio λ : µ isµa + λb

λ + µ

Vector product: a × b = |a| |b| sin θ n̂ = ∣∣∣∣∣i a1 b1j a2 b2k a

3b

3

∣∣∣∣∣ = (a2b3 − a3b2a3b1 − a1b3a

1b

2− a

2b

1

)If A is the point with position vector a = a1i + a2j + a3k and the direction vector b is given by

b = b1i + b

2j + b

3k, then the straight line through A with direction vector b has cartesian equation

x − a1

b1

= y − a2

b2

= � − a3

b3

(= λ )The plane through A with normal vector n = n1i + n2j + n3k has cartesian equation

n1x + n2y + n3� + d = 0, where d = −a.n

The plane through non-collinear points A, B and C has vector equation

r = a + λ (b − a) + µ(c − a) = (1 − λ − µ)a + λb + µc

The plane through the point with position vector a and parallel to b and c has equation r = a + sb + tc

The perpendicular distance of (α, β , γ ) from n1x + n2y + n3� + d = 0 is∣∣n1α + n2β + n3γ + d ∣∣√(n2

1 + n22 + n2

3)Matrix transformations

Anticlockwise rotation through θ about O: ( cos θ − sin θsin θ cos θ )

Reflection in the line y = (tan θ)x: ( cos 2θ sin 2θsin 2θ − cos 2θ )

Differentiation

f(x) f ′(x)tan kx k sec2 kx

sin−1 x1√(1 − x2)

cos−1 x − 1√(1 − x2)tan−1 x

1

1 + x2

sec x sec x tan xcot x − cosec2 x

cosec x − cosec x cot xsinh x cosh xcosh x sinh xtanh x sech2 x

sinh−1 x1√(1 + x2)

cosh−1 x1√(x2 − 1)

tanh−1 x1

1 − x2

If y = f(x)g(x) then

dydx

= f ′(x)g(x) − f(x)g′(x){g(x)}2

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Integration ( + constant; a > 0 where relevant)

f(x) � f(x) dx

sec2 kx1k

tan kx

tan x ln |sec x|cot x ln |sin x|

cosec x − ln |cosec x + cot x| = ln ∣∣tan 12x∣∣

sec x ln |sec x + tan x| = ln ∣∣tan(12x + 1

4π)∣∣

sinh x cosh xcosh x sinh xtanh x ln cosh x

1√(a2 − x2) sin−1(xa) (|x| < a)

1

a2 + x2

1a

tan−1(xa)

1√(x2 − a2) cosh−1(xa) or ln{x + √(x2 − a2)} (x > a)

1√(a2 + x2) sinh−1(xa) or ln{x + √(x2 + a2)}

1

a2 − x2

12a

ln ∣∣∣ a + xa − x

∣∣∣ = 1a

tanh−1(xa) (|x| < a)

1

x2 − a2

12a

ln ∣∣∣ x − ax + a

∣∣∣� u

dvdx

dx = uv − � vdudx

dx

Area of a sector

A = 12� r2 dθ (polar coordinates)

A = 12� (x dy

dt− y

dxdt) dt (parametric form)

Numerical Mathematics

Numerical integration

The trapezium rule: � b

ay dx ≈ 1

2h{(y0 + yn) + 2(y1 + y2 + . . . + yn−1)}, where h = b − a

n

Simpson’s Rule: � b

ay dx ≈ 1

3h{(y0 + yn) + 4(y1 + y3 + . . . + yn−1) + 2(y2 + y4 + . . . + yn−2)},

where h = b − an

and n is even

Numerical Solution of Equations

The Newton-Raphson iteration for solving f(x) = 0: xn+1 = xn − f(xn)f′(xn)

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Mechanics

Motion in a circle

Transverse velocity: v = rθ̇

Transverse acceleration: v̇ = rθ̈

Radial acceleration: − rθ̇2 = −v2

r

Centres of Mass (for uniform bodies)

Triangular lamina: 23

along median from vertex

Solid hemisphere, radius r: 38r from centre

Hemispherical shell, radius r: 12r from centre

Circular arc, radius r, angle at centre 2α:r sin α

αfrom centre

Sector of circle, radius r, angle at centre 2α:2r sin α

3αfrom centre

Solid cone or pyramid of height h: 14h above the base on the line from centre of base to vertex

Conical shell of height h: 13h above the base on the line from centre of base to vertex

Moments of Inertia (for uniform bodies of mass m)

Thin rod, length 2l, about perpendicular axis through centre: 13ml2

Rectangular lamina about axis in plane bisecting edges of length 2l: 13ml2

Thin rod, length 2l, about perpendicular axis through end: 43ml2

Rectangular lamina about edge perpendicular to edges of length 2l: 43ml2

Rectangular lamina, sides 2a and 2b, about perpendicular axis through centre: 13m(a2 + b2)

Hoop or cylindrical shell of radius r about axis: mr2

Hoop of radius r about a diameter: 12mr2

Disc or solid cylinder of radius r about axis: 12mr2

Disc of radius r about a diameter: 14mr2

Solid sphere, radius r, about diameter: 25mr2

Spherical shell of radius r about a diameter: 23mr2

Parallel axes theorem: IA = IG +m(AG)2

Perpendicular axes theorem: I� = Ix + Iy (for a lamina in the x-y plane)

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Probability & Statistics

Probability

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)P(A ∩ B) = P(A)P(B | A)P(A | B) = P(B | A)P(A)

P(B | A)P(A) + P(B | A′)P(A′)Bayes’ Theorem: P(Aj | B) = P(Aj)P(B | Aj)

ΣP(Ai)P(B | Ai)Discrete distributions

For a discrete random variable X taking values xi with probabilities pi

Expectation (mean): E(X) = µ = Σ xip

i

Variance: Var(X) = σ2 = Σ(xi − µ)2pi = Σ x2i pi − µ2

For a function g(X): E(g(X)) = Σ g(xi)pi

The probability generating function of X is GX(t) = E(tX), and

E(X) = G′X(1)

Var(X) = G′′X(1) + G′

X(1) − {G′X(1)}2

For Z = X + Y , where X and Y are independent: GZ(t) = GX(t)GY(t)Standard discrete distributions

Distribution of X P(X = x) Mean Variance P.G.F.

Binomial B(n, p) (nx)px(1 − p)n−x np np(1 − p) (1 − p + pt)n

Poisson Po(λ ) e−λ λ x

x!λ λ eλ (t−1)

Geometric Geo(p) on 1, 2, … p(1 − p)x−1 1p

1 − pp2

pt1 − (1 − p)t

Continuous distributions

For a continuous random variable X having probability density function f

Expectation (mean): E(X) = µ = � xf(x) dx

Variance: Var(X) = σ2 = � (x − µ)2f(x) dx = � x2f(x) dx − µ2

For a function g(X): E(g(X)) = � g(x)f(x) dx

Cumulative distribution function: F(x) = P(X ≤ x) = � x

−∞f(t) dt

The moment generating function of X is MX(t) = E(etX) and

E(X) = M′X(0)

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

Var(X) = M′′X(0) − {M′

X(0)}2

For Z = X + Y , where X and Y are independent: MZ(t) = MX(t)MY(t)

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Standard continuous distributions

Distribution of X P.D.F. Mean Variance M.G.F.

Uniform (Rectangular) on [a, b] 1b − a

12(a + b) 1

12(b − a)2 ebt − eat

(b − a)tExponential λe−λx 1

λ1

λ 2

λλ − t

Normal N(µ, σ2) 1σ√(2π)e−1

2( x−µ

σ )2

µ σ2 eµt+12σ2t2

Expectation algebra

Covariance: Cov(X, Y) = E((X − µX)(Y − µY)) = E(XY) − µXµY

Var(aX ± bY) = a2 Var(X) + b2 Var(Y) ± 2ab Cov(X, Y)Product moment correlation coefficient: ρ = Cov(X, Y)

σX

σY

If X = aX′ + b and Y = cY ′ + d, then Cov(X, Y) = ac Cov(X′, Y ′)For independent random variables X and Y

E(XY) = E(X)E(Y)Var(aX ± bY) = a2 Var(X) + b2 Var(Y)

Sampling distributions

For a random sample X1, X2, . . . , Xn of n independent observations from a distribution having mean µand variance σ2

X is an unbiased estimator of µ, with Var(X) = σ2

n

S2 is an unbiased estimator of σ2, where S2 = Σ(Xi − X)2

n − 1

For a random sample of n observations from N(µ, σ2)X − µσ/√n

∼ N(0, 1)X − µS/√n

∼ tn−1 (also valid in matched-pairs situations)

If X is the observed number of successes in n independent Bernoulli trials in each of which the

probability of success is p, and Y = Xn

, then

E(Y) = p and Var(Y) = p(1 − p)n

For a random sample of nx observations from N(µx, σ2x ) and, independently, a random sample of

ny observations from N(µy, σ2y )(X − Y) − (µx − µy)√(σ2

x

nx

+ σ2y

ny

) ∼ N(0, 1)

If σ2x = σ2

y = σ2 (unknown) then(X − Y) − (µx − µy)√{S2

p( 1

nx

+ 1ny

)} ∼ tnx+n

y−2,

where S2p = (nx − 1)S2

x + (ny − 1)S2y

nx + ny − 2

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Correlation and regression

For a set of n pairs of values (xi, yi)Sxx = Σ(xi − x)2 = Σ x2

i − (Σ xi)2

n

Syy = Σ(yi − y)2 = Σ y2i − (Σ yi)2

n

Sxy= Σ(x

i− x)(y

i− y) = Σ x

iy

i− (Σ xi)(Σ yi)

nThe product moment correlation coefficient is

r = Sxy√(SxxSyy) =Σ(xi − x)(yi − y)√{(Σ(x

i− x)2)(Σ(y

i− y)2)} = Σ xiyi − (Σ xi)(Σ yi)

n√{(Σ x2i − (Σ xi)2

n)(Σ y2

i − (Σ yi)2

n)}

Spearman’s rank correlation coefficient is rs = 1 − 6Σ d2

n(n2 − 1)The regression coefficient of y on x is b = Sxy

Sxx

= Σ(xi − x)(yi − y)Σ(xi − x)2

Least squares regression line of y on x is y = a + bx where a = y − bx

Distribution-free (non-parametric) tests

Goodness-of-fit test and contingency tables: ∑ (Oi − Ei)2

Ei

∼ χ2ν

Approximate distributions for large samples

Wilcoxon Signed Rank test: T ∼ N(14n(n + 1), 1

24n(n + 1)(2n + 1))

Wilcoxon Rank Sum test (samples of sizes m and n, with m ≤ n):

W ∼ N(12m(m + n + 1), 1

12mn(m + n + 1))

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CUMULATIVEBINOMIALPROBABILITIES

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4305

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2326

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0390

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3154

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0000

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9693

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7414

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640.

5941

0.47

700.

3633

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9954

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5318

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3215

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1348

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00

10

Page 11: ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION · PDF fileADVANCED GENERAL CERTIFICATE OF EDUCATION MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES (List MF1) MF1 CST252

CUMULATIVEBINOMIALPROBABILITIES

n=9 p

0.05

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20.

250.

31/3

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0.55

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0.65

2/30.

70.

750.

85/6

0.85

0.9

0.95

x=0

0.63

020.

3874

0.23

160.

1938

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420.

0751

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110.

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280.

3772

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730.

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4826

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2539

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8552

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9576

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7461

0.63

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3497

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560.

0480

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390.

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066

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0000

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000.

9999

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9987

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9750

0.95

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9102

0.85

050.

7682

0.66

270.

6228

0.53

720.

3993

0.26

180.

1783

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0530

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0.96

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9295

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890.

8569

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6997

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9740

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8062

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6126

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0p

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20.

250.

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70.

750.

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0.59

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3487

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1615

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5593

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3823

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1719

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200.

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9845

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720.

9219

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970.

7869

0.75

150.

6331

0.50

440.

3770

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1662

0.09

490.

0766

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9976

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9803

0.95

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9234

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510.

8338

0.73

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6230

0.49

560.

3669

0.24

850.

2131

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0781

0.03

280.

0155

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990.

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0.00

016

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001.

0000

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9997

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910.

9965

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940.

9803

0.97

400.

9452

0.89

800.

8281

0.73

400.

6177

0.48

620.

4407

0.35

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2241

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090.

0697

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0128

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107

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0000

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9877

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260.

9453

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8327

0.73

840.

7009

0.61

720.

4744

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220.

2248

0.17

980.

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158

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0000

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0000

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001.

0000

0.99

990.

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950.

9983

0.99

550.

9893

0.97

670.

9536

0.91

400.

8960

0.85

070.

7560

0.62

420.

5155

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570.

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9940

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8385

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310.

6513

0.40

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001.

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001.

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001.

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001.

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00

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2p

0.05

0.1

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20.

250.

31/3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

2/30.

70.

750.

85/6

0.85

0.9

0.95

x=0

0.54

040.

2824

0.14

220.

1122

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0317

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380.

0077

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160.

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350.

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490.

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8891

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6774

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830.

3907

0.25

280.

1811

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6488

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3931

0.34

670.

2253

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9636

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6315

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330.

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8223

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3872

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540.

8418

0.73

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6128

0.47

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001.

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9812

0.97

450.

9427

0.88

830.

8062

0.69

560.

5618

0.41

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11

Page 12: ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION · PDF fileADVANCED GENERAL CERTIFICATE OF EDUCATION MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES (List MF1) MF1 CST252

CUMULATIVEBINOMIALPROBABILITIES

n=1

4p

0.05

0.1

0.15

1/60.

20.

250.

31/3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

2/30.

70.

750.

85/6

0.85

0.9

0.95

x=0

0.48

770.

2288

0.10

280.

0779

0.04

400.

0178

0.00

680.

0034

0.00

240.

0008

0.00

020.

0001

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

001

0.84

700.

5846

0.35

670.

2960

0.19

790.

1010

0.04

750.

0274

0.02

050.

0081

0.00

290.

0009

0.00

030.

0001

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

002

0.96

990.

8416

0.64

790.

5795

0.44

810.

2811

0.16

080.

1053

0.08

390.

0398

0.01

700.

0065

0.00

220.

0006

0.00

010.

0001

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

003

0.99

580.

9559

0.85

350.

8063

0.69

820.

5213

0.35

520.

2612

0.22

050.

1243

0.06

320.

0287

0.01

140.

0039

0.00

110.

0007

0.00

020.

0000

0.00

000.

0000

0.00

000.

0000

0.00

004

0.99

960.

9908

0.95

330.

9310

0.87

020.

7415

0.58

420.

4755

0.42

270.

2793

0.16

720.

0898

0.04

260.

0175

0.00

600.

0040

0.00

170.

0003

0.00

000.

0000

0.00

000.

0000

0.00

005

1.00

000.

9985

0.98

850.

9809

0.95

610.

8883

0.78

050.

6898

0.64

050.

4859

0.33

730.

2120

0.11

890.

0583

0.02

430.

0174

0.00

830.

0022

0.00

040.

0001

0.00

000.

0000

0.00

006

1.00

000.

9998

0.99

780.

9959

0.98

840.

9617

0.90

670.

8505

0.81

640.

6925

0.54

610.

3953

0.25

860.

1501

0.07

530.

0576

0.03

150.

0103

0.00

240.

0007

0.00

030.

0000

0.00

007

1.00

001.

0000

0.99

970.

9993

0.99

760.

9897

0.96

850.

9424

0.92

470.

8499

0.74

140.

6047

0.45

390.

3075

0.18

360.

1495

0.09

330.

0383

0.01

160.

0041

0.00

220.

0002

0.00

008

1.00

001.

0000

1.00

000.

9999

0.99

960.

9978

0.99

170.

9826

0.97

570.

9417

0.88

110.

7880

0.66

270.

5141

0.35

950.

3102

0.21

950.

1117

0.04

390.

0191

0.01

150.

0015

0.00

009

1.00

001.

0000

1.00

001.

0000

1.00

000.

9997

0.99

830.

9960

0.99

400.

9825

0.95

740.

9102

0.83

280.

7207

0.57

730.

5245

0.41

580.

2585

0.12

980.

0690

0.04

670.

0092

0.00

0410

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

0.99

980.

9993

0.99

890.

9961

0.98

860.

9713

0.93

680.

8757

0.77

950.

7388

0.64

480.

4787

0.30

180.

1937

0.14

650.

0441

0.00

4211

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

000.

9999

0.99

990.

9994

0.99

780.

9935

0.98

300.

9602

0.91

610.

8947

0.83

920.

7189

0.55

190.

4205

0.35

210.

1584

0.03

0112

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

000.

9999

0.99

970.

9991

0.99

710.

9919

0.97

950.

9726

0.95

250.

8990

0.80

210.

7040

0.64

330.

4154

0.15

3013

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

000.

9999

0.99

980.

9992

0.99

760.

9966

0.99

320.

9822

0.95

600.

9221

0.89

720.

7712

0.51

2314

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

00

n=1

6p

0.05

0.1

0.15

1/60.

20.

250.

31/3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

2/30.

70.

750.

85/6

0.85

0.9

0.95

x=0

0.44

010.

1853

0.07

430.

0541

0.02

810.

0100

0.00

330.

0015

0.00

100.

0003

0.00

010.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

001

0.81

080.

5147

0.28

390.

2272

0.14

070.

0635

0.02

610.

0137

0.00

980.

0033

0.00

100.

0003

0.00

010.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

002

0.95

710.

7892

0.56

140.

4868

0.35

180.

1971

0.09

940.

0594

0.04

510.

0183

0.00

660.

0021

0.00

060.

0001

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

003

0.99

300.

9316

0.78

990.

7291

0.59

810.

4050

0.24

590.

1659

0.13

390.

0651

0.02

810.

0106

0.00

350.

0009

0.00

020.

0001

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

004

0.99

910.

9830

0.92

090.

8866

0.79

820.

6302

0.44

990.

3391

0.28

920.

1666

0.08

530.

0384

0.01

490.

0049

0.00

130.

0008

0.00

030.

0000

0.00

000.

0000

0.00

000.

0000

0.00

005

0.99

990.

9967

0.97

650.

9622

0.91

830.

8103

0.65

980.

5469

0.49

000.

3288

0.19

760.

1051

0.04

860.

0191

0.00

620.

0040

0.00

160.

0003

0.00

000.

0000

0.00

000.

0000

0.00

006

1.00

000.

9995

0.99

440.

9899

0.97

330.

9204

0.82

470.

7374

0.68

810.

5272

0.36

600.

2272

0.12

410.

0583

0.02

290.

0159

0.00

710.

0016

0.00

020.

0000

0.00

000.

0000

0.00

007

1.00

000.

9999

0.99

890.

9979

0.99

300.

9729

0.92

560.

8735

0.84

060.

7161

0.56

290.

4018

0.25

590.

1423

0.06

710.

0500

0.02

570.

0075

0.00

150.

0004

0.00

020.

0000

0.00

008

1.00

001.

0000

0.99

980.

9996

0.99

850.

9925

0.97

430.

9500

0.93

290.

8577

0.74

410.

5982

0.43

710.

2839

0.15

940.

1265

0.07

440.

0271

0.00

700.

0021

0.00

110.

0001

0.00

009

1.00

001.

0000

1.00

001.

0000

0.99

980.

9984

0.99

290.

9841

0.97

710.

9417

0.87

590.

7728

0.63

400.

4728

0.31

190.

2626

0.17

530.

0796

0.02

670.

0101

0.00

560.

0005

0.00

0010

1.00

001.

0000

1.00

001.

0000

1.00

000.

9997

0.99

840.

9960

0.99

380.

9809

0.95

140.

8949

0.80

240.

6712

0.51

000.

4531

0.34

020.

1897

0.08

170.

0378

0.02

350.

0033

0.00

0111

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

0.99

970.

9992

0.99

870.

9951

0.98

510.

9616

0.91

470.

8334

0.71

080.

6609

0.55

010.

3698

0.20

180.

1134

0.07

910.

0170

0.00

0912

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

000.

9999

0.99

980.

9991

0.99

650.

9894

0.97

190.

9349

0.86

610.

8341

0.75

410.

5950

0.40

190.

2709

0.21

010.

0684

0.00

7013

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

000.

9999

0.99

940.

9979

0.99

340.

9817

0.95

490.

9406

0.90

060.

8029

0.64

820.

5132

0.43

860.

2108

0.04

2914

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

0.99

990.

9997

0.99

900.

9967

0.99

020.

9863

0.97

390.

9365

0.85

930.

7728

0.71

610.

4853

0.18

9215

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

0.99

990.

9997

0.99

900.

9985

0.99

670.

9900

0.97

190.

9459

0.92

570.

8147

0.55

9916

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

00

12

Page 13: ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION · PDF fileADVANCED GENERAL CERTIFICATE OF EDUCATION MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES (List MF1) MF1 CST252

CUMULATIVEBINOMIALPROBABILITIES

n=1

8p

0.05

0.1

0.15

1/60.

20.

250.

31/3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

2/30.

70.

750.

85/6

0.85

0.9

0.95

x=0

0.39

720.

1501

0.05

360.

0376

0.01

800.

0056

0.00

160.

0007

0.00

040.

0001

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

001

0.77

350.

4503

0.22

410.

1728

0.09

910.

0395

0.01

420.

0068

0.00

460.

0013

0.00

030.

0001

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

002

0.94

190.

7338

0.47

970.

4027

0.27

130.

1353

0.06

000.

0326

0.02

360.

0082

0.00

250.

0007

0.00

010.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

003

0.98

910.

9018

0.72

020.

6479

0.50

100.

3057

0.16

460.

1017

0.07

830.

0328

0.01

200.

0038

0.00

100.

0002

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

004

0.99

850.

9718

0.87

940.

8318

0.71

640.

5187

0.33

270.

2311

0.18

860.

0942

0.04

110.

0154

0.00

490.

0013

0.00

030.

0001

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

005

0.99

980.

9936

0.95

810.

9347

0.86

710.

7175

0.53

440.

4122

0.35

500.

2088

0.10

770.

0481

0.01

830.

0058

0.00

140.

0009

0.00

030.

0000

0.00

000.

0000

0.00

000.

0000

0.00

006

1.00

000.

9988

0.98

820.

9794

0.94

870.

8610

0.72

170.

6085

0.54

910.

3743

0.22

580.

1189

0.05

370.

0203

0.00

620.

0039

0.00

140.

0002

0.00

000.

0000

0.00

000.

0000

0.00

007

1.00

000.

9998

0.99

730.

9947

0.98

370.

9431

0.85

930.

7767

0.72

830.

5634

0.39

150.

2403

0.12

800.

0576

0.02

120.

0144

0.00

610.

0012

0.00

020.

0000

0.00

000.

0000

0.00

008

1.00

001.

0000

0.99

950.

9989

0.99

570.

9807

0.94

040.

8924

0.86

090.

7368

0.57

780.

4073

0.25

270.

1347

0.05

970.

0433

0.02

100.

0054

0.00

090.

0002

0.00

010.

0000

0.00

009

1.00

001.

0000

0.99

990.

9998

0.99

910.

9946

0.97

900.

9567

0.94

030.

8653

0.74

730.

5927

0.42

220.

2632

0.13

910.

1076

0.05

960.

0193

0.00

430.

0011

0.00

050.

0000

0.00

0010

1.00

001.

0000

1.00

001.

0000

0.99

980.

9988

0.99

390.

9856

0.97

880.

9424

0.87

200.

7597

0.60

850.

4366

0.27

170.

2233

0.14

070.

0569

0.01

630.

0053

0.00

270.

0002

0.00

0011

1.00

001.

0000

1.00

001.

0000

1.00

000.

9998

0.99

860.

9961

0.99

380.

9797

0.94

630.

8811

0.77

420.

6257

0.45

090.

3915

0.27

830.

1390

0.05

130.

0206

0.01

180.

0012

0.00

0012

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

0.99

970.

9991

0.99

860.

9942

0.98

170.

9519

0.89

230.

7912

0.64

500.

5878

0.46

560.

2825

0.13

290.

0653

0.04

190.

0064

0.00

0213

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

000.

9999

0.99

970.

9987

0.99

510.

9846

0.95

890.

9058

0.81

140.

7689

0.66

730.

4813

0.28

360.

1682

0.12

060.

0282

0.00

1514

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

000.

9998

0.99

900.

9962

0.98

800.

9672

0.92

170.

8983

0.83

540.

6943

0.49

900.

3521

0.27

980.

0982

0.01

0915

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

0.99

990.

9993

0.99

750.

9918

0.97

640.

9674

0.94

000.

8647

0.72

870.

5973

0.52

030.

2662

0.05

8116

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

000.

9999

0.99

970.

9987

0.99

540.

9932

0.98

580.

9605

0.90

090.

8272

0.77

590.

5497

0.22

6517

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

000.

9999

0.99

960.

9993

0.99

840.

9944

0.98

200.

9624

0.94

640.

8499

0.60

2818

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

001.

0000

1.00

00

n=2

0p

0.05

0.1

0.15

1/60.

20.

250.

31/3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

2/30.

70.

750.

85/6

0.85

0.9

0.95

x=0

0.35

850.

1216

0.03

880.

0261

0.01

150.

0032

0.00

080.

0003

0.00

020.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

000.

0000

0.00

001

0.73

580.

3917

0.17

560.

1304

0.06

920.

0243

0.00

760.

0033

0.00

210.

0005

0.00

010.

0000

0.00

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13

Page 14: ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION · PDF fileADVANCED GENERAL CERTIFICATE OF EDUCATION MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES (List MF1) MF1 CST252

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n=2

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15

Page 16: ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION · PDF fileADVANCED GENERAL CERTIFICATE OF EDUCATION MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES (List MF1) MF1 CST252

CUMULATIVE POISSON PROBABILITIES

λ 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09x = 0 0.9900 0.9802 0.9704 0.9608 0.9512 0.9418 0.9324 0.9231 0.9139

1 1.0000 0.9998 0.9996 0.9992 0.9988 0.9983 0.9977 0.9970 0.99622 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.99993 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90x = 0 0.9048 0.8187 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066

1 0.9953 0.9825 0.9631 0.9384 0.9098 0.8781 0.8442 0.8088 0.77252 0.9998 0.9989 0.9964 0.9921 0.9856 0.9769 0.9659 0.9526 0.93713 1.0000 0.9999 0.9997 0.9992 0.9982 0.9966 0.9942 0.9909 0.98654 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9986 0.99775 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.99976 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90x = 0 0.3679 0.3329 0.3012 0.2725 0.2466 0.2231 0.2019 0.1827 0.1653 0.1496

1 0.7358 0.6990 0.6626 0.6268 0.5918 0.5578 0.5249 0.4932 0.4628 0.43372 0.9197 0.9004 0.8795 0.8571 0.8335 0.8088 0.7834 0.7572 0.7306 0.70373 0.9810 0.9743 0.9662 0.9569 0.9463 0.9344 0.9212 0.9068 0.8913 0.87474 0.9963 0.9946 0.9923 0.9893 0.9857 0.9814 0.9763 0.9704 0.9636 0.95595 0.9994 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.9920 0.9896 0.98686 0.9999 0.9999 0.9997 0.9996 0.9994 0.9991 0.9987 0.9981 0.9974 0.99667 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996 0.9994 0.99928 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.99989 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90x = 0 0.1353 0.1225 0.1108 0.1003 0.0907 0.0821 0.0743 0.0672 0.0608 0.0550

1 0.4060 0.3796 0.3546 0.3309 0.3084 0.2873 0.2674 0.2487 0.2311 0.21462 0.6767 0.6496 0.6227 0.5960 0.5697 0.5438 0.5184 0.4936 0.4695 0.44603 0.8571 0.8386 0.8194 0.7993 0.7787 0.7576 0.7360 0.7141 0.6919 0.66964 0.9473 0.9379 0.9275 0.9162 0.9041 0.8912 0.8774 0.8629 0.8477 0.83185 0.9834 0.9796 0.9751 0.9700 0.9643 0.9580 0.9510 0.9433 0.9349 0.92586 0.9955 0.9941 0.9925 0.9906 0.9884 0.9858 0.9828 0.9794 0.9756 0.97137 0.9989 0.9985 0.9980 0.9974 0.9967 0.9958 0.9947 0.9934 0.9919 0.99018 0.9998 0.9997 0.9995 0.9994 0.9991 0.9989 0.9985 0.9981 0.9976 0.99699 1.0000 0.9999 0.9999 0.9999 0.9998 0.9997 0.9996 0.9995 0.9993 0.9991

10 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9998 0.999811 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.999912 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90x = 0 0.0498 0.0450 0.0408 0.0369 0.0334 0.0302 0.0273 0.0247 0.0224 0.0202

1 0.1991 0.1847 0.1712 0.1586 0.1468 0.1359 0.1257 0.1162 0.1074 0.09922 0.4232 0.4012 0.3799 0.3594 0.3397 0.3208 0.3027 0.2854 0.2689 0.25313 0.6472 0.6248 0.6025 0.5803 0.5584 0.5366 0.5152 0.4942 0.4735 0.45324 0.8153 0.7982 0.7806 0.7626 0.7442 0.7254 0.7064 0.6872 0.6678 0.64845 0.9161 0.9057 0.8946 0.8829 0.8705 0.8576 0.8441 0.8301 0.8156 0.80066 0.9665 0.9612 0.9554 0.9490 0.9421 0.9347 0.9267 0.9182 0.9091 0.89957 0.9881 0.9858 0.9832 0.9802 0.9769 0.9733 0.9692 0.9648 0.9599 0.95468 0.9962 0.9953 0.9943 0.9931 0.9917 0.9901 0.9883 0.9863 0.9840 0.98159 0.9989 0.9986 0.9982 0.9978 0.9973 0.9967 0.9960 0.9952 0.9942 0.9931

10 0.9997 0.9996 0.9995 0.9994 0.9992 0.9990 0.9987 0.9984 0.9981 0.997711 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9996 0.9995 0.9994 0.999312 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9998 0.999813 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.999914 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

16

Page 17: ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION · PDF fileADVANCED GENERAL CERTIFICATE OF EDUCATION MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES (List MF1) MF1 CST252

CUMULATIVE POISSON PROBABILITIES

λ 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90

x = 0 0.0183 0.0166 0.0150 0.0136 0.0123 0.0111 0.0101 0.0091 0.0082 0.00741 0.0916 0.0845 0.0780 0.0719 0.0663 0.0611 0.0563 0.0518 0.0477 0.04392 0.2381 0.2238 0.2102 0.1974 0.1851 0.1736 0.1626 0.1523 0.1425 0.13333 0.4335 0.4142 0.3954 0.3772 0.3594 0.3423 0.3257 0.3097 0.2942 0.27934 0.6288 0.6093 0.5898 0.5704 0.5512 0.5321 0.5132 0.4946 0.4763 0.45825 0.7851 0.7693 0.7531 0.7367 0.7199 0.7029 0.6858 0.6684 0.6510 0.63356 0.8893 0.8786 0.8675 0.8558 0.8436 0.8311 0.8180 0.8046 0.7908 0.77677 0.9489 0.9427 0.9361 0.9290 0.9214 0.9134 0.9049 0.8960 0.8867 0.87698 0.9786 0.9755 0.9721 0.9683 0.9642 0.9597 0.9549 0.9497 0.9442 0.93829 0.9919 0.9905 0.9889 0.9871 0.9851 0.9829 0.9805 0.9778 0.9749 0.9717

10 0.9972 0.9966 0.9959 0.9952 0.9943 0.9933 0.9922 0.9910 0.9896 0.988011 0.9991 0.9989 0.9986 0.9983 0.9980 0.9976 0.9971 0.9966 0.9960 0.995312 0.9997 0.9997 0.9996 0.9995 0.9993 0.9992 0.9990 0.9988 0.9986 0.998313 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997 0.9997 0.9996 0.9995 0.999414 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.999815 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.999916 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00 9.50

x = 0 0.0067 0.0041 0.0025 0.0015 0.0009 0.0006 0.0003 0.0002 0.0001 0.00011 0.0404 0.0266 0.0174 0.0113 0.0073 0.0047 0.0030 0.0019 0.0012 0.00082 0.1247 0.0884 0.0620 0.0430 0.0296 0.0203 0.0138 0.0093 0.0062 0.00423 0.2650 0.2017 0.1512 0.1118 0.0818 0.0591 0.0424 0.0301 0.0212 0.01494 0.4405 0.3575 0.2851 0.2237 0.1730 0.1321 0.0996 0.0744 0.0550 0.04035 0.6160 0.5289 0.4457 0.3690 0.3007 0.2414 0.1912 0.1496 0.1157 0.08856 0.7622 0.6860 0.6063 0.5265 0.4497 0.3782 0.3134 0.2562 0.2068 0.16497 0.8666 0.8095 0.7440 0.6728 0.5987 0.5246 0.4530 0.3856 0.3239 0.26878 0.9319 0.8944 0.8472 0.7916 0.7291 0.6620 0.5925 0.5231 0.4557 0.39189 0.9682 0.9462 0.9161 0.8774 0.8305 0.7764 0.7166 0.6530 0.5874 0.5218

10 0.9863 0.9747 0.9574 0.9332 0.9015 0.8622 0.8159 0.7634 0.7060 0.645311 0.9945 0.9890 0.9799 0.9661 0.9467 0.9208 0.8881 0.8487 0.8030 0.752012 0.9980 0.9955 0.9912 0.9840 0.9730 0.9573 0.9362 0.9091 0.8758 0.836413 0.9993 0.9983 0.9964 0.9929 0.9872 0.9784 0.9658 0.9486 0.9261 0.898114 0.9998 0.9994 0.9986 0.9970 0.9943 0.9897 0.9827 0.9726 0.9585 0.940015 0.9999 0.9998 0.9995 0.9988 0.9976 0.9954 0.9918 0.9862 0.9780 0.966516 1.0000 0.9999 0.9998 0.9996 0.9990 0.9980 0.9963 0.9934 0.9889 0.982317 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9984 0.9970 0.9947 0.991118 1.0000 1.0000 1.0000 0.9999 0.9999 0.9997 0.9993 0.9987 0.9976 0.995719 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9995 0.9989 0.998020 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.999121 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.999622 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.999923 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.999924 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

17

Page 18: ADVANCED SUBSIDIARY GENERAL CERTIFICATE OF EDUCATION · PDF fileADVANCED GENERAL CERTIFICATE OF EDUCATION MATHEMATICS LIST OF FORMULAE AND STATISTICAL TABLES (List MF1) MF1 CST252

CUMULATIVE POISSON PROBABILITIES

λ 10.00 11.00 12.00 13.00 14.00 15.00 16.00 17.00 18.00 19.00

x = 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00001 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00002 0.0028 0.0012 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.00003 0.0103 0.0049 0.0023 0.0011 0.0005 0.0002 0.0001 0.0000 0.0000 0.00004 0.0293 0.0151 0.0076 0.0037 0.0018 0.0009 0.0004 0.0002 0.0001 0.00005 0.0671 0.0375 0.0203 0.0107 0.0055 0.0028 0.0014 0.0007 0.0003 0.00026 0.1301 0.0786 0.0458 0.0259 0.0142 0.0076 0.0040 0.0021 0.0010 0.00057 0.2202 0.1432 0.0895 0.0540 0.0316 0.0180 0.0100 0.0054 0.0029 0.00158 0.3328 0.2320 0.1550 0.0998 0.0621 0.0374 0.0220 0.0126 0.0071 0.00399 0.4579 0.3405 0.2424 0.1658 0.1094 0.0699 0.0433 0.0261 0.0154 0.0089

10 0.5830 0.4599 0.3472 0.2517 0.1757 0.1185 0.0774 0.0491 0.0304 0.018311 0.6968 0.5793 0.4616 0.3532 0.2600 0.1848 0.1270 0.0847 0.0549 0.034712 0.7916 0.6887 0.5760 0.4631 0.3585 0.2676 0.1931 0.1350 0.0917 0.060613 0.8645 0.7813 0.6815 0.5730 0.4644 0.3632 0.2745 0.2009 0.1426 0.098414 0.9165 0.8540 0.7720 0.6751 0.5704 0.4657 0.3675 0.2808 0.2081 0.149715 0.9513 0.9074 0.8444 0.7636 0.6694 0.5681 0.4667 0.3715 0.2867 0.214816 0.9730 0.9441 0.8987 0.8355 0.7559 0.6641 0.5660 0.4677 0.3751 0.292017 0.9857 0.9678 0.9370 0.8905 0.8272 0.7489 0.6593 0.5640 0.4686 0.378418 0.9928 0.9823 0.9626 0.9302 0.8826 0.8195 0.7423 0.6550 0.5622 0.469519 0.9965 0.9907 0.9787 0.9573 0.9235 0.8752 0.8122 0.7363 0.6509 0.560620 0.9984 0.9953 0.9884 0.9750 0.9521 0.9170 0.8682 0.8055 0.7307 0.647221 0.9993 0.9977 0.9939 0.9859 0.9712 0.9469 0.9108 0.8615 0.7991 0.725522 0.9997 0.9990 0.9970 0.9924 0.9833 0.9673 0.9418 0.9047 0.8551 0.793123 0.9999 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.8989 0.849024 1.0000 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.9317 0.893325 1.0000 0.9999 0.9997 0.9990 0.9974 0.9938 0.9869 0.9748 0.9554 0.926926 1.0000 1.0000 0.9999 0.9995 0.9987 0.9967 0.9925 0.9848 0.9718 0.951427 1.0000 1.0000 0.9999 0.9998 0.9994 0.9983 0.9959 0.9912 0.9827 0.968728 1.0000 1.0000 1.0000 0.9999 0.9997 0.9991 0.9978 0.9950 0.9897 0.980529 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9989 0.9973 0.9941 0.988230 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9986 0.9967 0.993031 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9993 0.9982 0.996032 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9990 0.997833 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0.998834 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.999435 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.999736 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.999837 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.999938 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

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THE NORMAL DISTRIBUTION FUNCTION

If Z has a normal distribution with mean 0 andvariance 1 then, for each value of �, the table givesthe value of Φ(�), where

Φ(�) = P(Z ≤ �).For negative values of � use Φ(−�) = 1 − Φ(�).

� 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9ADD

0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 4 8 12 16 20 24 28 32 360.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 4 8 12 16 20 24 28 32 360.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 4 8 12 15 19 23 27 31 350.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 4 7 11 15 19 22 26 30 340.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 4 7 11 14 18 22 25 29 32

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 3 7 10 14 17 20 24 27 310.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 3 7 10 13 16 19 23 26 290.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 3 6 9 12 15 18 21 24 270.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 3 5 8 11 14 16 19 22 250.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 3 5 8 10 13 15 18 20 23

1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 2 5 7 9 12 14 16 19 211.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 2 4 6 8 10 12 14 16 181.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 2 4 6 7 9 11 13 15 171.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 2 3 5 6 8 10 11 13 141.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1 3 4 6 7 8 10 11 13

1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1 2 4 5 6 7 8 10 111.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 1 2 3 4 5 6 7 8 91.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1 2 3 4 4 5 6 7 81.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1 1 2 3 4 4 5 6 61.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 1 1 2 2 3 4 4 5 5

2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817 0 1 1 2 2 3 3 4 42.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857 0 1 1 2 2 2 3 3 42.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890 0 1 1 1 2 2 2 3 32.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916 0 1 1 1 1 2 2 2 22.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936 0 0 1 1 1 1 1 2 2

2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952 0 0 0 1 1 1 1 1 12.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964 0 0 0 0 1 1 1 1 12.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974 0 0 0 0 0 1 1 1 12.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981 0 0 0 0 0 0 0 1 12.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986 0 0 0 0 0 0 0 0 0

Critical values for the normal distribution

If Z has a normal distribution with mean 0 andvariance 1 then, for each value of p, the table givesthe value of � such that

P(Z ≤ �) = p.

p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995

� 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291

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CRITICAL VALUES FOR THE t DISTRIBUTION

If T has a t distribution with v degrees of freedomthen, for each pair of values of p and v, the tablegives the value of t such that

P(T ≤ t) = p.

p 0.75 0.90 0.95 0.975 0.99 0.995 0.9975 0.999 0.9995

v = 1 1.000 3.078 6.314 12.71 31.82 63.66 127.3 318.3 636.62 0.816 1.886 2.920 4.303 6.965 9.925 14.09 22.33 31.603 0.765 1.638 2.353 3.182 4.541 5.841 7.453 10.21 12.924 0.741 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8.610

5 0.727 1.476 2.015 2.571 3.365 4.032 4.773 5.894 6.8696 0.718 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5.9597 0.711 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5.4088 0.706 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5.0419 0.703 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.781

10 0.700 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.58711 0.697 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.43712 0.695 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.31813 0.694 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.22114 0.692 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.140

15 0.691 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.07316 0.690 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4.01517 0.689 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3.96518 0.688 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.92219 0.688 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.883

20 0.687 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.85021 0.686 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3.81922 0.686 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3.79223 0.685 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.76824 0.685 1.318 1.711 2.064 2.492 2.797 3.091 3.467 3.745

25 0.684 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3.72526 0.684 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.70727 0.684 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.68928 0.683 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3.67429 0.683 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.660

30 0.683 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3.64640 0.681 1.303 1.684 2.021 2.423 2.704 2.971 3.307 3.55160 0.679 1.296 1.671 2.000 2.390 2.660 2.915 3.232 3.460

120 0.677 1.289 1.658 1.980 2.358 2.617 2.860 3.160 3.373∞ 0.674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.291

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CRITICAL VALUES FOR THE χχχχχ2 DISTRIBUTION

If X has a χ2 distribution with v degrees of freedomthen, for each pair of values of p and v, the tablegives the value of x such that

P(X ≤ x) = p.

p 0.01 0.025 0.05 0.90 0.95 0.975 0.99 0.995 0.999

v = 1 0.031571 0.039821 0.023932 2.706 3.841 5.024 6.635 7.879 10.832 0.02010 0.05064 0.1026 4.605 5.991 7.378 9.210 10.60 13.823 0.1148 0.2158 0.3518 6.251 7.815 9.348 11.34 12.84 16.274 0.2971 0.4844 0.7107 7.779 9.488 11.14 13.28 14.86 18.47

5 0.5543 0.8312 1.145 9.236 11.07 12.83 15.09 16.75 20.516 0.8721 1.237 1.635 10.64 12.59 14.45 16.81 18.55 22.467 1.239 1.690 2.167 12.02 14.07 16.01 18.48 20.28 24.328 1.647 2.180 2.733 13.36 15.51 17.53 20.09 21.95 26.129 2.088 2.700 3.325 14.68 16.92 19.02 21.67 23.59 27.88

10 2.558 3.247 3.940 15.99 18.31 20.48 23.21 25.19 29.5911 3.053 3.816 4.575 17.28 19.68 21.92 24.73 26.76 31.2612 3.571 4.404 5.226 18.55 21.03 23.34 26.22 28.30 32.9113 4.107 5.009 5.892 19.81 22.36 24.74 27.69 29.82 34.5314 4.660 5.629 6.571 21.06 23.68 26.12 29.14 31.32 36.12

15 5.229 6.262 7.261 22.31 25.00 27.49 30.58 32.80 37.7016 5.812 6.908 7.962 23.54 26.30 28.85 32.00 34.27 39.2517 6.408 7.564 8.672 24.77 27.59 30.19 33.41 35.72 40.7918 7.015 8.231 9.390 25.99 28.87 31.53 34.81 37.16 42.3119 7.633 8.907 10.12 27.20 30.14 32.85 36.19 38.58 43.82

20 8.260 9.591 10.85 28.41 31.41 34.17 37.57 40.00 45.3121 8.897 10.28 11.59 29.62 32.67 35.48 38.93 41.40 46.8022 9.542 10.98 12.34 30.81 33.92 36.78 40.29 42.80 48.2723 10.20 11.69 13.09 32.01 35.17 38.08 41.64 44.18 49.7324 10.86 12.40 13.85 33.20 36.42 39.36 42.98 45.56 51.18

25 11.52 13.12 14.61 34.38 37.65 40.65 44.31 46.93 52.6230 14.95 16.79 18.49 40.26 43.77 46.98 50.89 53.67 59.7040 22.16 24.43 26.51 51.81 55.76 59.34 63.69 66.77 73.4050 29.71 32.36 34.76 63.17 67.50 71.42 76.15 79.49 86.6660 37.48 40.48 43.19 74.40 79.08 83.30 88.38 91.95 99.61

70 45.44 48.76 51.74 85.53 90.53 95.02 100.4 104.2 112.380 53.54 57.15 60.39 96.58 101.9 106.6 112.3 116.3 124.890 61.75 65.65 69.13 107.6 113.1 118.1 124.1 128.3 137.2

100 70.06 74.22 77.93 118.5 124.3 129.6 135.8 140.2 149.4

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WILCOXON SIGNED RANK TEST

P is the sum of the ranks corresponding to the positive differences,

Q is the sum of the ranks corresponding to the negative differences,

T is the smaller of P and Q.

For each value of n the table gives the largest value of T which will lead to rejection of the null hypothesis at the levelof significance indicated.

Critical values of T

Level of significance

One Tail 0.05 0.025 0.01 0.005Two Tail 0.10 0.05 0.02 0.01

n = 6 2 07 3 2 08 5 3 1 09 8 5 3 1

10 10 8 5 311 13 10 7 512 17 13 9 713 21 17 12 914 25 21 15 1215 30 25 19 1516 35 29 23 1917 41 34 27 2318 47 40 32 2719 53 46 37 3220 60 52 43 37

For larger values of n, each of P and Q can be approximated by the normal distribution with mean 14n(n + 1) and

variance 124

n(n + 1)(2n + 1).

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WILCOXON RANK SUM TEST

The two samples have sizes m and n, where m ≤ n.

Rm is the sum of the ranks of the items in the sample of size m.

W is the smaller of Rm and m(m + n + 1) − Rm.

For each pair of values of m and n, the table gives the largest value of W which will lead to rejection of the nullhypothesis at the level of significance indicated.

Critical values ofW

Level of significance

One Tail 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01Two Tail 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.02

n m = 3 m = 4 m = 5 m = 6

3 6 − −4 6 − − 11 10 −5 7 6 − 12 11 10 19 17 166 8 7 − 13 12 11 20 18 17 28 26 24

7 8 7 6 14 13 11 21 20 18 29 27 258 9 8 6 15 14 12 23 21 19 31 29 279 10 8 7 16 14 13 24 22 20 33 31 28

10 10 9 7 17 15 13 26 23 21 35 32 29

Level of significance

One Tail 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01 0.05 0.025 0.01Two Tail 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.02 0.1 0.05 0.02

n m = 7 m = 8 m = 9 m = 10

7 39 36 348 41 38 35 51 49 459 43 40 37 54 51 47 66 62 59

10 45 42 39 56 53 49 69 65 61 82 78 74

For larger values of m and n, the normal distribution with mean 12m(m + n + 1) and variance 1

12mn(m + n + 1) should

be used as an approximation to the distribution of Rm.

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SP (NF) T41021/4