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GCE Edexcel GCE in Mathematics Mathematical Formulae and Statistical Tables

For use in Ede

xcel Advanced Subsidiary GCE and Advanced GCE examinations
Core Mathematics C1 – C4 Further Pure Mathematics FP1 – FP3 Mechanics M1 – M5 Statistics S1 – S4
For use from January 2008

UA018598

TABLE OF CONTENTS Page 4 4 4 5 5 5 5 5 5 6 6 6 6 7 7 8 8 8 8 8 8 9 9 9 10 11 11 13 14 14 15 Core Mathematics C1 Mensuration Arithmetic series Core Mathematics C2 Cosine rule Binomial series Logarithms and exponentials Geometric series Numerical integration Core Mathematics C3 Logarithms and exponentials Trigonometric identities Differentiation Core Mathematics C4 Integration Further Pure Mathematics FP1 Summations Numerical solution of equations Coordinate geometry Conics Matrix transformations Further Pure Mathematics FP2 Area of sector Maclaurin’s and Taylor’s Series Taylor polynomials Further Pure Mathematics FP3 Vectors Hyperbolics Integration Arc length Surface area of revolution

M21521RRA – Edexcel AS/A level Mathematics Formulae List – Issue 1 – November 2004

16 16 16 16 16 16 16 16 17 17 17 17 17 18 18 18 18 19 20 21 22 22 22 23 28 29 29 29 29 29 30 31 32 33 33 34 35

Mechanics M1 There are no formulae given for M1 in addition to those candidates are expected to know. Mechanics M2 Centres of mass Mechanics M3 Motion in a circle Centres of mass Universal law of gravitation Mechanics M4 There are no formulae given for M4 in addition to those candidates are expected to know. Mechanics M5 Moments of inertia Moments as vectors Statistics S1 Probability Discrete distributions Continuous distributions Correlation and regression The Normal distribution function Percentage points of the Normal distribution Statistics S2 Discrete distributions Continuous distributions Binomial cumulative distribution function Poisson cumulative distribution function Statistics S3 Expectation algebra Sampling distributions Correlation and regression Non-parametric tests Percentage points of the ? 2 distribution Critical values for correlation coefficients Random numbers Statistics S4 Sampling distributions Percentage points of Student’s t distribution Percentage points of the F distribution

There are no formulae provided for Decision Mathematics units D1 and D2.

The formulae in this booklet have been arranged according to the unit in which they are first introduced. Thus a candidate sitting a unit may be required to use the formulae that were introduced in a preceding unit (e.g. candidates sitting C3 might be expected to use formulae first introduced in C1 or C2). It may also be the case that candidates sitting Mechanics and Statistics units need to use formulae introduced in appropriate Core Mathematics units, as outlined in the specification.

UA018598 – Edexcel AS/A level Mathematics Formulae List – Issue 1 – September 2007

3

Core Mathematics C1
Mensuration Surface area of sphere = 4? r 2 Area of curved surface of cone = ? r ? slant height

Arithmetic series un = a + (n – 1)d Sn =
1 1 n(a + l) = n[2a + (n ? 1)d] 2 2

4

UA018598 – Edexcel AS/A level Mathematics Formulae List: Core Mathematics C1 – Issue 1 – September 2007

Core Mathematics C2
Candidates sitting C2 may also require those formulae listed under Core Mathematics C1. Cosine rule a2 = b2 + c2 – 2bc cos A Binomial series

? n? ? n? ? n? (a ? b) n ? a n ? ? ? a n ?1b ? ? ? a n ? 2 b 2 ? ? ? ? ? a n ? r b r ? ? ? b n ? 1? ? 2? ?r? ? ? ? ? ? ? ? n? n! where ? ? ? n C r ? ?r? r! (n ? r )! ? ?
(1 ? x) n ? 1 ? nx ?

(n ? ?)

n(n ? 1) 2 n(n ? 1) ?(n ? r ? 1) r x ??? x ? ? ( x ? 1, n ??) 1? 2 1? 2 ? ?? r

Logarithms and exponentials

loga x ?

logb x logb a

Geometric series un = arn ? 1 Sn =

a (1 ? r n ) 1? r
a for ?r? < 1 1? r

S? =

Numerical integration

? The trapezium rule: ? y dx ? ?a

b

1 2

h{(y0 + yn) + 2(y1 + y2 + ... + yn – 1)}, where h ?

b?a n

UA018598 – Edexcel AS/A level Mathematics Formulae List: Core Mathematics C2 – Issue 1 – September 2007

5

Core Mathematics C3
Candidates sitting C3 may also require those formulae listed under Core Mathematics C1 and C2. Logarithms and exponentials
e x ln a ? a x

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 ? tan B ( A ? B ? (k ? 1 )? ) 2 1 ? tan A tan B A? B A? B sin A ? sin B ? 2 sin cos 2 2 A? B A? B sin A ? sin B ? 2 cos sin 2 2 A? B A? B cos A ? cos B ? 2 cos cos 2 2 A? B A? B cos A ? cos B ? ?2 sin sin 2 2 tan ( A ? B) ?

Differentiation f(x) tan kx sec x cot x cosec x f ?(x) k sec2 kx sec x tan x –cosec2 x –cosec x cot x

f(x) g( x)

f ?( x) g( x) ? f(x) g?( x) (g( x) )2

6

UA018598 – Edexcel AS/A level Mathematics Formulae List: Core Mathematics C3 – Issue 1 – September 2007

Core Mathematics C4
Candidates sitting C4 may also require those formulae listed under Core Mathematics C1, C2 and C3. Integration (+ constant) f(x) sec2 kx
tan x cot x

? ? f( x ) dx ?
1 tan kx k

ln sec x ln sin x
? ln cosecx ? cot x ? ln tan(1 x) 2

cosec x
sec x
? u dv dx ? uv ? ? v du dx ? ? ? dx ? dx

ln sec x ? tan x ? ln tan(1 x ? 1 ? ) 2 4

UA018598 – Edexcel AS/A level Mathematics Formulae List: Core Mathematics C4 – Issue 1 – September 2007

7

Further Pure Mathematics FP1
Candidates sitting FP1 may also require those formulae listed under Core Mathematics C1 and C2. Summations

?r
r ?1
n

n

2

? 1 n(n ? 1)(2n ? 1) 6
? 1 n 2 (n ? 1) 2 4

?r
r ?1

3

Numerical solution of equations The Newton-Raphson iteration for solving f(x) ? 0 : x n ?1 ? x n ?

f(x n ) f ?( x n )

Coordinate geometry The perpendicular distance from (h, k) to ax ? by ? c ? 0 is

ah ? bk ? c a2 ? b2 m1 ? m 2 1 ? m1 m 2

The acute angle between lines with gradients m1 and m2 is arctan

Conics Rectangular Hyperbola xy = c2
c? ? ? ct, ? t? ?

Parabola Standard Form Parametric Form Foci

y 2 ? 4ax

(at2, 2at)

(a, 0)

Not required

Directrices

x ? ?a

Not required

8

UA018598 – Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 – Issue 1 – September 2007

Matrix transformations

? cos? Anticlockwise rotation through ? about O: ? ? sin ? ? ? cos 2? Reflection in the line y ? (tan? ) x : ? ? sin 2? ?

? sin ? ? ? cos? ? ?

sin 2? ? ? ? cos 2? ? ?

UA018598 – Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP1 – Issue 1 – September 2007

9

Further Pure Mathematics FP2
Candidates sitting FP2 may also require those formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1–C4. Area of a sector A=
1? 2 ? r d? 2?

(polar coordinates)

Complex numbers
e i? ? cos? ? i sin ? {r (cos? ? i sin ? )}n ? r n (cosn? ? i sin n? )

The roots of z n ? 1 are given by z ? e Maclaurin’s and Taylor’s Series
f(x) ? f(0) ? x f ?(0) ?

2? k i n

, for k ? 0, 1, 2, ? , n ? 1

x2 x r (r ) f ??(0) ? ? ? f (0) ? ? 2! r! ( x ? a) 2 ( x ? a) r ( r ) f ??(a) ? ? ? f (a) ? ? 2! r!

f(x) ? f(a) ? ( x ? a) f ?(a) ? f(a ? x) ? f(a) ? x f ?(a) ? e x ? exp(x) ? 1 ? x ? ln (1 ? x) ? x ?

x2 x r (r ) f ??(a) ? ? ? f (a) ? ? 2! r! for all x (?1 ? x ? 1)

x2 xr ??? ?? 2! r!

x2 x3 xr ? ? ? ? (?1) r ?1 ?? 2 3 r

sin x ? x ?

x3 x5 x 2r ?1 ? ? ? ? (?1) r ?? 3! 5! (2r ? 1)!

for all x for all x
(?1 ? x ? 1)

2r x2 x4 r x cos x ? 1 ? ? ? ? ? (?1) ?? 2! 4! (2r )!

arctan x ? x ?

x3 x5 x 2 r ?1 ? ? ? ? (?1) r ?? 3 5 2r ? 1 h2 f ??(a) ? error 2! h2 f ??(a ? ? ) 2!

Taylor polynomials
f(a ? h) ? f(a) ? h f ?(a) ? f(a ? h) ? f(a) ? h f ?(a) ?

(0 ? ? ? h )

f(x) ? f(a) ? ( x ? a) f ?(a) ? f(x) ? f(a) ? ( x ? a) f ?(a) ?

( x ? a) 2 f ??(a) ? error 2! ( x ? a) 2 f ??(? ) 2! ( a ? ? ? x)

10

UA018598 – Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP2 – Issue 1 – September 2007

Further Pure Mathematics FP3
Candidates sitting FP3 may also require those formulae listed under Further Pure Mathematics FP1, and Core Mathematics C1–C4. Vectors

The resolved part of a in the direction of b is

a.b b

The point dividing AB in the ratio ? : ? is

?a ? ?b ???
j a2 b2 ? a 2 b3 ? a 3 b2 ? ? ? a 3 ? ? a 3 b1 ? a1b3 ? b3 ? a1b2 ? a 2 b1 ? ? ? k

i ? Vector product: a ? b ? a b sin ? n ? a1 b1 a1 a.(b ? c) ? b1 c1 a2 b2 c2 a3

b3 ? b.(c ? a) ? c.(a ? b) c3

a ? (b ? c) ? (a.c)b ? (a.b)c

If A is the point with position vector a ? a1i ? a2 j ? a3k and the direction vector b is given by b ? b1i ? b2 j ? b3k , then the straight line through A with direction vector b has cartesian equation x ? a1 y ? a 2 z ? a 3 ? ? (? ? ) b1 b2 b3 The plane through A with normal vector n ? n1i ? n2 j ? n3 k has cartesian equation

n1 x ? n2 y ? n3 z ? d ? 0 where d ? ?a.n
The plane through non-collinear points A, B and C has vector equation r ? a ? ? (b ? a) ? ? (c ? a) ? (1 ? ? ? ? )a ? ?b ? ?c 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 (? , ? , ? ) from n1 x ? n2 y ? n3 z ? d ? 0 is

n1? ? n 2 ? ? n3? ? d
2 2 n12 ? n 2 ? n3

.

UA018598 – Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 2007

11

Hyperbolic functions
cosh 2 x ? sinh 2 x ? 1 sinh 2 x ? 2 sinh x cosh x cosh 2 x ? cosh 2 x ? sinh 2 x
arcosh x ? ln{x ? x 2 ? 1} arsinh x ? ln{x ? x 2 ? 1} ( x ? 1)

?1? x ? artanh x ? 1 ln? ? 2 ?1? x ?

( x ? 1)

Conics Rectangular Hyperbola

Ellipse Standard Form Parametric Form Eccentricity
x2 y2 ? ?1 a2 b2
(a cos? , b sin ? )
e ?1 b ? a 2 (1 ? e 2 )
2

Parabola

Hyperbola
x2 y2 ? ?1 a2 b2

y ? 4ax
2

xy ? c 2
c? ? ? ct, ? t? ?

(at 2 , 2at)
e ?1

(a sec ?, b tan ? ) (?a cosh ?, b sinh ? )
e ?1 b ? a 2 (e 2 ?1)
2

e = ?2 (??2c, ??2c) x + y = ??2c
x ? 0, y ? 0

Foci

(?ae, 0)
a e

(a, 0)

(?ae, 0)
a e

Directrices

x??

x ? ?a

x??

Asymptotes

none

none

y x ?? a b

12

UA018598 – Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 2007

Differentiation f(x)
arcsin x

f ?(x)

1 1? x 2 1

arccosx
arctan x

sinh x cosh x tanh x

1? x 2 1 1? x2 cosh x sinh x
sech 2 x 1

?

arsinh x arcosh x

1? x2 1

artanh x

x2 ?1 1 1? x2

Integration (+ constant; a ? 0 where relevant) f(x)
sinh x cosh x tanh x
1 a2 ? x2
1 a ?x
2 2

? ? f( x ) dx ?
cosh x sinh x ln cosh x

? x? arcsin ? ? ?a? 1 ? x? arctan ? ? a ?a?

( x ? a)

1 x2 ? a2 1 a2 ? x2
1 a ?x
2 2

? x? arcosh ? ? ? ln{x ? x 2 ? a 2 } ?a? ? x? arsinh ? ? ? ln{x ? x 2 ? a 2 } ?a?
1 a?x 1 ? x? ln ? artanh ? ? 2a a ? x a ?a? 1 x?a ln 2a x?a

( x ? a)

( x ? a)

1 x ?a
2 2

UA018598 – Edexcel AS/A level Mathematics Formulae List – Issue 1 – September 2007

13

Arc length
2 ? ? dy ? s ? ? 1 ? ? ? dx ? dx ? ?

(cartesian coordinates)

? ? dx ? 2 ? dy ? 2 s ? ? ? ? ? ? ? dt ? ? dt ? ? dt ?

(parametric form)

Surface area of revolution
S x ? 2? ? y ds ?
? 2? ? ? ? ?

? dx ? ? dy ? y ? ? ? ? ? dt ? dt ? ? dt ?

2

2

14

UA018598 – Edexcel AS/A level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 2007

BLANK PAGE TURN OVER FOR MECHANICS & STATISTICS FORMULAE

UA018598 – Edexcel AS/A level Mathematics Formulae List – Issue 1 – September 2007

15

Mechanics M1
There are no formulae given for M1 in addition to those candidates are expected to know. Candidates sitting M1 may also require those formulae listed under Core Mathematics C1.

Mechanics M2
Candidates sitting M2 may also require those formulae listed under Core Mathematics C1, C2 and C3. Centres of mass For uniform bodies: Triangular lamina:
2 3

along median from vertex
r sin ?

Circular arc, radius r, angle at centre 2? :

from centre ? 2r sin ? Sector of circle, radius r, angle at centre 2? : from centre 3?

Mechanics M3
Candidates sitting M3 may also require those formulae listed under Mechanics M2, and also those formulae listed under Core Mathematics C1–C4. Motion in a circle
? Transverse velocity: v ? r? ? ? Transverse acceleration: v ? r??

v ? Radial acceleration: ? r? 2 ? ? r

2

Centres of mass For uniform bodies:
3 Solid hemisphere, radius r: 8 r from centre

Hemispherical shell, radius r:

1 2

r from centre
1 4

Solid cone or pyramid of height h:

h above the base on the line from centre of base to vertex

Conical shell of height h: 1 h above the base on the line from centre of base to vertex 3 Universal law of gravitation
Force ? Gm1 m2 d2

16

UA018598 – Edexcel AS/A level Mathematics Formulae List: Mechanics M1–M3 – Issue 1 – September 2007

Mechanics M4
There are no formulae given for M4 in addition to those candidates are expected to know. Candidates sitting M4 may also require those formulae listed under Mechanics M2 and M3, and also those formulae listed under Core Mathematics C1–C4 and Further Pure Mathematics FP1.

Mechanics M5
Candidates sitting M5 may also require those formulae listed under Mechanics M2 and M3, and also those formulae listed under Core Mathematics C1–C4 and Further Pure Mathematics FP1. Moments of inertia For uniform bodies of mass m: Thin rod, length 2l, about perpendicular axis through centre:
1 3

m l2
1 3

Rectangular lamina about axis in plane bisecting edges of length 2l: Thin rod, length 2l, about perpendicular axis through end:
4 3

m l2

m l2
4 3

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

m l2
1 3

Rectangular lamina, sides 2a and 2b, about perpendicular axis through centre: Hoop or cylindrical shell of radius r about axis through centre: m r2 Hoop of radius r about a diameter:
1 2

m(a 2 ? b 2 )

mr2
1 2

Disc or solid cylinder of radius r about axis through centre: Disc of radius r about a diameter:
1 4

mr2

mr2
2 5

Solid sphere, radius r, about diameter:

mr2
2 3

Spherical shell of radius r about a diameter: Parallel axes theorem: I A ? I G ? m(AG) 2 Perpendicular axes theorem: I z ? I x ? I y Moments as vectors The moment about O of F acting at r is r ? F

mr2

(for a lamina in the x-y plane)

UA018598 – Edexcel AS/A level Mathematics Formulae List: Mechanics M4–M5 – Issue 1 – September 2007

17

Statistics S1
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?)

Discrete distributions For a discrete random variable X taking values x i with probabilities P(X = xi) Expectation (mean): E(X) = ? = ?xi P(X = xi) Variance: Var(X) = ? 2 = ?(xi – ? )2 P(X = xi) = ? xi2 P(X = xi) – ? 2 For a function g( X ) : E(g(X)) = ?g(xi) P(X = xi)

Continuous distributions Standard continuous distribution: Distribution of X Normal N(? , ? )
2

P.D.F.
1 e
2 ? x?? ? ?1? ? 2? ? ?

Mean

Variance

? 2?

?

?2

18

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S1 – Issue 1 – September 2007

Correlation and regression For a set of n pairs of values ( xi , yi )
S xx ? ?( xi ? x ) ? ?x ?
2 2 i

(?xi ) 2 n (?yi ) 2 n
(?xi )(?yi ) n

S yy ? ?( yi ? y ) ? ?y ?
2 2 i

S xy ? ?( xi ? x )( yi ? y ) ? ?xi yi ?

The product moment correlation coefficient is
r? S xy S xx S yy ? ?( xi ? x )( yi ? y ) ? ?xi yi ? ? 2 (?xi ) ? ?x ? ? i n ? (?xi )(?yi ) n ? ? 2 (?yi ) 2 ? ? ? ?y ? ? ?? i n ? ?? ?

{?( xi ? x ) }{?( yi ? y ) }
2 2

2

The regression coefficient of y on x is b ?

S xy S xx

?

?( xi ? x )( yi ? y ) ?( x i ? x ) 2

Least squares regression line of y on x is y ? a ? bx where a ? y ? bx

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S1 – Issue 1 – September 2007

19

THE NORMAL DISTRIBUTION FUNCTION The function tabulated below is ?(z), defined as ?(z) =
z
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50

1 ? ? 12 t 2 dt . ? e 2? ???
z
1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00

z

?(z)
0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.6915

z
0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00

?(z)
0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 0.8413

z
1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50

?(z)
0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 0.9332

?(z)
0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 0.9772

z
2.00 2.02 2.04 2.06 2.08 2.10 2.12 2.14 2.16 2.18 2.20 2.22 2.24 2.26 2.28 2.30 2.32 2.34 2.36 2.38 2.40 2.42 2.44 2.46 2.48 2.50 2.55 2.60 2.65 2.70 2.75 2.80 2.85 2.90 2.95 3.00 3.05 3.10 3.15 3.20 3.25 3.30 3.35 3.40 3.50 3.60 3.70 3.80 3.90 4.00

?(z)
0.9772 0.9783 0.9793 0.9803 0.9812 0.9821 0.9830 0.9838 0.9846 0.9854 0.9861 0.9868 0.9875 0.9881 0.9887 0.9893 0.9898 0.9904 0.9909 0.9913 0.9918 0.9922 0.9927 0.9931 0.9934 0.9938 0.9946 0.9953 0.9960 0.9965 0.9970 0.9974 0.9978 0.9981 0.9984 0.9987 0.9989 0.9990 0.9992 0.9993 0.9994 0.9995 0.9996 0.9997 0.9998 0.9998 0.9999 0.9999 1.0000 1.0000

20

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S1 – Issue 1 – September 2007

PERCENTAGE POINTS OF THE NORMAL DISTRIBUTION The values z in the table are those which a random variable Z ? N(0, 1) exceeds with probability p; that is, P(Z > z) = 1 ? ?(z) = p. p
0.5000 0.4000 0.3000 0.2000 0.1500 0.1000

z
0.0000 0.2533 0.5244 0.8416 1.0364 1.2816

p
0.0500 0.0250 0.0100 0.0050 0.0010 0.0005

z
1.6449 1.9600 2.3263 2.5758 3.0902 3.2905

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S1 – Issue 1 – September 2007

21

Statistics S2
Candidates sitting S2 may also require those formulae listed under Statistics S1, and also those listed under Core Mathematics C1 and C2. Discrete distributions Standard discrete distributions: Distribution of X Binomial B(n, p) Poisson Po(? )
P(X ? x)

Mean
np

Variance
np(1 ? p)

? n? x ? ? p (1 ? p) n ? x ? x? ? ?
e ??

?x
x!

?

?

Continuous distributions For a continuous random variable X having probability density function f Expectation (mean): E( X ) ? ? ? ? x f( x) dx Variance: Var( X ) ? ? 2 ? ? ( x ? ? ) 2 f( x) dx ? ? x 2 f( x) dx ? ? 2 For a function g( X ) : E(g( X )) ? ? g( x) f( x) dx

?0 Cumulative distribution function: F( x0 ) ? P(X ? x0 ) ? ? f (t ) dt ??? Standard continuous distribution:
Distribution of X Uniform (Rectangular) on [a, b] P.D.F.
1 b?a
1 2

x

Mean

Variance
1 12

(a ? b)

(b ? a) 2

22

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2007

BINOMIAL CUMULATIVE DISTRIBUTION FUNCTION The tabulated value is P(X ? x), where X has a binomial distribution with index n and parameter p.
p= n = 5, x = 0 1 2 3 4 n = 6, x = 0 1 2 3 4 5 n = 7, x = 0 1 2 3 4 5 6 n = 8, x = 0 1 2 3 4 5 6 7 n = 9, x = 0 1 2 3 4 5 6 7 8 n = 10, x = 0 1 2 3 4 5 6 7 8 9 0.05 0.7738 0.9774 0.9988 1.0000 1.0000 0.7351 0.9672 0.9978 0.9999 1.0000 1.0000 0.6983 0.9556 0.9962 0.9998 1.0000 1.0000 1.0000 0.6634 0.9428 0.9942 0.9996 1.0000 1.0000 1.0000 1.0000 0.6302 0.9288 0.9916 0.9994 1.0000 1.0000 1.0000 1.0000 1.0000 0.5987 0.9139 0.9885 0.9990 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 0.10 0.5905 0.9185 0.9914 0.9995 1.0000 0.5314 0.8857 0.9842 0.9987 0.9999 1.0000 0.4783 0.8503 0.9743 0.9973 0.9998 1.0000 1.0000 0.4305 0.8131 0.9619 0.9950 0.9996 1.0000 1.0000 1.0000 0.3874 0.7748 0.9470 0.9917 0.9991 0.9999 1.0000 1.0000 1.0000 0.3487 0.7361 0.9298 0.9872 0.9984 0.9999 1.0000 1.0000 1.0000 1.0000 0.15 0.4437 0.8352 0.9734 0.9978 0.9999 0.3771 0.7765 0.9527 0.9941 0.9996 1.0000 0.3206 0.7166 0.9262 0.9879 0.9988 0.9999 1.0000 0.2725 0.6572 0.8948 0.9786 0.9971 0.9998 1.0000 1.0000 0.2316 0.5995 0.8591 0.9661 0.9944 0.9994 1.0000 1.0000 1.0000 0.1969 0.5443 0.8202 0.9500 0.9901 0.9986 0.9999 1.0000 1.0000 1.0000 0.20 0.3277 0.7373 0.9421 0.9933 0.9997 0.2621 0.6554 0.9011 0.9830 0.9984 0.9999 0.2097 0.5767 0.8520 0.9667 0.9953 0.9996 1.0000 0.1678 0.5033 0.7969 0.9437 0.9896 0.9988 0.9999 1.0000 0.1342 0.4362 0.7382 0.9144 0.9804 0.9969 0.9997 1.0000 1.0000 0.1074 0.3758 0.6778 0.8791 0.9672 0.9936 0.9991 0.9999 1.0000 1.0000 0.25 0.2373 0.6328 0.8965 0.9844 0.9990 0.1780 0.5339 0.8306 0.9624 0.9954 0.9998 0.1335 0.4449 0.7564 0.9294 0.9871 0.9987 0.9999 0.1001 0.3671 0.6785 0.8862 0.9727 0.9958 0.9996 1.0000 0.0751 0.3003 0.6007 0.8343 0.9511 0.9900 0.9987 0.9999 1.0000 0.0563 0.2440 0.5256 0.7759 0.9219 0.9803 0.9965 0.9996 1.0000 1.0000 0.30 0.1681 0.5282 0.8369 0.9692 0.9976 0.1176 0.4202 0.7443 0.9295 0.9891 0.9993 0.0824 0.3294 0.6471 0.8740 0.9712 0.9962 0.9998 0.0576 0.2553 0.5518 0.8059 0.9420 0.9887 0.9987 0.9999 0.0404 0.1960 0.4628 0.7297 0.9012 0.9747 0.9957 0.9996 1.0000 0.0282 0.1493 0.3828 0.6496 0.8497 0.9527 0.9894 0.9984 0.9999 1.0000 0.35 0.1160 0.4284 0.7648 0.9460 0.9947 0.0754 0.3191 0.6471 0.8826 0.9777 0.9982 0.0490 0.2338 0.5323 0.8002 0.9444 0.9910 0.9994 0.0319 0.1691 0.4278 0.7064 0.8939 0.9747 0.9964 0.9998 0.0207 0.1211 0.3373 0.6089 0.8283 0.9464 0.9888 0.9986 0.9999 0.0135 0.0860 0.2616 0.5138 0.7515 0.9051 0.9740 0.9952 0.9995 1.0000 0.40 0.0778 0.3370 0.6826 0.9130 0.9898 0.0467 0.2333 0.5443 0.8208 0.9590 0.9959 0.0280 0.1586 0.4199 0.7102 0.9037 0.9812 0.9984 0.0168 0.1064 0.3154 0.5941 0.8263 0.9502 0.9915 0.9993 0.0101 0.0705 0.2318 0.4826 0.7334 0.9006 0.9750 0.9962 0.9997 0.0060 0.0464 0.1673 0.3823 0.6331 0.8338 0.9452 0.9877 0.9983 0.9999 0.45 0.0503 0.2562 0.5931 0.8688 0.9815 0.0277 0.1636 0.4415 0.7447 0.9308 0.9917 0.0152 0.1024 0.3164 0.6083 0.8471 0.9643 0.9963 0.0084 0.0632 0.2201 0.4770 0.7396 0.9115 0.9819 0.9983 0.0046 0.0385 0.1495 0.3614 0.6214 0.8342 0.9502 0.9909 0.9992 0.0025 0.0233 0.0996 0.2660 0.5044 0.7384 0.8980 0.9726 0.9955 0.9997 0.50 0.0312 0.1875 0.5000 0.8125 0.9688 0.0156 0.1094 0.3438 0.6563 0.8906 0.9844 0.0078 0.0625 0.2266 0.5000 0.7734 0.9375 0.9922 0.0039 0.0352 0.1445 0.3633 0.6367 0.8555 0.9648 0.9961 0.0020 0.0195 0.0898 0.2539 0.5000 0.7461 0.9102 0.9805 0.9980 0.0010 0.0107 0.0547 0.1719 0.3770 0.6230 0.8281 0.9453 0.9893 0.9990

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2007

23

p= n = 12, x = 0 1 2 3 4 5 6 7 8 9 10 11 n = 15, x = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 n = 20, x = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.05 0.5404 0.8816 0.9804 0.9978 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.4633 0.8290 0.9638 0.9945 0.9994 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.3585 0.7358 0.9245 0.9841 0.9974 0.9997 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

0.10 0.2824 0.6590 0.8891 0.9744 0.9957 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 0.2059 0.5490 0.8159 0.9444 0.9873 0.9978 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.1216 0.3917 0.6769 0.8670 0.9568 0.9887 0.9976 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.15 0.1422 0.4435 0.7358 0.9078 0.9761 0.9954 0.9993 0.9999 1.0000 1.0000 1.0000 1.0000 0.0874 0.3186 0.6042 0.8227 0.9383 0.9832 0.9964 0.9994 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0388 0.1756 0.4049 0.6477 0.8298 0.9327 0.9781 0.9941 0.9987 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.20 0.0687 0.2749 0.5583 0.7946 0.9274 0.9806 0.9961 0.9994 0.9999 1.0000 1.0000 1.0000 0.0352 0.1671 0.3980 0.6482 0.8358 0.9389 0.9819 0.9958 0.9992 0.9999 1.0000 1.0000 1.0000 1.0000 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 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.25 0.0317 0.1584 0.3907 0.6488 0.8424 0.9456 0.9857 0.9972 0.9996 1.0000 1.0000 1.0000 0.0134 0.0802 0.2361 0.4613 0.6865 0.8516 0.9434 0.9827 0.9958 0.9992 0.9999 1.0000 1.0000 1.0000 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 1.0000 1.0000 1.0000 1.0000 1.0000

0.30 0.0138 0.0850 0.2528 0.4925 0.7237 0.8822 0.9614 0.9905 0.9983 0.9998 1.0000 1.0000 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 1.0000 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 1.0000 1.0000 1.0000 1.0000

0.35 0.0057 0.0424 0.1513 0.3467 0.5833 0.7873 0.9154 0.9745 0.9944 0.9992 0.9999 1.0000 0.0016 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 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 0.9997 1.0000 1.0000 1.0000 1.0000

0.40 0.0022 0.0196 0.0834 0.2253 0.4382 0.6652 0.8418 0.9427 0.9847 0.9972 0.9997 1.0000 0.0005 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 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 1.0000 1.0000

0.45 0.0008 0.0083 0.0421 0.1345 0.3044 0.5269 0.7393 0.8883 0.9644 0.9921 0.9989 0.9999 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 0.0000 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 1.0000

0.50 0.0002 0.0032 0.0193 0.0730 0.1938 0.3872 0.6128 0.8062 0.9270 0.9807 0.9968 0.9998 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 0.0000 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

24

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2007

p= n = 25, x = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 n = 30, x = 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

0.05 0.2774 0.6424 0.8729 0.9659 0.9928 0.9988 0.9998 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 0.2146 0.5535 0.8122 0.9392 0.9844 0.9967 0.9994 0.9999 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

0.10 0.0718 0.2712 0.5371 0.7636 0.9020 0.9666 0.9905 0.9977 0.9995 0.9999 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 0.0424 0.1837 0.4114 0.6474 0.8245 0.9268 0.9742 0.9922 0.9980 0.9995 0.9999 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

0.15 0.0172 0.0931 0.2537 0.4711 0.6821 0.8385 0.9305 0.9745 0.9920 0.9979 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0076 0.0480 0.1514 0.3217 0.5245 0.7106 0.8474 0.9302 0.9722 0.9903 0.9971 0.9992 0.9998 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

0.20 0.0038 0.0274 0.0982 0.2340 0.4207 0.6167 0.7800 0.8909 0.9532 0.9827 0.9944 0.9985 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0012 0.0105 0.0442 0.1227 0.2552 0.4275 0.6070 0.7608 0.8713 0.9389 0.9744 0.9905 0.9969 0.9991 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.25 0.0008 0.0070 0.0321 0.0962 0.2137 0.3783 0.5611 0.7265 0.8506 0.9287 0.9703 0.9893 0.9966 0.9991 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0002 0.0020 0.0106 0.0374 0.0979 0.2026 0.3481 0.5143 0.6736 0.8034 0.8943 0.9493 0.9784 0.9918 0.9973 0.9992 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.30 0.0001 0.0016 0.0090 0.0332 0.0905 0.1935 0.3407 0.5118 0.6769 0.8106 0.9022 0.9558 0.9825 0.9940 0.9982 0.9995 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0003 0.0021 0.0093 0.0302 0.0766 0.1595 0.2814 0.4315 0.5888 0.7304 0.8407 0.9155 0.9599 0.9831 0.9936 0.9979 0.9994 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.35 0.0000 0.0003 0.0021 0.0097 0.0320 0.0826 0.1734 0.3061 0.4668 0.6303 0.7712 0.8746 0.9396 0.9745 0.9907 0.9971 0.9992 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 0.0000 0.0000 0.0003 0.0019 0.0075 0.0233 0.0586 0.1238 0.2247 0.3575 0.5078 0.6548 0.7802 0.8737 0.9348 0.9699 0.9876 0.9955 0.9986 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000

0.40 0.0000 0.0001 0.0004 0.0024 0.0095 0.0294 0.0736 0.1536 0.2735 0.4246 0.5858 0.7323 0.8462 0.9222 0.9656 0.9868 0.9957 0.9988 0.9997 0.9999 1.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0003 0.0015 0.0057 0.0172 0.0435 0.0940 0.1763 0.2915 0.4311 0.5785 0.7145 0.8246 0.9029 0.9519 0.9788 0.9917 0.9971 0.9991 0.9998 1.0000 1.0000 1.0000 1.0000

0.45 0.0000 0.0000 0.0001 0.0005 0.0023 0.0086 0.0258 0.0639 0.1340 0.2424 0.3843 0.5426 0.6937 0.8173 0.9040 0.9560 0.9826 0.9942 0.9984 0.9996 0.9999 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0011 0.0040 0.0121 0.0312 0.0694 0.1350 0.2327 0.3592 0.5025 0.6448 0.7691 0.8644 0.9286 0.9666 0.9862 0.9950 0.9984 0.9996 0.9999 1.0000 1.0000

0.50 0.0000 0.0000 0.0000 0.0001 0.0005 0.0020 0.0073 0.0216 0.0539 0.1148 0.2122 0.3450 0.5000 0.6550 0.7878 0.8852 0.9461 0.9784 0.9927 0.9980 0.9995 0.9999 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0007 0.0026 0.0081 0.0214 0.0494 0.1002 0.1808 0.2923 0.4278 0.5722 0.7077 0.8192 0.8998 0.9506 0.9786 0.9919 0.9974 0.9993 0.9998 1.0000

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2007

25

p= n = 40, x = 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 27 28 29 30 31 32

0.05 0.1285 0.3991 0.6767 0.8619 0.9520 0.9861 0.9966 0.9993 0.9999 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 1.0000

0.10 0.0148 0.0805 0.2228 0.4231 0.6290 0.7937 0.9005 0.9581 0.9845 0.9949 0.9985 0.9996 0.9999 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

0.15 0.0015 0.0121 0.0486 0.1302 0.2633 0.4325 0.6067 0.7559 0.8646 0.9328 0.9701 0.9880 0.9957 0.9986 0.9996 0.9999 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

0.20 0.0001 0.0015 0.0079 0.0285 0.0759 0.1613 0.2859 0.4371 0.5931 0.7318 0.8392 0.9125 0.9568 0.9806 0.9921 0.9971 0.9990 0.9997 0.9999 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

0.25 0.0000 0.0001 0.0010 0.0047 0.0160 0.0433 0.0962 0.1820 0.2998 0.4395 0.5839 0.7151 0.8209 0.8968 0.9456 0.9738 0.9884 0.9953 0.9983 0.9994 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.30 0.0000 0.0000 0.0001 0.0006 0.0026 0.0086 0.0238 0.0553 0.1110 0.1959 0.3087 0.4406 0.5772 0.7032 0.8074 0.8849 0.9367 0.9680 0.9852 0.9937 0.9976 0.9991 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.35 0.0000 0.0000 0.0000 0.0001 0.0003 0.0013 0.0044 0.0124 0.0303 0.0644 0.1215 0.2053 0.3143 0.4408 0.5721 0.6946 0.7978 0.8761 0.9301 0.9637 0.9827 0.9925 0.9970 0.9989 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.40 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0006 0.0021 0.0061 0.0156 0.0352 0.0709 0.1285 0.2112 0.3174 0.4402 0.5681 0.6885 0.7911 0.8702 0.9256 0.9608 0.9811 0.9917 0.9966 0.9988 0.9996 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000

0.45 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0009 0.0027 0.0074 0.0179 0.0386 0.0751 0.1326 0.2142 0.3185 0.4391 0.5651 0.6844 0.7870 0.8669 0.9233 0.9595 0.9804 0.9914 0.9966 0.9988 0.9996 0.9999 1.0000 1.0000 1.0000

0.50 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0003 0.0011 0.0032 0.0083 0.0192 0.0403 0.0769 0.1341 0.2148 0.3179 0.4373 0.5627 0.6821 0.7852 0.8659 0.9231 0.9597 0.9808 0.9917 0.9968 0.9989 0.9997 0.9999 1.0000

26

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2007

p= n = 50, x = 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 27 28 29 30 31 32 33 34 35 36 37 38

0.05 0.0769 0.2794 0.5405 0.7604 0.8964 0.9622 0.9882 0.9968 0.9992 0.9998 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 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.10 0.0052 0.0338 0.1117 0.2503 0.4312 0.6161 0.7702 0.8779 0.9421 0.9755 0.9906 0.9968 0.9990 0.9997 0.9999 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 1.0000

0.15 0.0003 0.0029 0.0142 0.0460 0.1121 0.2194 0.3613 0.5188 0.6681 0.7911 0.8801 0.9372 0.9699 0.9868 0.9947 0.9981 0.9993 0.9998 0.9999 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

0.20 0.0000 0.0002 0.0013 0.0057 0.0185 0.0480 0.1034 0.1904 0.3073 0.4437 0.5836 0.7107 0.8139 0.8894 0.9393 0.9692 0.9856 0.9937 0.9975 0.9991 0.9997 0.9999 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

0.25 0.0000 0.0000 0.0001 0.0005 0.0021 0.0070 0.0194 0.0453 0.0916 0.1637 0.2622 0.3816 0.5110 0.6370 0.7481 0.8369 0.9017 0.9449 0.9713 0.9861 0.9937 0.9974 0.9990 0.9996 0.9999 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

0.30 0.0000 0.0000 0.0000 0.0000 0.0002 0.0007 0.0025 0.0073 0.0183 0.0402 0.0789 0.1390 0.2229 0.3279 0.4468 0.5692 0.6839 0.7822 0.8594 0.9152 0.9522 0.9749 0.9877 0.9944 0.9976 0.9991 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.35 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0008 0.0025 0.0067 0.0160 0.0342 0.0661 0.1163 0.1878 0.2801 0.3889 0.5060 0.6216 0.7264 0.8139 0.8813 0.9290 0.9604 0.9793 0.9900 0.9955 0.9981 0.9993 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.40 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0008 0.0022 0.0057 0.0133 0.0280 0.0540 0.0955 0.1561 0.2369 0.3356 0.4465 0.5610 0.6701 0.7660 0.8438 0.9022 0.9427 0.9686 0.9840 0.9924 0.9966 0.9986 0.9995 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000

0.45 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0002 0.0006 0.0018 0.0045 0.0104 0.0220 0.0427 0.0765 0.1273 0.1974 0.2862 0.3900 0.5019 0.6134 0.7160 0.8034 0.8721 0.9220 0.9556 0.9765 0.9884 0.9947 0.9978 0.9991 0.9997 0.9999 1.0000 1.0000 1.0000

0.50 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0005 0.0013 0.0033 0.0077 0.0164 0.0325 0.0595 0.1013 0.1611 0.2399 0.3359 0.4439 0.5561 0.6641 0.7601 0.8389 0.8987 0.9405 0.9675 0.9836 0.9923 0.9967 0.9987 0.9995 0.9998 1.0000

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2007

27

POISSON CUMULATIVE DISTRIBUTION FUNCTION The tabulated value is P(X ? x), where X has a Poisson distribution with parameter ?.
?= x=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 ?= x=0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
0.5 0.6065 0.9098 0.9856 0.9982 0.9998 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 5.5 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 1.0000 1.0000 1.0000 1.0000 1.0000 1.0 0.3679 0.7358 0.9197 0.9810 0.9963 0.9994 0.9999 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 6.0 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 1.0000 1.0000 1.0000 1.0000 1.5 0.2231 0.5578 0.8088 0.9344 0.9814 0.9955 0.9991 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 6.5 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 1.0000 1.0000 1.0000 2.0 0.1353 0.4060 0.6767 0.8571 0.9473 0.9834 0.9955 0.9989 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 7.0 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 1.0000 1.0000 1.0000 2.5 0.0821 0.2873 0.5438 0.7576 0.8912 0.9580 0.9858 0.9958 0.9989 0.9997 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 7.5 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 1.0000 1.0000 3.0 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 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 8.0 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 1.0000 3.5 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 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 8.5 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 4.0 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 1.0000 1.0000 1.0000 1.0000 1.0000 9.0 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 4.5 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 1.0000 1.0000 1.0000 1.0000 9.5 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 5.0 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 1.0000 1.0000 1.0000 10.0 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

28

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2007

Statistics S3
Candidates sitting S3 may also require those formulae listed under Statistics S1 and S2. Expectation algebra For independent random variables X and Y
E( XY ) ? E( X ) E(Y ) , Var(aX ? bY) ? a 2 Var( X ) ? b 2 Var(Y )

Sampling distributions For a random sample X 1 , X 2 , ?, X n of n independent observations from a distribution having mean ? and variance ? 2
X is an unbiased estimator of ? , with Var( X ) ?

?2
n

S is an unbiased estimator of ? , where S ?
2 2

2

?( X i ? X ) 2 n ?1

For a random sample of n observations from N(? , ? 2 )
X ??

?/ n

~ N(0, 1)

2 For a random sample of n x observations from N(? x , ? x ) and, independently, a random

2 sample of n y observations from N(? y , ? y )

( X ? Y ) ? (? x ? ? y )
2 ?x

nx

?

2 ?y

~ N(0, 1)

ny

Correlation and regression

Spearman’s rank correlation coefficient is rs ? 1 ?

6?d 2 n(n 2 ? 1)

Non-parametric tests Goodness-of-fit test and contingency tables:

?

(Oi ? Ei ) 2 Ei

~ ??2

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2007

29

PERCENTAGE POINTS OF THE ? 2 DISTRIBUTION The values in the table are those which a random variable with the ? 2 distribution on ? degrees of freedom exceeds with the probability shown.
?
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 0.995 0.000 0.010 0.072 0.207 0.412 0.676 0.989 1.344 1.735 2.156 2.603 3.074 3.565 4.075 4.601 5.142 5.697 6.265 6.844 7.434 8.034 8.643 9.260 9.886 10.520 11.160 11.808 12.461 13.121 13.787 0.990 0.000 0.020 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 10.196 10.856 11.524 12.198 12.879 13.565 14.256 14.953 0.975 0.001 0.051 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 10.283 10.982 11.689 12.401 13.120 13.844 14.573 15.308 16.047 16.791 0.950 0.004 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 10.117 10.851 11.591 12.338 13.091 13.848 14.611 15.379 16.151 16.928 17.708 18.493 0.900 0.016 0.211 0.584 1.064 1.610 2.204 2.833 3.490 4.168 4.865 5.580 6.304 7.042 7.790 8.547 9.312 10.085 10.865 11.651 12.443 13.240 14.042 14.848 15.659 16.473 17.292 18.114 18.939 19.768 20.599 0.100 2.705 4.605 6.251 7.779 9.236 10.645 12.017 13.362 14.684 15.987 17.275 18.549 19.812 21.064 22.307 23.542 24.769 25.989 27.204 28.412 29.615 30.813 32.007 33.196 34.382 35.563 36.741 37.916 39.088 40.256 0.050 3.841 5.991 7.815 9.488 11.070 12.592 14.067 15.507 16.919 18.307 19.675 21.026 22.362 23.685 24.996 26.296 27.587 28.869 30.144 31.410 32.671 33.924 35.172 36.415 37.652 38.885 40.113 41.337 42.557 43.773 0.025 5.024 7.378 9.348 11.143 12.832 14.449 16.013 17.535 19.023 20.483 21.920 23.337 24.736 26.119 27.488 28.845 30.191 31.526 32.852 34.170 35.479 36.781 38.076 39.364 40.646 41.923 43.194 44.461 45.722 46.979 0.010 6.635 9.210 11.345 13.277 15.086 16.812 18.475 20.090 21.666 23.209 24.725 26.217 27.688 29.141 30.578 32.000 33.409 34.805 36.191 37.566 38.932 40.289 41.638 42.980 44.314 45.642 46.963 48.278 49.588 50.892 0.005 7.879 10.597 12.838 14.860 16.750 18.548 20.278 21.955 23.589 25.188 26.757 28.300 29.819 31.319 32.801 34.267 35.718 37.156 38.582 39.997 41.401 42.796 44.181 45.558 46.928 48.290 49.645 50.993 52.336 53.672

30

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2007

CRITICAL VALUES FOR CORRELATION COEFFICIENTS These tables concern tests of the hypothesis that a population correlation coefficient ? is 0. The values in the tables are the minimum values which need to be reached by a sample correlation coefficient in order to be significant at the level shown, on a one-tailed test.
Product Moment Coefficient Level 0.05 0.025 0.01 0.9000 0.9500 0.9800 0.8054 0.8783 0.9343 0.7293 0.8114 0.8822 0.6694 0.7545 0.8329 0.6215 0.7067 0.7887 0.5822 0.6664 0.7498 0.5494 0.6319 0.7155 0.5214 0.6021 0.6851 0.4973 0.5760 0.6581 0.4762 0.5529 0.6339 0.4575 0.5324 0.6120 0.4409 0.5140 0.5923 0.4259 0.4973 0.5742 0.4124 0.4821 0.5577 0.4000 0.4683 0.5425 0.3887 0.4555 0.5285 0.3783 0.4438 0.5155 0.3687 0.4329 0.5034 0.3598 0.4227 0.4921 0.3515 0.4133 0.4815 0.3438 0.4044 0.4716 0.3365 0.3961 0.4622 0.3297 0.3882 0.4534 0.3233 0.3809 0.4451 0.3172 0.3739 0.4372 0.3115 0.3673 0.4297 0.3061 0.3610 0.4226 0.2638 0.3120 0.3665 0.2353 0.2787 0.3281 0.2144 0.2542 0.2997 0.1982 0.2352 0.2776 0.1852 0.2199 0.2597 0.1745 0.2072 0.2449 0.1654 0.1966 0.2324 Sample Level 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 40 50 60 70 80 90 100 Spearman’s Coefficient Level 0.05 0.025 0.01 1.0000 0.9000 1.0000 1.0000 0.8286 0.8857 0.9429 0.7143 0.7857 0.8929 0.6429 0.7381 0.8333 0.6000 0.7000 0.7833 0.5636 0.6485 0.7455 0.5364 0.6182 0.7091 0.5035 0.5874 0.6783 0.4835 0.5604 0.6484 0.4637 0.5385 0.6264 0.4464 0.5214 0.6036 0.4294 0.5029 0.5824 0.4142 0.4877 0.5662 0.4014 0.4716 0.5501 0.3912 0.4596 0.5351 0.3805 0.4466 0.5218 0.3701 0.4364 0.5091 0.3608 0.4252 0.4975 0.3528 0.4160 0.4862 0.3443 0.4070 0.4757 0.3369 0.3977 0.4662 0.3306 0.3901 0.4571 0.3242 0.3828 0.4487 0.3180 0.3755 0.4401 0.3118 0.3685 0.4325 0.3063 0.3624 0.4251 0.2640 0.3128 0.3681 0.2353 0.2791 0.3293 0.2144 0.2545 0.3005 0.1982 0.2354 0.2782 0.1852 0.2201 0.2602 0.1745 0.2074 0.2453 0.1654 0.1967 0.2327

0.10 0.8000 0.6870 0.6084 0.5509 0.5067 0.4716 0.4428 0.4187 0.3981 0.3802 0.3646 0.3507 0.3383 0.3271 0.3170 0.3077 0.2992 0.2914 0.2841 0.2774 0.2711 0.2653 0.2598 0.2546 0.2497 0.2451 0.2407 0.2070 0.1843 0.1678 0.1550 0.1448 0.1364 0.1292

0.005 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 0.4026 0.3610 0.3301 0.3060 0.2864 0.2702 0.2565

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2007

31

RANDOM NUMBERS
86 60 78 80 99 56 66 31 98 50 90 31 22 06 08 86 94 63 11 01 13 78 48 56 09 32 02 77 79 97 51 99 96 84 64 87 44 25 22 70 84 48 06 90 39 32 49 53 72 92 94 52 23 55 89 62 97 55 83 10 10 12 37 79 25 72 93 94 43 15 50 24 34 41 30 43 13 14 98 83 07 99 82 66 66 91 97 05 14 10 12 13 46 27 25 15 77 66 15 94 30 47 26 94 31 65 44 93 76 01 48 43 12 06 25 11 04 47 21 71 39 09 01 18 70 97 99 56 54 57 88 27 67 74 71 76 35 99 18 13 05 46 06 40 56 36 15 14 77 01 95 88 11 59 35 49 02 90 57 67 97 91 64 97 30 56 56 71 66 87 09 11 48 14 33 79 12 02 53 11 96 33 65 79 15 61 86 23 29 33 34 39 06 29 31 13 76 90 42 12 88 17 94 93 52 12 80 60 84 73 09 41 99 20 04 78 60 83 91 36 07 21 41 20 17 79 57 46 09 17 30 65 24 14 56 80 91 43 91 54 37 03 17 41 87 95 11 05 88 70 22 00 14 45 12 93 93 16 26 53 26 94 26 51 55 17 78 33 56 18 27 84 83 75 67 89 40 01 52 32 04 79 74 25 31 57 40 23 75 40 25 13 78 31 03 09 81 19 89 90 89 00 66 04 11 16 23 72 86 21 56 06 37 16 74 57 06 69 13 43 13 08 61 20 10 53 58 93 41 24 40 31 65 05 07 07 85 11 86 79 48 50 93 71 68 58 40 10 67 20 76 79 73 41 16 14 85 78 00 01 19 40 24 76 98 96 86 81 04 66 01 74 39 22 49 40 72 87 47 95 20 16 97 04 23 36 14 23 42 62 59 97 47 96 32 74 84 16 61 15

32

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2007

Statistics S4
Candidates sitting S4 may also require those formulae listed under Statistics S1, S2 and S3.

Sampling distributions For a random sample of n observations from N(? , ? 2 )
(n ? 1) S 2

?

2

2 ~ ? n?1

X ?? S/ n

~ t n ?1

(also valid in matched-pairs situations)

2 For a random sample of n x observations from N(? x , ? x ) and, independently, a random

2 sample of n y observations from N(? y , ? y )
2 2 Sx /? x 2 2 S y /? y

~ Fn

x ?1, n y ?1

2 2 If ? x ? ? y ? ? 2 (unknown) then

( X ? Y ) ? (? x ? ? y ) ? 1 1 ? 2 Sp? ? ? ?n ? ? x ny ?

~ tn

x ?n y ?2

where S ?
2 p

2 2 (n x ? 1) S x ? (n y ? 1) S y

nx ? n y ? 2

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S4 – Issue 1 – September 2007

33

PERCENTAGE POINTS OF STUDENT’S t DISTRIBUTION The values in the table are those which a random variable with Student’s t distribution on ? degrees of freedom exceeds with the probability shown.
?
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 32 34 36 38 40 45 50 55 60 70 80 90 100 110 120 0.10 3.078 1.886 1.638 1.533 1.476 1.440 1.415 1.397 1.383 1.372 1.363 1.356 1.350 1.345 1.341 1.337 1.333 1.330 1.328 1.325 1.323 1.321 1.319 1.318 1.316 1.315 1.314 1.313 1.311 1.310 1.309 1.307 1.306 1.304 1.303 1.301 1.299 1.297 1.296 1.294 1.292 1.291 1.290 1.289 1.289 0.05 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 1.746 1.740 1.734 1.729 1.725 1.721 1.717 1.714 1.711 1.708 1.706 1.703 1.701 1.699 1.697 1.694 1.691 1.688 1.686 1.684 1.679 1.676 1.673 1.671 1.667 1.664 1.662 1.660 1.659 1.658 0.025 12.706 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 2.120 2.110 2.101 2.093 2.086 2.080 2.074 2.069 2.064 2.060 2.056 2.052 2.048 2.045 2.042 2.037 2.032 2.028 2.024 2.021 2.014 2.009 2.004 2.000 1.994 1.990 1.987 1.984 1.982 1.980 0.01 31.821 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 2.583 2.567 2.552 2.539 2.528 2.518 2.508 2.500 2.492 2.485 2.479 2.473 2.467 2.462 2.457 2.449 2.441 2.435 2.429 2.423 2.412 2.403 2.396 2.390 2.381 2.374 2.369 2.364 2.361 2.358 0.005 63.657 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 2.921 2.898 2.878 2.861 2.845 2.831 2.819 2.807 2.797 2.787 2.779 2.771 2.763 2.756 2.750 2.738 2.728 2.719 2.712 2.704 2.690 2.678 2.668 2.660 2.648 2.639 2.632 2.626 2.621 2.617

34

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S4 – Issue 1 – September 2007

PERCENTAGE POINTS OF THE F DISTRIBUTION The values in the table are those which a random variable with the F distribution on ?1 and ?2 degrees of freedom exceeds with probability 0.05 or 0.01.
Probability

?2/?1
1 2 3 4 5 6 7 8 9 10 11 12 14 16 18 20 25 30 40 60 120 ? 1 2 3 4 5 6 7 8 9 10 11 12 14 16 18 20 25 30 40 60 120 ?

0.05

1 161.4 18.51 10.13 7.71 6.61 5.99 5.59 5.32 5.12 4.96 4.84 4.75 4.60 4.49 4.41 4.35 4.24 4.17 4.08 4.00 3.92 3.84 4052. 98.50 34.12 21.20 16.26 13.70 12.20 11.30 10.60 10.00 9.65 9.33 8.86 8.53 8.29 8.10 7.77 7.56 7.31 7.08 6.85 6.63

2 199.5 19.00 9.55 6.94 5.79 5.14 4.74 4.46 4.26 4.10 3.98 3.89 3.74 3.63 3.55 3.49 3.39 3.32 3.23 3.15 3.07 3.00 5000. 99.00 30.82 18.00 13.27 10.90 9.55 8.65 8.02 7.56 7.21 6.93 6.51 6.23 6.01 5.85 5.57 5.39 5.18 4.98 4.79 4.61

3 215.7 19.16 9.28 6.59 5.41 4.76 4.35 4.07 3.86 3.71 3.59 3.49 3.34 3.24 3.16 3.10 2.99 2.92 2.84 2.76 2.68 2.60 5403. 99.17 29.46 16.69 12.06 9.78 8.45 7.59 6.99 6.55 6.22 5.95 5.56 5.29 5.09 4.94 4.68 4.51 4.31 4.13 3.95 3.78

4 224.6 19.25 9.12 6.39 5.19 4.53 4.12 3.84 3.63 3.48 3.36 3.26 3.11 3.01 2.93 2.87 2.76 2.69 2.61 2.53 2.45 2.37 5625. 99.25 28.71 15.98 11.39 9.15 7.85 7.01 6.42 5.99 5.67 5.41 5.04 4.77 4.58 4.43 4.18 4.02 3.83 3.65 3.48 3.32

5 230.2 19.30 9.01 6.26 5.05 4.39 3.97 3.69 3.48 3.33 3.20 3.11 2.96 2.85 2.77 2.71 2.60 2.53 2.45 2.37 2.29 2.21 5764. 99.30 28.24 15.52 10.97 8.75 7.46 6.63 6.06 5.64 5.32 5.06 4.70 4.44 4.25 4.10 3.86 3.70 3.51 3.34 3.17 3.02

6 234.0 19.33 8.94 6.16 4.95 4.28 3.87 3.58 3.37 3.22 3.09 3.00 2.85 2.74 2.66 2.60 2.49 2.42 2.34 2.25 2.18 2.10 5859. 99.33 27.91 15.21 10.67 8.47 7.19 6.37 5.80 5.39 5.07 4.82 4.46 4.20 4.01 3.87 3.63 3.47 3.29 3.12 2.96 2.80

8 238.9 19.37 8.85 6.04 4.82 4.15 3.73 3.44 3.23 3.07 2.95 2.85 2.70 2.59 2.51 2.45 2.34 2.27 2.18 2.10 2.02 1.94 5982. 99.37 27.49 14.80 10.29 8.10 6.84 6.03 5.47 5.06 4.74 4.50 4.14 3.89 3.71 3.56 3.32 3.17 2.99 2.82 2.66 2.51

10 241.9 19.40 8.79 5.96 4.74 4.06 3.64 3.35 3.14 2.98 2.85 2.75 2.60 2.49 2.41 2.35 2.24 2.16 2.08 1.99 1.91 1.83 6056. 99.40 27.23 14.55 10.05 7.87 6.62 5.81 5.26 4.85 4.54 4.30 3.94 3.69 3.51 3.37 3.13 2.98 2.80 2.63 2.47 2.32

12 243.9 19.41 8.74 5.91 4.68 4.00 3.57 3.28 3.07 2.91 2.79 2.69 2.53 2.42 2.34 2.28 2.16 2.09 2.00 1.92 1.83 1.75 6106. 99.42 27.05 14.37 9.89 7.72 6.47 5.67 5.11 4.17 4.40 4.16 3.80 3.55 3.37 3.23 2.99 2.84 2.66 2.50 2.34 2.18

24 249.1 19.46 8.64 5.77 4.53 3.84 3.41 3.12 2.90 2.74 2.61 2.51 2.35 2.24 2.15 2.08 1.96 1.89 1.79 1.70 1.61 1.52 6235. 99.46 26.60 13.93 9.47 7.31 6.07 5.28 4.73 4.33 4.02 3.78 3.43 3.18 3.00 2.86 2.62 2.47 2.29 2.12 1.95 1.79

? 254.3 19.50 8.53 5.63 4.37 3.67 3.23 2.93 2.71 2.54 2.40 2.30 2.13 2.01 1.92 1.84 1.71 1.62 1.51 1.39 1.25 1.00 6366. 99.50 26.13 13.45 9.02 6.88 5.65 4.86 4.31 3.91 3.60 3.36 3.00 2.75 2.57 2.42 2.17 2.01 1.80 1.60 1.38 1.00

0.01

If an upper percentage point of the F distribution on ?1 and ?2 degrees of freedom is f , then the corresponding lower percentage point of the F distribution on ?2 and ?1 degrees of freedom is 1/ f .

UA018598 – Edexcel AS/A level Mathematics Formulae List: Statistics S4 – Issue 1 – September 2007

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