A_Level数学公式书

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GCE
Edexcel GCE in Mathematics

Mathematical Formulae and Statistical Tables

For use in Edexcel 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

Core Mathematics C1
Mensuration
Arithmetic series
5
5
5
5
5
5

6
6
6
6

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
7
7

8
8
8
8
8
8

9
9
9
10

11
11
13
14
14
15
M21521RRA – Edexcel ASA level Mathematics Formulae List – Issue 1 – November 2004

16
16

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
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 Sampling distributions
34 Percentage points of Student’s t distribution
35 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 ASA 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


u
n
= a + (n – 1)d
S
n
= n(a + l) = n[2a + (n ? 1)d]
1
2
1
2
4
UA018598 – Edexcel ASA 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
a
2
= b
2
+ c
2
– 2bc cos A
Binomial series
(a ?b)
n
?a
n
?
?
?
n
?
?< br>?
1
?
?
?
a
n?1
b?
??
n
?
?
?
2
?
?
?
a
n?2
b
2
?
?
?
?
?
n
?
?
?
r
?
?
?
a
n?r
b
r
?
?
?b
n

(n ?
?
)
where
?
?
n
?
?
?
r
?
?
n
n!
?
? C
r
?
r!(n?r)!

(1?x)
n
?1?n x?
n(n?1)
2
n(
1?2
x?
?
?n?1)
?
(n?r?1)
1?2?
?
?r
x
r
?
?
(x?1, n?
?)
Logarithms and exponentials
log
x
a
x?
log
b
log

b
a

Geometric series

u
n
= ar
n ? 1

S
n
=
a(1?r
n
)
1? r

S
?
=
a
1 ? r
for ?r? < 1

Numerical integration
b
The trapezium rule:
?
?
?
ydx
?
1
2
h{(y
0
+ y
n
) + 2(y
1
+ y
2
+ ... + y
n – 1
)}, where
h?
b?a

a
n

UA018598 – Edexcel ASA 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
xlna
?a
x


Trigonometric identities
sin(A?B)?sinAcosB?cosAsinB

cos(A?B)?cosAcosB?sinAsinB

tanA?tanB
(A?B?(k?
1
2
)
?
)

1
?tanAtanB
A?BA?B
sinA?sinB?2sincos

22
A?BA?B
sinA?sinB?2cossin

22
A?BA?B
cosA?cosB?2coscos

22
A?BA?B
cosA?cosB??2sinsin

22
tan(A?B)?

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

f?(x)

k sec
2
kx
sec x tan x
–cosec
2
x
–cosec x cot x


f(x)

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

6
UA018598 – Edexcel ASA 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)
?
?
?
f(x)dx

sec
2
kx
1
k
tan kx
tanx

lnsecx

cotx

lnsinx

cosecx

?lncosecx?cotx?lntan(
1
2
x)

secx

lnsecx?tanx?lntan(
11
2
x?
4
?
)

?
?
u
dv
?dx
dx?uv?
?
?
?
v
du
dx
dx


UA018598 – Edexcel ASA 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
?

Coordinate geometry
The perpendicular distance from (h, k) to
ax?by?c?0
is
f(x
n
)
< br>f
?
(x
n
)
ah?bk?c
a?b
22< br>
The acute angle between lines with gradients
m
1
and
m
2
is
arctan

Conics

m
1
?m
2

1?m
1
m
2

Parabola
Standard
Form
Parametric
Form
Foci
Rectangular
Hyperbola
xy = c
2

c
??
?
ct,
?

t
??
y
2
?4ax

(at
2
, 2at)
(a, 0)

Not required
Directrices

x??a

Not required
8
UA018598 – Edexcel ASA level Mathematics Formulae List: Further Pure Mathematics FP1 – Issue 1 – September 2007

Matrix transformations
?
co s
?
?sin
?
?
Anticlockwise rotation through
?
about O:
?
?
?
sin
?
cos
?
?
?
?


Reflection in the line
y?(tan
?
)x
:
?
?
cos2
?
sin2
?
?
??
sin2
?
?cos2
?
?
?
?

UA018598

– Edexcel ASA 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
1
2
A =
?
?
rd
?
(polar coordinates)
2
?

Complex numbers

e
i
?
?cos
?
?isin
?

{r(cos
?
?isin
?
)}
n
?r
n(cosn
?
?isinn
?
)

The roots of
z?1
are given by
z?e
Maclaurin’s and Taylor’s Series
n
2
?
ki
n
, for
k?0, 1, 2, ? , n?1

x
2
x
r
(r)
f(x)?f( 0)?xf
?
(0)?f
??
(0)?
?
?
f(0)
?
?

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

2!r!
x
2
x
r
(r)
f(a?x)?f(a)?xf
?
(a)?f
??
(a)?
?
?
f(
a
)
?
?

2!r!
x
2
x
r
e
?
exp(x)
?
1
?
x
??

?

??

?
for all x

2!r!
x
r
x
2
x
3
r?1
x
ln(1?x)?x???
?
?(?1)?
?
(?1?x?1)

23r
x
3
x
5
x
2r?1
r
sinx?x???
?
?(?1)
?

?
for all x

3! 5!(2r?1)!
2r
x
2
x
4
r
x
c osx
?
1
???

?

?
(
?
1)
?

?
for all x

2!4!(2r)!
2r?1
x
3
x
5
r
x
arctanx?x???
?
? (?1)?
?
(?1?x?1)

352r?1
Taylor polynomials
h
2
f(a?h)?f(a)?hf
?
( a)?f
??
(a)?error

2!
h
2
f( a?h)?f(a)?hf
?
(a)?f
??
(a?
?
) (0?
?
?h)

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

2!
10
UA018598 – Edexcel ASA 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

The point dividing AB in the ratio
?
:
?
is

i
?
?a
1
Vector product:
a?b?ab
sin
?

n
b
1
j
a
2
b
2
?
a
2
b
3
?a3
b
2
?
??
a
3
?
?
a< br>3
b
1
?a
1
b
3
?

?
b
3
?
?
a
1
b
2
?a
2
b
1
?
k
a.b

b
?
a?
?
b

?
?
?

a
1
a.(b?c)?b
1
c
1
a
2b
2
c
2
a
3
b
3
?b.(c?a) ?c.(a?b)

c
3

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


If A is the point with position vector
a?a
1
i?a
2
j?a
3
k
and the direction vector b is given by
b?b
1
i?b
2
j?b
3
k
, then the straight line through A with direction vector b has cartesian
equation
x?a
1
y?a
2
z?a
3
?? (?
?
)

b
1
b
2
b
3

The plane through A with normal vector
n?n
1
i?n
2
j?n
3
k
has cartesian equation
n
1
x?n
2
y?n
3
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?
?
?< br>?
)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
n
1
x?n
2
y?n
3
z?d?0
is
n
1
?
?n
2
?
?n
3
?
?d
n?n?n
2
1
2
2
2
3
.
UA018598 – Edexcel ASA level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 2007
11

Hyperbolic functions
cosh
2
x?sinh
2
x?1

sinh2x?2sinhxcoshx

cosh2x?cosh
2
x?sinh
2
x

arcoshx?ln
{
x?x
2
?1
}
(x?1)

arsinhx?ln
{
x?x
2
?1
}

?
1?x
?
artanhx?
1
ln
??
(x?1)

2
1?x
??

Conics


Ellipse
Standard
Form
Parametric
Form
Eccentricity
x
2
y
2
?
2
?1

2
ab
(acos
?
, bsin
?
)

e?1

b
2
?a
2
(1?e
2
)

Parabola Hyperbola
x
2
y
2
?
2
?1

2
ab
Rectangular
Hyperbola
y?4ax

2
xy?c
2

c
??
?
ct,
?

t
??
(at
2
, 2at)

(a sec
?
, b tan
?
)
(?a cosh
?
, b sinh

?

)

e?1

b
2
?a
2
(e
2
?1)

e?1

e = ?2
Foci
(?ae, 0)

a

e
(a, 0)

(?ae, 0)

a

e
(??2c, ??2c)
Directrices
x??
x??a

x??
x + y = ??2c
Asymptotes none none
y
x
??

ab
x?0, y?0

12
UA018598 – Edexcel ASA level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 2007

Differentiation
f(x) f

?(x)
arcsinx

1
1?x
2

arccosx

?
1
1?x
2

arctanx

1
1?x
2

sinhx

coshx

coshx

sinhx

tanhx

sech
2
x

arsinhx

1
1?x
2

arcoshx

1
x
2
?1

artanh x
1
1?x
2

Integration (+ constant;
a?0
where relevant)
f(x)
?
?
?
f(x)dx

sinhx

coshx

coshx

sinhx

tanhx

lncoshx

1

arcsin
?
a
2
?x
2
?
x
?
?
a
?
?
(x?a)

1
a
2
?x
2

1
a
arcta n
?
?
x
?
?
a
?
?

1
2

arcosh
?
x?a
2
?
x
?
?
a
?
?
?ln
{
x?x
2
?a
2
}
(x?a)
1
2

arsinh
?
a
2
?x
?
x
?
?a
?
?
?ln
{
x?x
2
?a
2}

1
a
2
?x
2

1
2a
ln
a?x
a?x
?
1
a
artanh
?
?
x
?
?
a
?
?
(x?a)

1
x
2
?a
2

1x
2a
ln
?a
x?a


UA018598 – Edexcel ASA level Mathematics Formulae List – Issue 1 – September 2007

13

Arc length
2
?
?
dy?
s?
?
1?
??
dx
(cartesian coordinates)
?
dx
?
?

?
?
dx
?
2
?
dy
?
2
s?
?
??
?
??
dt
(parametric form)
?
?
dt
??
dt
?


Surface area of revolution
S
x
?2
?
?
yds?
?
2
?
?
?
?
?
?
dx
??
dy
?
y
??
?
??
dt

?
dt
??
dt
?
22


14
UA018598 – Edexcel ASA level Mathematics Formulae List: Further Pure Mathematics FP3 – Issue 1 – September 2007

BLANK PAGE

TURN OVER FOR MECHANICS & STATISTICS FORMULAE

UA018598 – Edexcel ASA 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
along median from vertex
3
Circular arc, radius r, angle at centre 2
?
: from centre
?
2rsin
?
Sector of circle, radius r, angle at centre 2
?
: from centre
3
?

rsin
?
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
?
??

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

r
2
Centres of mass
For uniform bodies:
3
Solid hemisphere, radius r:
8
r
from centre
Hemispherical shell, radius r:
1
r
from centre
2
Solid cone or pyramid of height h:
1
h
above the base on the line from centre of base to vertex
4
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?
Gm
1
m
2
d
2

16
UA018598 – Edexcel ASA 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:
2
Thin rod, length 2l, about perpendicular axis through centre:
1

ml
3
2
Rectangular lamina about axis in plane bisecting edges of length 2l:
1

ml
3
Thin rod, length 2l, about perpendicular axis through end:
4
ml
2

3
Rectangular lamina about edge perpendicular to edges of length 2l:
4
ml
2

3
Rectangular lamina, sides 2a and 2b, about perpendicular axis through centre:
1
m(a
2
?b
2
)

3
Hoop or cylindrical shell of radius r about axis through centre:
mr
2

2
Hoop of radius r about a diameter:
1

mr
2
Disc or solid cylinder of radius r about axis through centre:
1
mr
2

2
Disc of radius r about a diameter:
1
mr
2

4
2
Solid sphere, radius r, about diameter:
5
mr
2

2
Spherical shell of radius r about a diameter:
2

mr
3

Parallel axes theorem:
I
A
?I
G
?m(AG)
2

Perpendicular axes theorem:
I
z
?I
x
?I
y
(for a lamina in the x-y plane)
Moments as vectors
The moment about O of F acting at r is
r?F


UA018598 – Edexcel ASA 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 = x
i
)
Expectation (mean): E(X) =
?
= ?x
i
P(X = x
i
)
Variance: Var(X) =
?
2
= ?(x
i
–
?
)
2
P(X = x
i
) = ?
x
i
2
P(X = x
i
) –
?
2
For a function
g(X)
: E(g(X)) = ?g(x
i
) P(X = x
i
)

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 ASA level Mathematics Formulae List: Statistics S1 – Issue 1 – September 2007

Correlation and regression
For a set of n pairs of values
(x
i
, y
i
)
< br>S
xx
??(x
i
?x)
2
??x
i
2
?
S
yy
??(y
i
?y)??y?
22i
(?x
i
)
2
n
(?y
i
)
2
n


(?x)(?y)
S??(xx)(y
iixyi
?
i
?y)??x
i
y
i
?
n


The product moment correlation coefficient is
(?x
r?
S
xy
y)
? xy
i
)(?y
i
)
S
?
?(x
i
?x)(y
i
?
{
?(x?x)
2
}{
?(y< br>2
?
ii
?
n
xx
S
yy
?
22
ii
?y)
}
?
2
(
??
?
?x?
?x
i
)
?
?

?
?
? y
2
?
(?y
i
)
?
?
i
n?
?
?
i
n
?
?

The regression coefficient of y on x is
b?
S
xy
?x)(y
i
?y)
S
?
?(x
i
xx
?(x
i
?x)
2


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



UA018598 – Edexcel ASA level Mathematics Formulae List: Statistics S1 – Issue 1 – September 2007
19

THE NORMAL DISTRIBUTION FUNCTION
1
?
?
1
2
t
2
The function tabulated below is ?(z), defined as ?(z) =
dt
.
?
e
2
?
?
??

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
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
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)
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 ASA 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 z p z
0.5000 0.0000 0.0500 1.6449
0.4000 0.2533 0.0250 1.9600
0.3000 0.5244 0.0100 2.3263
0.2000 0.8416 0.0050 2.5758
0.1500 1.0364 0.0010 3.0902
0.1000 1.2816 0.0005 3.2905

UA018598 – Edexcel ASA 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(
?
)


Continuous distributions
For a continuous random variable X having probability density function f
Expectation (mean):
E(X)?
?
?
?
xf(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

P(X?x)

Mean
np

Variance
np(1?p)

?
n
?
xn?x
??

p(1?p)
?
x
?
??
e
?
?
?
x
x!

?

?

?
0
Cumulative distribution function: F(x
0
)?P(X?x
0
)?
?
f(t)dt

?
??
Standard continuous distribution:
Distribution of X
Uniform (Rectangular) on [a, b]

x
P.D.F.
1

b?a
1
2
Mean Variance
1
12
(a?b)

(b?a)
2


22
UA018598 – Edexcel ASA 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 = 0.05
n = 5, x = 0 0.7738
1 0.9774
2 0.9988
3 1.0000
4 1.0000
n = 6, x = 0 0.7351
1 0.9672
2 0.9978
3 0.9999
4 1.0000
5 1.0000
n = 7, x = 0 0.6983
1 0.9556
2 0.9962
3 0.9998
4 1.0000
5 1.0000
6 1.0000
n = 8, x = 0 0.6634
1 0.9428
2 0.9942
3 0.9996
4 1.0000
5 1.0000
6 1.0000
7 1.0000
n = 9, x = 0 0.6302
1 0.9288
2 0.9916
3 0.9994
4 1.0000
5 1.0000
6 1.0000
7 1.0000
8 1.0000
n = 10, x = 0 0.5987
1 0.9139
2 0.9885
3 0.9990
4 0.9999
5 1.0000
6 1.0000
7 1.0000
8 1.0000
9 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 ASA level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2007
23

p = 0.05
n = 12, x = 0 0.5404
1 0.8816
2 0.9804
3 0.9978
4 0.9998
5 1.0000
6 1.0000
7 1.0000
8 1.0000
9 1.0000
10 1.0000
11 1.0000
n = 15, x = 0 0.4633
1 0.8290
2 0.9638
3 0.9945
4 0.9994
5 0.9999
6 1.0000
7 1.0000
8 1.0000
9 1.0000
10 1.0000
11 1.0000
12 1.0000
13 1.0000
14 1.0000
n = 20, x = 0 0.3585
1 0.7358
2 0.9245
3 0.9841
4 0.9974
5 0.9997
6 1.0000
7 1.0000
8 1.0000
9 1.0000
10 1.0000
11 1.0000
12 1.0000
13 1.0000
14 1.0000
15 1.0000
16 1.0000
17 1.0000
18 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 ASA level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2007

p = 0.05
n = 25, x = 0 0.2774
1 0.6424
2 0.8729
3 0.9659
4 0.9928
5 0.9988
6 0.9998
7 1.0000
8 1.0000
9 1.0000
10 1.0000
11 1.0000
12 1.0000
13 1.0000
14 1.0000
15 1.0000
16 1.0000
17 1.0000
18 1.0000
19 1.0000
20 1.0000
21 1.0000
22 1.0000
n = 30, x = 0 0.2146
1 0.5535
2 0.8122
3 0.9392
4 0.9844
5 0.9967
6 0.9994
7 0.9999
8 1.0000
9 1.0000
10 1.0000
11 1.0000
12 1.0000
13 1.0000
14 1.0000
15 1.0000
16 1.0000
17 1.0000
18 1.0000
19 1.0000
20 1.0000
21 1.0000
22 1.0000
23 1.0000
24 1.0000
25 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 ASA level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2007
25

p = 0.05
n = 40, x = 0 0.1285
1 0.3991
2 0.6767
3 0.8619
4 0.9520
5 0.9861
6 0.9966
7 0.9993
8 0.9999
9 1.0000
10 1.0000
11 1.0000
12 1.0000
13 1.0000
14 1.0000
15 1.0000
16 1.0000
17 1.0000
18 1.0000
19 1.0000
20 1.0000
21 1.0000
22 1.0000
23 1.0000
24 1.0000
25 1.0000
26 1.0000
27 1.0000
28 1.0000
29 1.0000
30 1.0000
31 1.0000
32 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 ASA level Mathematics Formulae List: Statistics S2 – Issue 1 – September 2007

p = 0.05
n = 50, x = 0 0.0769
1 0.2794
2 0.5405
3 0.7604
4 0.8964
5 0.9622
6 0.9882
7 0.9968
8 0.9992
9 0.9998
10 1.0000
11 1.0000
12 1.0000
13 1.0000
14 1.0000
15 1.0000
16 1.0000
17 1.0000
18 1.0000
19 1.0000
20 1.0000
21 1.0000
22 1.0000
23 1.0000
24 1.0000
25 1.0000
26 1.0000
27 1.0000
28 1.0000
29 1.0000
30 1.0000
31 1.0000
32 1.0000
33 1.0000
34 1.0000
35 1.0000
36 1.0000
37 1.0000
38 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 ASA 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
?
.

?

=
0.5
x = 0 0.6065
1 0.9098
2 0.9856
3 0.9982
4 0.9998
5 1.0000
6 1.0000
7 1.0000
8 1.0000
9 1.0000
10 1.0000
11 1.0000
12 1.0000
13 1.0000
14 1.0000
15 1.0000
16 1.0000
17 1.0000
18 1.0000
19 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










?
=
5.5
x = 0 0.0041
1 0.0266
2 0.0884
3 0.2017
4 0.3575
5 0.5289
6 0.6860
7 0.8095
8 0.8944
9 0.9462
10 0.9747
11 0.9890
12 0.9955
13 0.9983
14 0.9994
15 0.9998
16 0.9999
17 1.0000
18 1.0000
19 1.0000
20 1.0000
21 1.0000
22 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 ASA 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?

22
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
n
x
?
?
2
y
~N(0, 1)

n
y
Correlation and regression

6?d
2
Spearman’s rank correlation coefficient is

r
s
?1?

2
n(n?1)

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

?
(O
i
?E
i
)
2
E
i
~
?
?
2

UA018598 – Edexcel ASA 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.

0.995 0.990 0.975 0.950 0.900 0.100 0.050 0.025 0.010 0.005
?

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

0.000
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
0.000
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
0.001
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
0.004
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
0.016
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






























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
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
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
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
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 ASA 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.

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
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
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
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
UA018598 – Edexcel ASA level Mathematics Formulae List: Statistics S3 – Issue 1 – September 2007
31

RANDOM NUMBERS
86 13
60 78
78 48
80 56
99 09
56 32
66 02
31 77
98 79
50 97










84 10
48 12
06 37
90 79
39 25
32 72
49 93
53 94
72 43
92 15










07 30
99 47
82 26
66 94
66 31
91 65
97 44
05 93
14 76
10 01










39 05
09 46
01 06
18 40
70 56
97 36
99 15
56 14
54 77
57 01











97 96
91 33
64 65
97 79
30 15
56 61
56 86
71 23
66 29
87 33










88 07
17 21
94 41
93 20
52 17
12 79
80 57
60 46
84 09
73 17










37 26
03 94
17 26
41 51
87 55
95 17
11 78
05 33
88 56
70 18










04 89
79 00
74 66
25 04
31 11
57 16
40 23
23 72
75 86
40 21










13 48
08 50
61 93
20 71
10 68
53 58
58 40
93 10
41 67
24 20










19 20
40 16
24 97
76 04
98 23
96 36
86 14
81 23
04 42
66 62
90 51
31 99
22 96
06 84
08 64
86 87
94 44
63 25
11 22
01 70

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

UA018598 – Edexcel ASA 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?
?
Sn
~t
n?1
(also valid in matched-pairs situations)

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

S
22
x

?
x
S
22
~F
n?1, n?1

y

?
y
xy

If
?
222
x
?
?
y
?
?
(unknown) then
(X?Y)?(
?
x
?
?
y
)
~t
(n
2
?(n
2
x
?1)S< br>xy
?1)S
y
n
x
?n
y
?2
where
S
2
p
?
n
S
2
?x
?n
y
?2

p
?
11
?
?
?
?
?
n
x
n
y
?
?
UA018598 – Edexcel ASA 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.

34
?

0.10 0.05 0.025 0.01 0.005
1 3.078 6.314 12.706 31.821

63.657
2 1.886 2.920 4.303 6.965 9.925
3 1.638 2.353 3.182 4.541 5.841
4 1.533 2.132 2.776 3.747 4.604
5 1.476 2.015 2.571 3.365 4.032
6 1.440 1.943 2.447 3.143 3.707
7 1.415 1.895 2.365 2.998 3.499
8 1.397 1.860 2.306 2.896 3.355
9 1.383 1.833 2.262 2.821 3.250
10 1.372 1.812 2.228 2.764 3.169
11 1.363 1.796 2.201 2.718 3.106
12 1.356 1.782 2.179 2.681 3.055
13 1.350 1.771 2.160 2.650 3.012
14 1.345 1.761 2.145 2.624 2.977
15 1.341 1.753 2.131 2.602 2.947
16 1.337 1.746 2.120 2.583 2.921
17 1.333 1.740 2.110 2.567 2.898
18 1.330 1.734 2.101 2.552 2.878
19 1.328 1.729 2.093 2.539 2.861
20 1.325 1.725 2.086 2.528 2.845
21 1.323 1.721 2.080 2.518 2.831
22 1.321 1.717 2.074 2.508 2.819
23 1.319 1.714 2.069 2.500 2.807
24 1.318 1.711 2.064 2.492 2.797
25 1.316 1.708 2.060 2.485 2.787
26 1.315 1.706 2.056 2.479 2.779
27 1.314 1.703 2.052 2.473 2.771
28 1.313 1.701 2.048 2.467 2.763
29 1.311 1.699 2.045 2.462 2.756
30 1.310 1.697 2.042 2.457 2.750
32 1.309 1.694 2.037 2.449 2.738
34 1.307 1.691 2.032 2.441 2.728
36 1.306 1.688 2.028 2.435 2.719
38 1.304 1.686 2.024 2.429 2.712
40 1.303 1.684 2.021 2.423 2.704
45 1.301 1.679 2.014 2.412 2.690
50 1.299 1.676 2.009 2.403 2.678
55 1.297 1.673 2.004 2.396 2.668
60 1.296 1.671 2.000 2.390 2.660
70 1.294 1.667 1.994 2.381 2.648
80 1.292 1.664 1.990 2.374 2.639
90 1.291 1.662 1.987 2.369 2.632
100 1.290 1.660 1.984 2.364 2.626
110 1.289 1.659 1.982 2.361 2.621
120 1.289 1.658 1.980 2.358 2.617

UA018598 – Edexcel ASA 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 2 3 4 5 6 8 10 12 24
?
161.4 199.5 215.7 224.6 230.2 234.0 238.9 241.9 243.9 249.1 254.3
18.51 19.00 19.16 19.25 19.30 19.33 19.37 19.40 19.41 19.46 19.50
10.13 9.55 9.28 9.12 9.01 8.94 8.85 8.79 8.74 8.64 8.53
7.71 6.94 6.59 6.39 6.26 6.16 6.04 5.96 5.91 5.77 5.63
6.61 5.79 5.41 5.19 5.05 4.95 4.82 4.74 4.68 4.53 4.37
5.99 5.14 4.76 4.53 4.39 4.28 4.15 4.06 4.00 3.84 3.67
5.59 4.74 4.35 4.12 3.97 3.87 3.73 3.64 3.57 3.41 3.23
5.32 4.46 4.07 3.84 3.69 3.58 3.44 3.35 3.28 3.12 2.93
5.12 4.26 3.86 3.63 3.48 3.37 3.23 3.14 3.07 2.90 2.71
4.96 4.10 3.71 3.48 3.33 3.22 3.07 2.98 2.91 2.74 2.54
4.84 3.98 3.59 3.36 3.20 3.09 2.95 2.85 2.79 2.61 2.40
4.75 3.89 3.49 3.26 3.11 3.00 2.85 2.75 2.69 2.51 2.30
4.60 3.74 3.34 3.11 2.96 2.85 2.70 2.60 2.53 2.35 2.13
4.49 3.63 3.24 3.01 2.85 2.74 2.59 2.49 2.42 2.24 2.01
4.41 3.55 3.16 2.93 2.77 2.66 2.51 2.41 2.34 2.15 1.92
4.35 3.49 3.10 2.87 2.71 2.60 2.45 2.35 2.28 2.08 1.84
4.24 3.39 2.99 2.76 2.60 2.49 2.34 2.24 2.16 1.96 1.71
4.17 3.32 2.92 2.69 2.53 2.42 2.27 2.16 2.09 1.89 1.62
4.08 3.23 2.84 2.61 2.45 2.34 2.18 2.08 2.00 1.79 1.51
4.00 3.15 2.76 2.53 2.37 2.25 2.10 1.99 1.92 1.70 1.39
3.92 3.07 2.68 2.45 2.29 2.18 2.02 1.91 1.83 1.61 1.25
3.84 3.00 2.60 2.37 2.21 2.10 1.94 1.83 1.75 1.52 1.00
0.01
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
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
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
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
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
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
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
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
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
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
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

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 ASA level Mathematics Formulae List: Statistics S4 – Issue 1 – September 2007
35