Cambridge International AS &A Level Mathematics 9709 syllabus for 2023, 2024 and 2025.
5 List of formulae and statistical tables {MF19)
PURE MATHEMATICS
Mensuration
Volume of sphere= fnr 3
Surface area of sphere = 4nr 2
Volume of cone or pyramid =
½x base area x height
Area of curved surface of cone = nr x slant height
Arc length of circle = r0 ( 0 in radians)
Area of sector of circle = ½r 2 0
( 0 in radians)
Algebra
For the quadratic equation ax 2 +bx+ c = 0:
-b± ✓b 2 -4ac
x=-----2a
For an arithmetic series:
Un
=a+ (n - l)d ,
Sn =½n(a+l)=½n{2a+(n-l)d}
For a geometric series:
Un
=ar
n-1
,
S =_!!:_
1-r
00
(lrl<l)
Binomial series:
and
(1 + X
Back to contents page
•
(n)r = r!(n-r)!
nl
r = 1+ nx + -n(n-2!-1)
-x
2
n(n - l)(n - 2)
+ - - - - - x 3 + ... ' where n is rational and
3!
IX I< 1
www.cambridgeinternational.org/alevel
39
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
Trigonometry
sin0
tan0=-cos0
cos 2 0 + sin 2 0 ==1,
cot 2 0 + I = cosec 2 0
l+tan 2 0=sec 2 0,
sin(A ± B) = sinAcosB ± cosAsinB
cos(A ± B) = cosAcosB + sinAsinB
tanA±tanB
tan(A ± B) = - - - - l + tanAtanB
sin2A = 2sinAcosA
cos2A = cos 2 A-sin 2 A= 2cos 2 A-1 = 1-2sin 2 A
tan2A = 2 tan A
1-tan2 A
Principal values:
0 ~cos -1 x
~
n,
Differentiation
f(x)
f'(x)
nx
lnx
n-1
1
X
sinx
cosx
cosx
-smx
tanx
sec 2 x
secx
secxtanx
cosecx
-cosecxcotx
cotx
-cosec 2 x
1
l+x 2
UV
du
dv
v-+udx
dx
u
du
dv
v--udx
dx
V
v2
dy dy dx
If x=f(t) and y =g(t) then-=-+dx dt dt
40
www.cambridgeinternational.org/alevel
Back to contents page
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
Integration
(Arbitrary constants are omitted; a denotes a positive constant.)
f f(x)dx
f(x)
Xn+l
xn
(n
:;t:
-1)
n+I
1
Inlxl
X
ex
ex
smx
-cosx
cosx
smx
sec 2 x
tanx
1
1 -1(X)
~tan ~
x2 +a2
1
2
X -a
2
1
2
a -x
2
1x-a1
-1l n
-2a
x+a
(x> a)
_l ln1~1
2a
a-x
fu dvdx dx=uv- f v dudx dx
dx = Inlf(x)I
f f'(x)
f(x)
Vectors
If a= a1i + a2 j + a3k and b = b1i + b2 j + b3k then
Back to contents page
www.cambridgeinternational.org/alevel
41
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
FURTHER PURE MATHEMATICS
Algebra
Summations:
~:> =½n(n+l),
n
n
r=I
n
~:> 2 = ¼n(n + 1)(2n + 1),
~:>3 =¼n2(n+l)2
r=I
r=I
Maclaurin's series:
x2
xr
f(x) = f(O) + xf'(O) +-f"(O) + ... +-f(r) (0) + ...
2!
r!
2
r
ex =exp ( x ) = l + x +X- + ... +X- + ...
2!
r!
x2
ln(l + x) = x - .
X3
x3
(allx)
xr
+ - - ... + (-1y+1 - + ...
r
2
3
X5
r
(allx)
2r
4
cos x = l - ~ + ~ - ... + (-1r _x_ + ...
2! 4!
(2r)!
X3
< X < 1)
X2r+I
smx=x--+-- ... +(-1) - - - + ...
3! 5!
(2r + 1)!
2
(-1
X5
(allx)
X2r+I
tan- 1 x = x - - + - - ... + ( - l Y - - + ...
3
5
2r+ 1
.
X3
X5
X2r+I
smhx = x +- +- + ... + - - - + ...
3! 5!
(2r+l)!
x2
x4
(allx)
x2r
coshx = 1 + - + - + ... + - - + ...
2! 4!
(2r)!
tanh
-1
(allx)
3
5
2r+l
X
X
X
x = x + - + - + ... + - - + ...
3
5
(-l<x<l)
2r +l
Trigonometry
If t = tan½x then:
.
2t
sinx=-2
l-t 2
cosx=-2
and
l+t
Hyperbolic Junctions
cosh 2 x - sinh 2 x = l ,
l+t
sinh 2x = 2 sinh x cosh x
cosh 2x = cosh 2 x + sinh 2 x
,
sinh - I x = In(x + ,Jx 2 + 1)
cosh- 1 x=ln(x+ ✓x 2 -1)
(l+x)
_ 11- n
tanh _1 X
-2
42
www.cambridgeinternational.org/alevel
l-x
(x ~ 1)
(Ix I< 1)
Back to contents page
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
Differentiation
f(x)
f'(x)
1
sin- 1 x
✓1-x 2
1
cos- 1 x
✓1-x 2
sinhx
coshx
coshx
sinhx
tanhx
sech 2 x
sinh-l X
1
✓I+x 2
1
cosh- 1 x
-J x 2 -1
1
l-x 2
tanh-l X
Integration
(Arbitrary constants are omitted; a denotes a positive constant.)
f(x)
ff(x)dx
secx
lnlsecx+tanxl =lnltan(½x+¼n)I
cosecx
- ln cosec x + cot x = ln tan(½ x)
sinhx
coshx
coshx
sinhx
sech 2 x
tanhx
1
-Jaz -xz
1
-Jx2 -a2
1
-Jaz +xz
Back to contents page
I
I
I
I
(O<x<n)
. -1 (XJ
a
sm
cosh
-t(XJ
~
(x>a)
1(~J
sinh-
www.cambridgeinternational.org/alevel
43
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
MECHANICS
Uniformly accelerated motion
v=u +at,
s =ut +l.at 2
s =½(u +v)t,
2
'
v2 =u 2 +2as
FURTHER MECHANICS
Motion of a projectile
Equation of trajectory is:
y=xtan0-
gx2
2
2
2V cos 0
Elastic strings and springs
E=AX2
21
Motion in a circle
For uniform circular motion, the acceleration is directed towards the centre and has magnitude
or
r
Centres of mass of uniform bodies
Triangular lamina: f along median from vertex
Solid hemisphere ofradius r:
¾r from centre
Hemispherical shell of radius r:
½r from centre
Circular arc of radius rand angle 2a: rsma from centre
a
2rsina
.i::
• 1ar sector of rad.ms r and ang1e 2 a: C1rcu
- - 1-rom
centre
3a
Solid cone or pyramid of height h:
44
www.cambridgeinternational.org/alevel
¾h
from vertex
Back to contents page
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
PROBABILITY & STATISTICS
Summary statistics
For ungrouped data:
LX
x=-,
n
standard deviation =
✓L(x: x)2 = ✓~ 2 x
-
2
For grouped data:
standard deviation =
E(x - X)' f = JD-' f Lf
Lf
x'
Discrete random variables
E(X)=LXp,
Var(X) = LX 2 p-{E(X)} 2
For the binomial distribution B(n,p):
a- 2 = np(I - p)
µ=np,
For the geometric distribution Geo(p):
1
p
Pr= p(I- py-1'
µ=-
For the Poisson distribution Po(,1,)
Continuous random variables
f
f
Var(X) = x 2 f(x) dx-{E(X)} 2
E(X) = xf(x) dx,
Sampling and testing
Unbiased estimators:
LX
x=-,
n
s2
= L(x-x) 2 =-1-(LX 2
n-1
n-1
_
(LX) 2 )
n
Central Limit Theorem:
Approximate distribution of sample proportion:
Back to contents page
www.cambridgeinternational.org/alevel
45
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
FURTHER PROBABILITY & STATISTICS
Sampling and testing
Two-sample estimate of a common variance:
L(x1 -:x1)2 +L(x2 -:x2 )2
2
s =--------nl +n 2 -2
Probability generating functions
Gx(t) = E(tx),
46
www.cambridgeinternational.org/alevel
E(X) = G'.r(l),
Var(X) = G'.'r(l) + G'.r(l)- {G'.r(1)} 2
Back to contents page
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
THE NORMAL DISTRIBUTION FUNCTION
If Z has a normal distribution with mean O and
variance 1, then, for each value of z, the table gives
the value of <l>(z), where
<l>(z) = P(Z,..; z).
LI
For negative values of z, use <l>(-z) = 1 - <l>(z).
z
0
1
2
4
3
5
6
7
8
9
1
2
3
4
8
8
12
12
12
16 20 24 28
16 20 24 28
15 19 23 27
0.5239 0.5279 0.5319 0.5359
0.5636 0.5675 0.5714 0.5753
0.6026 0.6064 0.6103 0.6141
4
4
4
0.6406 0.6443 0.6480 0.6517
0.6772 0.6808 0.6844 0.6879
4
4
0.7
0.8
0.9
0.6915 0.6950 0.6985 0.7019 0.7054 0.7088
0.7257 0.7291 0.7324 0.7357 0.7389 0.7422
0.7580 0.7611 0.7642 0.7673 0.7704 0.7734
0.7881 0.7910 0.7939 0.7967 0.7995 0.8023
0.8159 0.8186 0.8212 0.8238 0.8264 0.8289
0.7123 0.7157 0.7190 0.7224
0.7454 0.7486 0.7517 0.7549
0.7764 0.7794 0.7823 0.7852
0.3
0.4
6
7
8
9
ADD
0.5000 0.5040 0.5080 0.5120 0.5160 0.5199
0.5398 0.5438 0.5478 0.5517 0.5557 0.5596
0.5793 0.5832 0.5871 0.5910 0.5948 0.5987
0.6179 0.6217 0.6255 0.6293 0.6331 0.6368
0.6554 0.6591 0.6628 0.6664 0.6700 0.6736
0.0
0.1
0.2
5
8
7
32 36
32 36
31 35
15 19 22 26 30 34
14 18 22 25 29 32
7
11
11
3
3
7
7
10
10
0.8051 0.8078 0.8106 0.8133
0.8315 0.8340 0.8365 0.8389
3
3
3
6
5
5
9
8
8
1.0
0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
2
5
12
14 16
19 21
0.8643
0.8849
0.9032
0.9192
0.8830
0.9015
0.9177
0.9319
2
2
2
1
4
4
3
3
7
6
6
5
4
9
1.1
1.2
1.3
1.4
8
7
6
6
10
12 14
11 13
10 11
8 10
16
15
13
11
18
17
14
13
1.5
1.6
1.7
1.8
1.9
0.9332 0.9345 0.9357
0.9452 0.9463 0.9474
0.9554 0.9564 0.9573
0.9641 0.9649 0.9656
0.9713 0.9719 0.9726
0.9418 0.9429 0.9441
1
2
4
1
1
2
2
6
5
4
1
1
1
1
3
3
2
2
5
4
4
3
2
4
3
7
6
5
4
4
8
7
6
5
4
10
8
7
6
5
11
0.9525 0.9535 0.9545
0.9616 0.9625 0.9633
0.9693 0.9699 0.9706
0.9756 0.9761 0.9767
2.0
2.1
2.2
0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
0
1
1
1
1
1
1
2
2
1
2
2
2
3
2
2
3
4
0
0
3
2
3
3
4
4
3
2.3
2.4
0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
0
0
1
0
1
1
1
1
1
1
2
1
2
1
2
2
2
2
2.5
2.6
2.7
0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
0
0
0
0
0
0
0
0
0
1
0
0
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
2.8
2.9
0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
0.5
0.6
0.8665
0.8869
0.9049
0.9207
0.8686
0.8888
0.9066
0.9222
0.8708
0.8907
0.9082
0.9236
0.8729
0.8925
0.9099
0.9251
0.8749
0.8944
0.9115
0.9265
0.8770
0.8962
0.9131
0.9279
0.9370 0.9382 0.9394 0.9406
0.9484 0.9495 0.9505 0.9515
0.9582 0.9591 0.9599 0.9608
0.9664 0.9671 0.9678 0.9686
0.9732 0.9738 0.9744 0.9750
0.8790
0.8980
0.9147
0.9292
0.8810
0.8997
0.9162
0.9306
14 17 20 24 27 31
13 16 19 23 26 29
12 15 18 21 24 27
11 14
10 13
9
8
7
16 19 22 25
15 18 20 23
9
8
6
5
Critical values for the normal distribution
If Z has a normal distribution with mean O and
variance 1, then, for each value ofp, the table
gives the value of z such that
P(Z,..; z) =p.
p
0.75
0.90
0.95
z
0.674
1.282
1.645
Back to contents page
0.975
0.99
0.995
0.9975
0.999
0.9995
1.960
2.326
2.576
2.807
3.090
3.291
www.cambridgeinternational.org/alevel
47
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
CRITICAL V ALOES FOR THE t-DISTRIBUTION
If T has a t-distribution with v degrees of freedom, then,
for each pair of values of p and v, the table gives the value
oft such that:
P(T <;. t) = p.
0.75
0.90
0.95
0.975
0.99
0.995
0.9975
0.999
0.9995
v= 1
1.000
3.078
6.314
12.71
31.82
63.66
127.3
318.3
636.6
2.920
2.353
2.132
4.303
3.182
2.776
6.965
4.541
3.747
9.925
5.841
4.604
14.09
7.453
5.598
22.33
10.21
7.173
31.60
12.92
8.610
2.015
1.943
2.571
2.447
2.365
3.365
3.143
2.998
4.032
3.707
5.894
5.208
4.785
6.869
5.959
2.306
2.262
2.896
2.821
3.499
3.355
3.250
4.773
4.317
4.029
3.833
3.690
4.501
4.297
2.764
2.718
2.681
2.650
2.624
3.169
3.106
3.055
3.012
2.977
3.581
3.497
3.428
3.372
3.326
4.144
4.025
3.930
3.852
3.787
4.587
4.437
4.318
4.221
4.140
p
2
0.816
1.886
3
4
0.765
0.741
1.638
1.533
5
6
0.727
0.718
0.711
1.476
1.440
1.415
0.706
0.703
1.397
1.383
10
11
12
13
14
0.700
0.697
0.695
0.694
0.692
1.372
1.363
1.356
1.350
1.345
1.812
1.796
1.782
1.771
1.761
2.228
2.201
2.179
2.160
2.145
15
16
17
18
0.691
0.690
0.689
0.688
1.341
1.337
1.333
1.330
1.753
1.746
1.740
1.734
2.131
2.120
2.110
2.101
2.602
2.583
2.567
2.552
2.947
2.921
2.898
2.878
3.286
3.252
3.222
3.197
3.733
3.686
3.646
3.610
4.073
4.015
3.965
3.922
19
0.688
1.328
1.729
2.093
2.539
2.861
3.174
3.579
3.883
20
21
22
0.687
0.686
1.325
1.323
1.321
1.725
1.721
1.717
2.086
2.080
2.074
2.528
2.518
2.508
2.845
2.831
2.819
3.153
3.135
3.119
3.552
3.527
3.850
3.819
3.792
0.685
1.319
1.318
1.714
1.711
2.069
2.064
2.500
2.492
2.807
2.797
3.104
3.091
25
26
27
0.684
0.684
0.684
1.316
1.315
1.314
1.708
1.706
1.703
2.060
2.056
2.052
2.485
2.479
2.473
2.787
2.779
2.771
28
29
0.683
0.683
1.313
1.311
1.701
1.699
2.048
2.045
2.467
2.462
30
40
60
120
0.683
0.681
0.679
0.677
1.310
1.697
1.684
1.671
1.658
2.042
1.303
1.296
1.289
2.021
2.000
1.980
00
0.674
1.282
1.645
1.960
7
8
9
23
24
48
0.686
0.685
www.cambridgeinternational.org/alevel
1.895
1.860
1.833
3.505
3.485
5.408
5.041
4.781
3.467
3.768
3.745
3.078
3.067
3.057
3.450
3.435
3.421
3.725
3.707
3.689
2.763
2.756
3.047
3.038
3.408
3.396
3.674
3.660
2.457
2.750
2.423
2.390
2.358
2.704
2.660
2.617
3.030
2.971
2.915
2.860
3.385
3.307
3.232
3.160
3.646
3.551
3.460
3.373
2.326
2.576
2.807
3.090
3.291
Back to contents page
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
CRITICAL VALUES FOR THE
If Xhas a
z 2 -DISTRIBUTION
z 2 -distribution with v degrees of
freedom then, for each pair of values ofp and v,
the table gives the value of x such that
P(X ~ x) = p.
p
.t
p
0.01
0.025
0.05
0.9
0.95
0.975
0.99
0.995
0.999
v= 1
2
3
4
0.0 3 1571
0.02010
0.1148
0.2971
0.0 39821
0.05064
0.2158
0.4844
0.0 23932
0.1026
0.3518
0.7107
2.706
4.605
6.251
7.779
3.841
5.991
7.815
9.488
5.024
7.378
9.348
11.14
6.635
9.210
11.34
13.28
7.879
10.60
12.84
14.86
10.83
13.82
16.27
18.47
5
6
7
8
9
0.5543
0.8721
1.239
1.647
2.088
0.8312
1.237
1.690
2.180
2.700
1.145
1.635
2.167
2.733
3.325
9.236
10.64
12.02
13.36
14.68
11.07
12.59
14.07
15.51
16.92
12.83
14.45
16.01
17.53
19.02
15.09
16.81
18.48
20.09
21.67
16.75
18.55
20.28
21.95
23.59
20.51
22.46
24.32
26.12
27.88
10
11
12
13
14
2.558
3.053
3.571
4.107
4.660
3.247
3.816
4.404
5.009
5.629
3.940
4.575
5.226
5.892
6.571
15.99
17.28
18.55
19.81
21.06
18.31
19.68
21.03
22.36
23.68
20.48
21.92
23.34
24.74
26.12
23.21
24.73
26.22
27.69
29.14
25.19
26.76
28.30
29.82
31.32
29.59
31.26
32.91
34.53
36.12
15
16
17
18
19
5.229
5.812
6.408
7.015
7.633
6.262
6.908
7.564
8.231
8.907
7.261
7.962
8.672
9.390
10.12
22.31
23.54
24.77
25.99
27.20
25.00
26.30
27.59
28.87
30.14
27.49
28.85
30.19
31.53
32.85
30.58
32.00
33.41
34.81
36.19
32.80
34.27
35.72
37.16
38.58
37.70
39.25
40.79
42.31
43.82
20
21
22
23
24
8.260
8.897
9.542
10.20
10.86
9.591
10.28
10.98
11.69
12.40
10.85
11.59
12.34
13.09
13.85
28.41
29.62
30.81
32.01
33.20
31.41
32.67
33.92
35.17
36.42
34.17
35.48
36.78
38.08
39.36
37.57
38.93
40.29
41.64
42.98
40.00
41.40
42.80
44.18
45.56
45.31
46.80
48.27
49.73
51.18
25
30
40
50
60
11.52
14.95
22.16
29.71
37.48
13.12
16.79
24.43
32.36
40.48
14.61
18.49
26.51
34.76
43.19
34.38
40.26
51.81
63.17
74.40
37.65
43.77
55.76
67.50
79.08
40.65
46.98
59.34
71.42
83.30
44.31
50.89
63.69
76.15
88.38
46.93
53.67
66.77
79.49
91.95
52.62
59.70
73.40
86.66
99.61
70
80
90
100
45.44
53.54
61.75
70.06
48.76
57.15
65.65
74.22
51.74
60.39
69.13
77.93
85.53
96.58
107.6
118.5
90.53
101.9
113.1
124.3
95.02
106.6
118.1
129.6
100.4
112.3
124.1
135.8
104.2
116.3
128.3
140.2
112.3
124.8
137.2
149.4
Back to contents page
www.cambridgeinternational.org/alevel
49
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
WILCOXON SIGNED-RANK TEST
The sample has size n.
P is the sum of the ranks corresponding to the positive differences.
Q is the sum of the ranks corresponding to the negative differences.
Tis the smaller of P and Q.
For each value of n the table gives the largest value of Twhich will lead to rejection of the null hypothesis at
the level of significance indicated.
Critical values of T
Level of significance
One-tailed
Two-tailed
n=6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.05
0.1
2
3
5
8
10
13
17
21
25
30
35
41
47
53
60
0.025
0.05
0
2
3
5
8
10
13
17
21
25
29
34
40
46
52
0.01
0.02
0.005
0.01
0
1
3
5
7
9
12
15
19
23
27
32
37
43
0
1
3
5
7
9
12
15
19
23
27
32
37
For larger values of n, each of P and Q can be approximated by the normal distribution with mean
and variance
so
¼n( n + 1)
J4 n(n + 1)(2n + 1).
www.cambridgeinternational.org/alevel
Back to contents page
Cambridge International AS & A Level Mathematics 9709 syllabus for 2023, 2024 and 2025. List of formulae and statistical tables (MF19)
WILCOXON RANK-SUM TEST
The two samples have sizes m and n, where m ,.;;; n.
Rm is the sum of the ranks of the items in the sample of size m.
Wis the smaller of Rm and m(n + m + 1) - Rm.
For each pair of values of m and n, the table gives the largest value of W which will lead to rejection of the
null hypothesis at the level of significance indicated.
Critical values of W
One-tailed
Two-tailed
0.05
0.1
n
3
4
5
6
7
8
9
10
6
6
7
8
8
9
10
10
One-tailed
Two-tailed
0.05
0.1
n
7
8
9
10
39
41
43
45
0.025
0.05
m=3
0.01
0.02
-
-
-
-
6
7
7
8
8
9
-
0.025
0.05
m=7
36
38
40
42
-
6
6
7
7
0.05
0.1
11
12
13
14
15
16
17
0.01
0.02
0.05
0.1
34
35
37
39
51
54
56
Level of significance
0.025 0.01
0.05 0.025
0.1
0.02
0.05
0.05
m=4
m=5
10
11
12
13
14
14
15
0.05
0.1
0.025
0.05
m=6
0.01
0.02
16
17
18
19
20
21
28
29
31
33
35
26
27
29
31
32
24
25
27
28
29
0.01
0.02
0.05
0.1
0.025
0.05
m= 10
0.01
0.02
59
61
82
78
74
-
10
11
11
12
13
13
19
20
21
23
24
26
17
18
20
21
22
23
Level of significance
0.025 0.01
0.05 0.025
0.1
0.02
0.05
0.05
m=8
m=9
49
51
53
0.01
0.02
45
47
49
For larger values of m and n, the normal distribution with mean
66
69
62
65
½m( m + n + 1)
and variance
1mn( m + n + 1)
1
should be used as an approximation to the distribution of Rm.
Back to contents page
www.cambridgeinternational.org/alevel
51