Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022.
5 List of formulae and statistical tables (MF19)
PURE MATHEMATICS
Mensuration
Volume of sphere =
4
3
πr 3
Surface area of sphere = 4πr 2
Volume of cone or pyramid = 13 × base area × height
Area of curved surface of cone = πr × slant height
Arc length of circle = rθ
( θ in radians)
Area of sector of circle = 12 r 2θ
( θ in radians)
Algebra
0:
For the quadratic equation ax 2 + bx + c =
x=
−b ± b 2 − 4ac
2a
For an arithmetic series:
un= a + (n − 1)d ,
S=
n
1
2
n( a + l=
)
1
2
n{2a + (n − 1) d }
For a geometric series:
un = ar n −1 ,
Sn =
a(1 − r n )
1− r
(r ≠ 1) ,
S∞ =
a
1− r
(
r <1 )
Binomial series:
n
 n
 n
(a + b) n = a n +   a n −1b +   a n − 2b 2 +   a n −3b3 + K + b n , where n is a positive integer
1
 2
 3
n
n!
and   =
 r  r!(n − r )!
(1 + x) n =+
1 nx +
36
n(n − 1) 2 n(n − 1)(n − 2) 3
x +
x + K , where n is rational and x < 1
2!
3!
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Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. List of formulae and statistical tables (MF19)
Trigonometry
tan θ ≡
cos 2 θ + sin 2 θ ≡ 1 ,
sin θ
cos θ
1 + tan 2 θ ≡ sec 2 θ ,
cot 2 θ + 1 ≡ cosec 2 θ
sin( A ± B) ≡ sin A cos B ± cos A sin B
cos( A ± B) ≡ cos A cos B m sin A sin B
tan( A ± B ) ≡
tan A ± tan B
1 m tan A tan B
sin 2 A ≡ 2sin A cos A
cos 2 A ≡ cos 2 A − sin 2 A ≡ 2cos 2 A − 1 ≡ 1 − 2sin 2 A
tan 2 A ≡
2 tan A
1 − tan 2 A
Principal values:
− 12 π ⩽ sin −1 x ⩽ 12 π ,
Differentiation
0 ⩽ cos −1 x ⩽ π ,
− 12 π < tan −1 x < 12 π
f( x )
f ′( x )
xn
nx n −1
ln x
1
x
ex
ex
sin x
cos x
cos x
− sin x
tan x
sec 2 x
sec x
sec x tan x
cosec x
− cosec x cot x
cot x
− cosec 2 x
tan −1 x
1
1 + x2
uv
u
v
v
v
du
dv
+u
dx
dx
du
dv
−u
dx
dx
v2
dy dy dx
÷
If x = f(t ) and y = g(t ) then =
dx dt dt
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Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. List of formulae and statistical tables (MF19)
Integration
(Arbitrary constants are omitted; a denotes a positive constant.)
f( x )
∫ f( x ) dx
xn
x n +1
n +1
1
x
ln x
ex
ex
sin x
− cos x
cos x
sin x
sec 2 x
tan x
1
x + a2
1
2
x − a2
1
x
tan −1  
a
a
1
x−a
ln
2a x + a
1
a − x2
1
a+x
ln
2a a − x
2
2
dv
(n ≠ −1)
( x > a)
(
x < a)
du
∫ u dx dx = uv −∫ v dx dx
f ′( x)
∫ f ( x) dx = ln f ( x)
Vectors
If a = a1i + a2 j + a3k and b = b1i + b2 j + b3k then
a.b = a1b1 + a2b2 + a3b3 = a b cos θ
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Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. List of formulae and statistical tables (MF19)
FURTHER PURE MATHEMATICS
Algebra
Summations:
n
∑
=
r
r =1
1
2
n
∑
n(n + 1) ,
r =1
n
∑r
r 2 = 16 n(n + 1)(2n + 1) ,
3
r =1
= 14 n 2 (n + 1) 2
Maclaurin’s series:
f( x) = f(0) + x f ′(0) +
x2
xr
f ′′(0) + K + f ( r ) (0) + K
r!
2!
e x =exp( x) =1 + x +
ln(1 + x) =
x−
sin x = x −
x2
xr
+K + +K
r!
2!
(all x)
x 2 x3
xr
+ − K + (−1) r +1 + K
2
3
r
x3 x5
x 2 r +1
+ − K + (−1) r
+K
3! 5!
(2r + 1)!
cos x = 1 −
x2 x4
x2r
+
− K + (−1) r
+K
2! 4!
(2r )!
tan −1 x = x −
x3 x5
x 2 r +1
+ +K +
+K
3! 5!
(2r + 1)!
cosh x =1 +
(–1 ⩽ x ⩽ 1)
(all x)
x2 x4
x2r
+
+K+
+K
2! 4!
(2r )!
tanh −1 x =x +
(all x)
(all x)
x3 x5
x 2 r +1
+ − K + (−1) r
+K
3
5
2r + 1
sinh x =x +
(–1 < x ⩽ 1)
(all x)
x3 x5
x 2 r +1
+ +K +
+K
3
5
2r + 1
(–1 < x < 1)
Trigonometry
If t = tan 12 x then:
sin x =
Hyperbolic functions
cosh 2 x − sinh 2 x ≡ 1 ,
2t
1+ t2
and
cos x =
1− t2
1+ t2
sinh 2 x ≡ 2sinh x cosh x ,
cosh 2 x ≡ cosh 2 x + sinh 2 x
sinh −1 x = ln( x + x 2 + 1)
−1
cosh =
x ln( x + x 2 − 1)
1+ x 
tanh −1 x = 12 ln 

1− x 
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(x ⩾ 1)
(| x | < 1)
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Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. List of formulae and statistical tables (MF19)
Differentiation
f( x )
f ′( x )
sin −1 x
1
cos −1 x
1 − x2
−
1
1 − x2
sinh x
cosh x
cosh x
sinh x
tanh x
sech 2 x
sinh −1 x
1
cosh −1 x
tanh −1 x
1 + x2
1
x2 − 1
1
1 − x2
Integration
(Arbitrary constants are omitted; a denotes a positive constant.)
f( x )
∫ f( x ) dx
sec x
ln| sec x=
+ tan x | ln| tan( 12 x + 14 π) |
(
cosec x
− ln| cosec x + cot x | =
ln| tan( 12 x) |
(0 < x < π)
sinh x
cosh x
cosh x
sinh x
sech 2 x
tanh x
1
x
sin −1  
a
a −x
2
2
1
x −a
2
2
1
a +x
2
40
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2
 x
cosh −1  
a
x < 12 π )
(
x < a)
( x > a)
 x
sinh −1  
a
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Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. List of formulae and statistical tables (MF19)
MECHANICS
Uniformly accelerated motion
v= u + at ,
=
s
1
(u
2
+ v)t ,
2
v=
u 2 + 2as
=
s ut + 12 at 2 ,
FURTHER MECHANICS
Motion of a projectile
Equation of trajectory is:
y = x tan θ −
gx 2
2V 2 cos 2 θ
Elastic strings and springs
T=
λx
l
E=
,
λ x2
2l
Motion in a circle
For uniform circular motion, the acceleration is directed towards the centre and has magnitude
ω 2r
v2
r
or
Centres of mass of uniform bodies
Triangular lamina: 23 along median from vertex
Solid hemisphere of radius r: 83 r from centre
Hemispherical shell of radius r:
1
2
r from centre
Circular arc of radius r and angle 2α:
r sin α
Circular sector of radius r and angle 2α:
Solid cone or pyramid of height h:
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3
4
α
from centre
2r sin α
from centre
3α
h from vertex
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Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. List of formulae and statistical tables (MF19)
PROBABILITY & STATISTICS
Summary statistics
For ungrouped data:
x=
Σx
,
n
x=
Σxf
,
Σf
Σ( x − x ) 2
=
n
standard=
deviation
Σx 2
− x2
n
For grouped data:
Σ( x − x ) 2 f
=
Σf
=
standard
deviation
Σx 2 f
− x2
Σf
Discrete random variables
Var( X ) = Σx 2 p − {E( X )}2
E( X ) = Σxp ,
For the binomial distribution B(n, p) :
n
pr =   p r (1 − p) n − r ,
r
µ = np ,
σ 2 = np(1 − p )
For the geometric distribution Geo(p):
pr = p(1 − p) r −1 ,
µ=
1
p
For the Poisson distribution Po(λ )
pr = e − λ
λr
r!
,
Continuous random variables
E( X ) = x f( x) dx ,
∫
σ2 =λ
µ =λ ,
∫
Var( X ) = x 2 f( x) dx − {E( X )}2
Sampling and testing
Unbiased estimators:
x=
Σx
,
n
s2
=
1  2 ( Σx ) 2 
Σ( x − x ) 2
=
 Σx −

n −1
n −1
n 
Central Limit Theorem:

σ2 
X ~ N  µ,

n 

Approximate distribution of sample proportion:
p (1 − p) 

N  p,

n


42
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Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. List of formulae and statistical tables (MF19)
FURTHER PROBABILITY & STATISTICS
Sampling and testing
Two-sample estimate of a common variance:
s2 =
Probability generating functions
G X (t ) = E(t X ) ,
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Σ( x1 − x1 ) 2 + Σ( x2 − x2 ) 2
n1 + n 2 − 2
E( X ) = G ′X (1) ,
Var(
=
X ) G ′′X (1) + G ′X (1) − {G ′X (1)}2
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Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. List of formulae and statistical tables (MF19)
THE NORMAL DISTRIBUTION FUNCTION
If Z has a normal distribution with mean 0 and
variance 1, then, for each value of z, the table gives
the value of Φ(z), where
Φ(z) = P(Z ⩽ z).
For negative values of z, use Φ(–z) = 1 – Φ(z).
1
2
3
4
0.5359
0.5753
0.6141
0.6517
0.6879
4
4
4
4
4
8
8
8
7
7
12
12
12
11
11
16
16
15
15
14
0.7190
0.7517
0.7823
0.8106
0.8365
0.7224
0.7549
0.7852
0.8133
0.8389
3
3
3
3
3
7
7
6
5
5
10
10
9
8
8
0.8577
0.8790
0.8980
0.9147
0.9292
0.8599
0.8810
0.8997
0.9162
0.9306
0.8621
0.8830
0.9015
0.9177
0.9319
2
2
2
2
1
5
4
4
3
3
0.9406
0.9515
0.9608
0.9686
0.9750
0.9418
0.9525
0.9616
0.9693
0.9756
0.9429
0.9535
0.9625
0.9699
0.9761
0.9441
0.9545
0.9633
0.9706
0.9767
1
1
1
1
1
0.9798
0.9842
0.9878
0.9906
0.9929
0.9803
0.9846
0.9881
0.9909
0.9931
0.9808
0.9850
0.9884
0.9911
0.9932
0.9812
0.9854
0.9887
0.9913
0.9934
0.9817
0.9857
0.9890
0.9916
0.9936
0.9946
0.9960
0.9970
0.9978
0.9984
0.9948
0.9961
0.9971
0.9979
0.9985
0.9949
0.9962
0.9972
0.9979
0.9985
0.9951
0.9963
0.9973
0.9980
0.9986
0.9952
0.9964
0.9974
0.9981
0.9986
z
0
1
2
3
4
5
6
7
8
9
0.0
0.1
0.2
0.3
0.4
0.5000
0.5398
0.5793
0.6179
0.6554
0.5040
0.5438
0.5832
0.6217
0.6591
0.5080
0.5478
0.5871
0.6255
0.6628
0.5120
0.5517
0.5910
0.6293
0.6664
0.5160
0.5557
0.5948
0.6331
0.6700
0.5199
0.5596
0.5987
0.6368
0.6736
0.5239
0.5636
0.6026
0.6406
0.6772
0.5279
0.5675
0.6064
0.6443
0.6808
0.5319
0.5714
0.6103
0.6480
0.6844
0.5
0.6
0.7
0.8
0.9
0.6915
0.7257
0.7580
0.7881
0.8159
0.6950
0.7291
0.7611
0.7910
0.8186
0.6985
0.7324
0.7642
0.7939
0.8212
0.7019
0.7357
0.7673
0.7967
0.8238
0.7054
0.7389
0.7704
0.7995
0.8264
0.7088
0.7422
0.7734
0.8023
0.8289
0.7123
0.7454
0.7764
0.8051
0.8315
0.7157
0.7486
0.7794
0.8078
0.8340
1.0
1.1
1.2
1.3
1.4
0.8413
0.8643
0.8849
0.9032
0.9192
0.8438
0.8665
0.8869
0.9049
0.9207
0.8461
0.8686
0.8888
0.9066
0.9222
0.8485
0.8708
0.8907
0.9082
0.9236
0.8508
0.8729
0.8925
0.9099
0.9251
0.8531
0.8749
0.8944
0.9115
0.9265
0.8554
0.8770
0.8962
0.9131
0.9279
1.5
1.6
1.7
1.8
1.9
0.9332
0.9452
0.9554
0.9641
0.9713
0.9345
0.9463
0.9564
0.9649
0.9719
0.9357
0.9474
0.9573
0.9656
0.9726
0.9370
0.9484
0.9582
0.9664
0.9732
0.9382
0.9495
0.9591
0.9671
0.9738
0.9394
0.9505
0.9599
0.9678
0.9744
2.0
2.1
2.2
2.3
2.4
0.9772
0.9821
0.9861
0.9893
0.9918
0.9778
0.9826
0.9864
0.9896
0.9920
0.9783
0.9830
0.9868
0.9898
0.9922
0.9788
0.9834
0.9871
0.9901
0.9925
0.9793
0.9838
0.9875
0.9904
0.9927
2.5
2.6
2.7
2.8
2.9
0.9938
0.9953
0.9965
0.9974
0.9981
0.9940
0.9955
0.9966
0.9975
0.9982
0.9941
0.9956
0.9967
0.9976
0.9982
0.9943
0.9957
0.9968
0.9977
0.9983
0.9945
0.9959
0.9969
0.9977
0.9984
5
6
7
8
9
20
20
19
19
18
24
24
23
22
22
28
28
27
26
25
32
32
31
30
29
36
36
35
34
32
14
13
12
11
10
17
16
15
14
13
20
19
18
16
15
24
23
21
19
18
27
26
24
22
20
31
29
27
25
23
7
6
6
5
4
9
8
7
6
6
12
10
9
8
7
14
12
11
10
8
16
14
13
11
10
19
16
15
13
11
21
18
17
14
13
2
2
2
1
1
4
3
3
2
2
5
4
4
3
2
6
5
4
4
3
7
6
5
4
4
8
7
6
5
4
10 11
8 9
7 8
6 6
5 5
0
0
0
0
0
1
1
1
1
0
1
1
1
1
1
2
2
1
1
1
2
2
2
1
1
3
2
2
2
1
3
3
2
2
1
4
3
3
2
2
4
4
3
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
1
0
0
0
1
1
1
0
0
1
1
1
0
0
1
1
1
1
0
1
1
1
1
0
ADD
Critical values for the normal distribution
If Z has a normal distribution with mean 0 and
variance 1, then, for each value of p, the table
gives the value of z such that
p
z
44
P(Z ⩽ z) = p.
90% CI
95% CI
98% CI
99% CI
0.75
0.90
0.95
0.975
0.99
0.995
0.9975
0.999
0.9995
0.674
1.282
1.645
1.960
2.326
2.576
2.807
3.090
3.291
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Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. List of formulae and statistical tables (MF19)
CRITICAL VALUES FOR THE t-DISTRIBUTION
If T has a t-distribution with ν degrees of freedom, then,
for each pair of values of p and ν, the table gives the value
of t such that:
P(T ⩽ t) = p.
p
0.75
0.90
0.95
0.975
0.99
0.995
0.9975
0.999
0.9995
ν=1
1.000
3.078
6.314
12.71
2
3
4
0.816
0.765
0.741
1.886
1.638
1.533
2.920
2.353
2.132
4.303
3.182
2.776
31.82
6.965
4.541
3.747
63.66
9.925
5.841
4.604
127.3
14.09
7.453
5.598
318.3
22.33
10.21
7.173
636.6
31.60
12.92
8.610
5
6
7
8
9
0.727
0.718
0.711
0.706
0.703
1.476
1.440
1.415
1.397
1.383
2.015
1.943
1.895
1.860
1.833
2.571
2.447
2.365
2.306
2.262
3.365
3.143
2.998
2.896
2.821
4.032
3.707
3.499
3.355
3.250
4.773
4.317
4.029
3.833
3.690
5.894
5.208
4.785
4.501
4.297
6.869
5.959
5.408
5.041
4.781
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
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
15
16
17
18
19
0.691
0.690
0.689
0.688
0.688
1.341
1.337
1.333
1.330
1.328
1.753
1.746
1.740
1.734
1.729
2.131
2.120
2.110
2.101
2.093
2.602
2.583
2.567
2.552
2.539
2.947
2.921
2.898
2.878
2.861
3.286
3.252
3.222
3.197
3.174
3.733
3.686
3.646
3.610
3.579
4.073
4.015
3.965
3.922
3.883
20
21
22
23
24
0.687
0.686
0.686
0.685
0.685
1.325
1.323
1.321
1.319
1.318
1.725
1.721
1.717
1.714
1.711
2.086
2.080
2.074
2.069
2.064
2.528
2.518
2.508
2.500
2.492
2.845
2.831
2.819
2.807
2.797
3.153
3.135
3.119
3.104
3.091
3.552
3.527
3.505
3.485
3.467
3.850
3.819
3.792
3.768
3.745
25
26
27
28
29
0.684
0.684
0.684
0.683
0.683
1.316
1.315
1.314
1.313
1.311
1.708
1.706
1.703
1.701
1.699
2.060
2.056
2.052
2.048
2.045
2.485
2.479
2.473
2.467
2.462
2.787
2.779
2.771
2.763
2.756
3.078
3.067
3.057
3.047
3.038
3.450
3.435
3.421
3.408
3.396
3.725
3.707
3.689
3.674
3.660
30
40
60
120
0.683
0.681
0.679
0.677
1.310
1.303
1.296
1.289
1.697
1.684
1.671
1.658
2.042
2.021
2.000
1.980
2.457
2.423
2.390
2.358
2.750
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
∞
0.674
1.282
1.645
1.960
2.326
2.576
2.807
3.090
3.291
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Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. List of formulae and statistical tables (MF19)
CRITICAL VALUES FOR THE
χ 2 -DISTRIBUTION
If X has a χ 2 -distribution with ν degrees of
freedom then, for each pair of values of p and ν,
the table gives the value of x such that
P(X ⩽ x) = p.
46
p
0.01
0.025
0.05
0.9
0.95
0.975
0.99
0.995
0.999
ν=1
2
3
4
0.031571
0.02010
0.1148
0.2971
0.039821
0.05064
0.2158
0.4844
0.023932
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
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Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. 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.
T is the smaller of P and Q.
For each value of n the table gives the largest value of T which will lead to rejection of the null hypothesis at
the level of significance indicated.
Critical values of T
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
Level of significance
0.025
0.01
0.05
0.02
0
2
0
3
1
5
3
8
5
10
7
13
9
17
12
21
15
25
19
29
23
34
27
40
32
46
37
52
43
0.005
0.01
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
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1
24
1
4
n(n + 1)
n(n + 1)(2n + 1) .
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47
Cambridge International AS & A Level Mathematics 9709 syllabus for 2020, 2021 and 2022. 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.
W is 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
n
3
4
5
6
7
8
9
10
One-tailed
Two-tailed
n
7
8
9
10
0.05
0.1
6
6
7
8
8
9
10
10
0.05
0.1
39
41
43
45
0.025
0.05
m=3
–
–
6
7
7
8
8
9
0.025
0.05
m=7
36
38
40
42
0.01
0.02
0.05
0.1
–
–
–
–
6
6
7
7
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
0.05
0.02
0.1
0.05
m=4
m=5
10
11
12
13
14
14
15
–
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
0.05
0.02
0.1
0.05
m=8
m=9
49
51
53
45
47
49
For larger values of m and n, the normal distribution with mean
66
69
1
2
62
65
0.01
0.02
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
78
74
59
61
82
m(m + n + 1) and variance
1
12
mn(m + n + 1)
should be used as an approximation to the distribution of Rm.
48
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