423527 list of formulae and statistical tables
List MF10
UNIVERSITY OF CAMBRIDGE LOCAL EXAMINATIONS SYNDICATE
General Certificate of Education Advanced Level
FURTHER MATHEMATICS (9231)
LIST OF FORMULAE
AND
STATISTICAL TABLES
PURE MATHEMATICS
Algebraic series
n
n
n
∑ r = 12 n(n + 1) ,
∑ r3 = 14 n2(n + 1)2
∑ r2 = 16 n(n + 1)(2n + 1) ,
r =1
r =1
r =1
Binomial expansion:
n
n
n
(a + b)n = a n + a n −1b + a n − 2b2 + a n −3b3 +
1
2
3
+ b , where n is a positive
n
integer
n!
n
and =
r r!(n − r)!
Maclaurin’s expansion:
+ xn! f (0) +
n(n − 1)
n(n − 1) (n − r + 1)
x +
x ++
= 1 + nx +
r!
2!
x
x
x
+ ++ +
e = 1+ x +
r!
2! 3!
x
x
(−1) x
+
sin x = x − + − +
3! 5!
(2r + 1)!
f( x) = f(0) + x f ′(0) +
(1 + x)n
n
x2
f ′′(0) +
2!
(n )
r
2
2
x
3
cos x = 1 −
r
3
+ (−(12)r)!x +
x 2 x3
+ −
2
3
+ (−1)r
r +1 x r
+
(all x)
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
1 tan A tan B
#
sin 3A ≡ 3 sin A − 4 sin3 A
cos 3A ≡ 4 cos3 A − 3 cos A
sin P + sin Q ≡ 2 sin 12 ( P + Q ) cos 12 ( P − Q )
sin P − sin Q ≡ 2 cos 12 ( P + Q ) sin 12 ( P − Q )
cos P + cos Q ≡ 2 cos 12 ( P + Q ) cos 12 ( P − Q )
cos P − cos Q ≡ −2 sin 12 ( P + Q ) sin 12 ( P − Q)
If t = tan 12 x then:
sin x =
2t
1 + t2
and
2
(all x)
r 2r
Trigonometry
tan( A ± B) ≡
(all x)
r 2r +1
5
x 2 x4
+ −
2! 4!
ln(1 + x) = x −
( x < 1)
cos x =
1 − t2
1 + t2
( −1 < x ≤ 1)
Principal values:
− 12 π ≤ sin −1 x ≤ 12 π
( x ≤ 1)
0 ≤ cos−1 x ≤ π
( x ≤ 1)
− 12 π < tan −1 x < 12 π
Integrals
(Arbitrary constants are omitted; a denotes a positive constant.)
∫ f( x) dx
f(x)
x2
1 −1 x
tan
a
a
1
+ a2
a2 − x2
x
sin−1
a
( x < a)
x2
1
− a2
1 x − a
ln
2a x + a
( x > a)
a2
1
− x2
1 a + x
ln
2a a − x
( x < a)
1
ln(sec x + tan x)
sec x
(x
Numerical methods
Trapezium rule:
∫a f( x) dx ≈ 12 h{y0 + 2( y1 + y2 + + yn −1) + yn} , where h =
b
The Newton-Raphson iteration for approximating a root of f( x) = 0 :
xr +1 = xr −
f( xr )
f ′( xr )
Vectors
The point dividing AB in the ratio λ : µ has position vector
3
µa + λb
λ+µ
b−a
n
< 12 π )
MECHANICS
Centres of mass of uniform bodies
Triangular lamina:
2
3
along median from vertex
Solid hemisphere of radius r:
3r
8
Hemispherical shell of radius r:
from centre
1r
2
from centre
r sin α
from centre
α
2r sin α
Circular sector of radius r and angle 2α :
from centre
3α
Circular arc of radius r and angle 2α :
Solid cone or pyramid of height h:
3h
4
from vertex
Moments of inertia for uniform bodies of mass m
Thin rod, length 2l, about perpendicular axis through centre:
1 ml 2
3
Rectangular lamina, sides 2a and 2b, about perpendicular axis through centre:
Disc or solid cylinder of radius r about axis:
Solid sphere of radius r about a diameter:
1 mr 2
2
2 mr 2
5
Spherical shell of radius r about a diameter:
2 mr 2
3
PROBABILITY AND STATISTICS
Sampling and testing
Unbiased variance estimate from a single sample:
s2 =
1 2 (Σx)2
1
Σ( x − x )2
=
Σx −
n n −1
n − 1
Two-sample estimate of a common variance:
s2 =
Σ( x1 − x1)2 + Σ( x2 − x2 )2
n1 + n2 − 2
Regression and correlation
Estimated product moment correlation coefficient:
Σ xΣ y
n
r=
=
{Σ( x − x )2}{Σ( y − y)2} Σx2 − (Σx)2 Σy 2 − (Σy)2
n
n
Σxy −
Σ( x − x )( y − y )
Estimated regression line of y on x:
y − y = b( x − x ),
where b =
4
Σ( x − x )( y − y )
Σ( x − x )2
1 m(a 2
3
+ b2 )
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
7
8 9
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
24
24
23
22
22
28
28
27
26
25
32
32
31
30
29
36
36
35
34
32
0.7190
0.7517
0.7823
0.8106
0.8365
0.7224
0.7549
0.7852
0.8133
0.8389
3 7 10 14 17 20
3 7 10 13 16 19
3 6 9 12 15 18
3 5 8 11 14 16
3 5 8 10 13 15
24
23
21
19
18
27
26
24
22
20
31
29
27
25
23
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
7
6
6
5
4
9 12 14 16 19 21
8 10 12 14 16 18
7 9 11 13 15 17
6 8 10 11 13 14
6 7 8 10 11 13
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
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 10 11
7 8 9
6 7 8
5 6 6
4 5 5
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
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
0.9946
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
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
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
ADD
20
20
19
19
18
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 ≤ z) = p .
p
z
0.75
0.674
0.90
1.282
0.95
1.645
0.975
1.960
0.99
2.326
5
0.995
2.576
0.9975
2.807
0.999
3.090
0.9995
3.291
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
2
3
4
1.000
0.816
0.765
0.741
3.078
1.886
1.638
1.533
6.314
2.920
2.353
2.132
12.71
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
0.674
1.310
1.303
1.296
1.289
1.282
1.697
1.684
1.671
1.658
1.645
2.042
2.021
2.000
1.980
1.960
2.457
2.423
2.390
2.358
2.326
2.750
2.704
2.660
2.617
2.576
3.030
2.971
2.915
2.860
2.807
3.385
3.307
3.232
3.160
3.090
3.646
3.551
3.460
3.373
3.291
6
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
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
7
CRITICAL VALUES FOR THE PRODUCT MOMENT CORRELATION COEFFICIENT
Significance level
One-tail
Two-tail
5%
10%
2.5%
5%
1%
2%
0.5%
1%
n=3
4
0.988
0.900
0.997
0.950
0.980
0.990
5
6
7
8
9
0.805
0.729
0.669
0.621
0.582
0.878
0.811
0.754
0.707
0.666
0.934
0.882
0.833
0.789
0.750
0.959
0.917
0.875
0.834
0.798
10
11
12
13
14
0.549
0.521
0.497
0.476
0.458
0.632
0.602
0.576
0.553
0.532
0.715
0.685
0.658
0.634
0.612
0.765
0.735
0.708
0.684
0.661
15
16
17
18
19
0.441
0.426
0.412
0.400
0.389
0.514
0.497
0.482
0.468
0.456
0.592
0.574
0.558
0.543
0.529
0.641
0.623
0.606
0.590
0.575
20
25
30
35
40
0.378
0.337
0.306
0.283
0.264
0.444
0.396
0.361
0.334
0.312
0.516
0.462
0.423
0.392
0.367
0.561
0.505
0.463
0.430
0.403
45
50
60
70
80
0.248
0.235
0.214
0.198
0.185
0.294
0.279
0.254
0.235
0.220
0.346
0.328
0.300
0.278
0.260
0.380
0.361
0.330
0.306
0.286
90
100
0.174
0.165
0.207
0.197
0.245
0.232
0.270
0.256
8
UNIVERSITY OF CAMBRIDGE LOCAL EXAMINATIONS SYNDICATE
General Certificate of Education Advanced Level
FURTHER MATHEMATICS (9231)
LIST OF FORMULAE
AND
STATISTICAL TABLES
PURE MATHEMATICS
Algebraic series
n
n
n
∑ r = 12 n(n + 1) ,
∑ r3 = 14 n2(n + 1)2
∑ r2 = 16 n(n + 1)(2n + 1) ,
r =1
r =1
r =1
Binomial expansion:
n
n
n
(a + b)n = a n + a n −1b + a n − 2b2 + a n −3b3 +
1
2
3
+ b , where n is a positive
n
integer
n!
n
and =
r r!(n − r)!
Maclaurin’s expansion:
+ xn! f (0) +
n(n − 1)
n(n − 1) (n − r + 1)
x +
x ++
= 1 + nx +
r!
2!
x
x
x
+ ++ +
e = 1+ x +
r!
2! 3!
x
x
(−1) x
+
sin x = x − + − +
3! 5!
(2r + 1)!
f( x) = f(0) + x f ′(0) +
(1 + x)n
n
x2
f ′′(0) +
2!
(n )
r
2
2
x
3
cos x = 1 −
r
3
+ (−(12)r)!x +
x 2 x3
+ −
2
3
+ (−1)r
r +1 x r
+
(all x)
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
1 tan A tan B
#
sin 3A ≡ 3 sin A − 4 sin3 A
cos 3A ≡ 4 cos3 A − 3 cos A
sin P + sin Q ≡ 2 sin 12 ( P + Q ) cos 12 ( P − Q )
sin P − sin Q ≡ 2 cos 12 ( P + Q ) sin 12 ( P − Q )
cos P + cos Q ≡ 2 cos 12 ( P + Q ) cos 12 ( P − Q )
cos P − cos Q ≡ −2 sin 12 ( P + Q ) sin 12 ( P − Q)
If t = tan 12 x then:
sin x =
2t
1 + t2
and
2
(all x)
r 2r
Trigonometry
tan( A ± B) ≡
(all x)
r 2r +1
5
x 2 x4
+ −
2! 4!
ln(1 + x) = x −
( x < 1)
cos x =
1 − t2
1 + t2
( −1 < x ≤ 1)
Principal values:
− 12 π ≤ sin −1 x ≤ 12 π
( x ≤ 1)
0 ≤ cos−1 x ≤ π
( x ≤ 1)
− 12 π < tan −1 x < 12 π
Integrals
(Arbitrary constants are omitted; a denotes a positive constant.)
∫ f( x) dx
f(x)
x2
1 −1 x
tan
a
a
1
+ a2
a2 − x2
x
sin−1
a
( x < a)
x2
1
− a2
1 x − a
ln
2a x + a
( x > a)
a2
1
− x2
1 a + x
ln
2a a − x
( x < a)
1
ln(sec x + tan x)
sec x
(x
Numerical methods
Trapezium rule:
∫a f( x) dx ≈ 12 h{y0 + 2( y1 + y2 + + yn −1) + yn} , where h =
b
The Newton-Raphson iteration for approximating a root of f( x) = 0 :
xr +1 = xr −
f( xr )
f ′( xr )
Vectors
The point dividing AB in the ratio λ : µ has position vector
3
µa + λb
λ+µ
b−a
n
< 12 π )
MECHANICS
Centres of mass of uniform bodies
Triangular lamina:
2
3
along median from vertex
Solid hemisphere of radius r:
3r
8
Hemispherical shell of radius r:
from centre
1r
2
from centre
r sin α
from centre
α
2r sin α
Circular sector of radius r and angle 2α :
from centre
3α
Circular arc of radius r and angle 2α :
Solid cone or pyramid of height h:
3h
4
from vertex
Moments of inertia for uniform bodies of mass m
Thin rod, length 2l, about perpendicular axis through centre:
1 ml 2
3
Rectangular lamina, sides 2a and 2b, about perpendicular axis through centre:
Disc or solid cylinder of radius r about axis:
Solid sphere of radius r about a diameter:
1 mr 2
2
2 mr 2
5
Spherical shell of radius r about a diameter:
2 mr 2
3
PROBABILITY AND STATISTICS
Sampling and testing
Unbiased variance estimate from a single sample:
s2 =
1 2 (Σx)2
1
Σ( x − x )2
=
Σx −
n n −1
n − 1
Two-sample estimate of a common variance:
s2 =
Σ( x1 − x1)2 + Σ( x2 − x2 )2
n1 + n2 − 2
Regression and correlation
Estimated product moment correlation coefficient:
Σ xΣ y
n
r=
=
{Σ( x − x )2}{Σ( y − y)2} Σx2 − (Σx)2 Σy 2 − (Σy)2
n
n
Σxy −
Σ( x − x )( y − y )
Estimated regression line of y on x:
y − y = b( x − x ),
where b =
4
Σ( x − x )( y − y )
Σ( x − x )2
1 m(a 2
3
+ b2 )
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
7
8 9
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
24
24
23
22
22
28
28
27
26
25
32
32
31
30
29
36
36
35
34
32
0.7190
0.7517
0.7823
0.8106
0.8365
0.7224
0.7549
0.7852
0.8133
0.8389
3 7 10 14 17 20
3 7 10 13 16 19
3 6 9 12 15 18
3 5 8 11 14 16
3 5 8 10 13 15
24
23
21
19
18
27
26
24
22
20
31
29
27
25
23
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
7
6
6
5
4
9 12 14 16 19 21
8 10 12 14 16 18
7 9 11 13 15 17
6 8 10 11 13 14
6 7 8 10 11 13
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
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 10 11
7 8 9
6 7 8
5 6 6
4 5 5
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
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
0.9946
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
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
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
ADD
20
20
19
19
18
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 ≤ z) = p .
p
z
0.75
0.674
0.90
1.282
0.95
1.645
0.975
1.960
0.99
2.326
5
0.995
2.576
0.9975
2.807
0.999
3.090
0.9995
3.291
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
2
3
4
1.000
0.816
0.765
0.741
3.078
1.886
1.638
1.533
6.314
2.920
2.353
2.132
12.71
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
0.674
1.310
1.303
1.296
1.289
1.282
1.697
1.684
1.671
1.658
1.645
2.042
2.021
2.000
1.980
1.960
2.457
2.423
2.390
2.358
2.326
2.750
2.704
2.660
2.617
2.576
3.030
2.971
2.915
2.860
2.807
3.385
3.307
3.232
3.160
3.090
3.646
3.551
3.460
3.373
3.291
6
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
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
7
CRITICAL VALUES FOR THE PRODUCT MOMENT CORRELATION COEFFICIENT
Significance level
One-tail
Two-tail
5%
10%
2.5%
5%
1%
2%
0.5%
1%
n=3
4
0.988
0.900
0.997
0.950
0.980
0.990
5
6
7
8
9
0.805
0.729
0.669
0.621
0.582
0.878
0.811
0.754
0.707
0.666
0.934
0.882
0.833
0.789
0.750
0.959
0.917
0.875
0.834
0.798
10
11
12
13
14
0.549
0.521
0.497
0.476
0.458
0.632
0.602
0.576
0.553
0.532
0.715
0.685
0.658
0.634
0.612
0.765
0.735
0.708
0.684
0.661
15
16
17
18
19
0.441
0.426
0.412
0.400
0.389
0.514
0.497
0.482
0.468
0.456
0.592
0.574
0.558
0.543
0.529
0.641
0.623
0.606
0.590
0.575
20
25
30
35
40
0.378
0.337
0.306
0.283
0.264
0.444
0.396
0.361
0.334
0.312
0.516
0.462
0.423
0.392
0.367
0.561
0.505
0.463
0.430
0.403
45
50
60
70
80
0.248
0.235
0.214
0.198
0.185
0.294
0.279
0.254
0.235
0.220
0.346
0.328
0.300
0.278
0.260
0.380
0.361
0.330
0.306
0.286
90
100
0.174
0.165
0.207
0.197
0.245
0.232
0.270
0.256
8