Formula Mf10 Further Mathematics 6i33
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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
∑ r = 12 n(n + 1) ,
n
∑ r2 = 16 n(n + 1)(2n + 1) ,
r =1
∑ r3 = 14 n2(n + 1)2
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 − 1 n 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
FURTHER MATHEMATICS (9231)
LIST OF FORMULAE AND STATISTICAL TABLES
PURE MATHEMATICS Algebraic series n
n
∑ r = 12 n(n + 1) ,
n
∑ r2 = 16 n(n + 1)(2n + 1) ,
r =1
∑ r3 = 14 n2(n + 1)2
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 − 1 n 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
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