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Tablas de dispersión, Apuntes de Estadística

Universidad Politécnica del Bicentenario (México)Estadística

Tablas que ayudan al cálculo de ciertos valores

Tipo: Apuntes

2021/2022

Subido el 11/06/2023

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652 APPENDIX A STATISTICAL TABLES AND CHARTS
Table I Summary of Common Probability Distributions
Probability Section
Name Distribution Mean Variance in Book
Discrete
Uniform 3-5
Binomial np 3-6
Geometric 3-7.1
Negative binomial 3-7.2
Hypergeometric np, 3-8
Poisson 3-9
Continuous
Uniform 4-5
Normal 
24-6
Exponential 4-8
Erlang 4-9.1
Gamma 4-9.2
Weibull 4-10
Lognormal 4-11
e221e212e22
1
x22
exp a3ln1x242
22b2c a11
bd2
0x, 0 , 0 
2 a12
b a11
b
ax
b1
e1x2,
r2
r
xr1ex
1r2, 0 x, 0 r, 0
r2
r
rxr1ex
1r12!, 0 x, r1, 2, . . .
12
1ex, 0 x, 0
 x, , 0 
1
12
e1
21x
22
1ba22
12
1ba2
2
1
ba, axb
ex
x!, x0, 1, 2, . . . , 0
min1K, n2, KN, nN
xmax10, nNK2, 1, . . .
where pK
N
np11p2 aNn
N1b
aK
x b aNK
nx b
aN
n b
xr, r1, r2, . . . , 0 p1
r11p2p2
rpax1
r1b 11p2xrpr
x1, 2, . . . , 0 p111p2p2
1p11p2x1p,
x0, 1, . . . , n, 0 p1
np11p2an
xb px11p2nx,
1ba1221
12
1ba2
2
1
n, ab
PQ220 6234F.pp [ 1 26].qxd 5/8/02 8:31 PM Page 652 RK UL 6 RK UL 6:Desktop Folder:TEMP WORK:PQ220 MONT 8/5/2002: PPENDIX :
pf3
pf4
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652 APPENDIX A STATISTICAL TABLES AND CHARTS

Table I Summary of Common Probability Distributions

Probability Section Name Distribution Mean Variance in Book Discrete

Uniform 3-

Binomial np 3-

Geometric 3-7.

Negative binomial 3-7.

Hypergeometric np , 3-

Poisson   3- Continuous

Uniform 4-

Normal  ^2 4-

Exponential 4-

Erlang 4-9.

Gamma 4-9.

Weibull 4-

Lognormal e^2 ^ 4- 2 1 e 2 e ^12

x  22

exp a

3 ln 1 x 2  42 2 2 b

^2 c  a 1

b d

2 0  x , 0  , 0  

^2  a 1

 a 1 b

b

a

x  b

 1 e^1 x^ ^2

 ,

r  r ^2

xr^^1 e x  1 r 2 , 0  x , 0  r , 0  

r  r ^2

rxr^^1 e x 1 r 12! , 0  x , r  1, 2,...

e x , 0  x , 0   1  1 ^2

  x  ,     , 0  

e 1 2 1 x^  22

1 b a 22 12

1 b a 2 2

b a , a  x  b

e ^ x x! , x  0, 1, 2,... , 0  

min 1 K , n 2 , K  N , n  N

x  max 1 0, n N K 2 , 1,...

where p 

K

N

np 11 p 2 a N n N 1 b

a

K

x b a

N K

n x b

a

N

n

b

x  r , r 1, r 2,... , 0  p  1

a r p r 11 p 2 p^2

x 1 r 1 b 11 p 2 x^ rpr

x  1, 2,... , 0  p  1

11 p 2 x^^1 p , 1 p 11 p 2 p^2

x  0, 1,... , n , 0  p  1

a np 11 p 2 n x

b px 11 p 2 n^ x ,

1 b a 122 1 12

1 b a 2 2

n ,^ a^ ^ b

Table II Cumulative Standard Normal Distribution

Φ (z)

 1 z 2  P 1 Z  z 2  

z



e

1 2 u

2

Table III Percentage Points ^2 , of the Chi-Squared Distribution

656 APPENDIX A STATISTICAL TABLES AND CHARTS

Table IV Percentage Points t , of the t -Distribution   degrees of freedom. 0 α

 - APPENDIX A
  • z 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0.
  • 3.9 0.000033 0.000034 0.000036 0.000037 0.000039 0.000041 0.000042 0.000044 0.000046 0.
  • 3.8 0.000050 0.000052 0.000054 0.000057 0.000059 0.000062 0.000064 0.000067 0.000069 0.
  • 3.7 0.000075 0.000078 0.000082 0.000085 0.000088 0.000092 0.000096 0.000100 0.000104 0.
  • 3.6 0.000112 0.000117 0.000121 0.000126 0.000131 0.000136 0.000142 0.000147 0.000153 0.
  • 3.5 0.000165 0.000172 0.000179 0.000185 0.000193 0.000200 0.000208 0.000216 0.000224 0.
  • 3.4 0.000242 0.000251 0.000260 0.000270 0.000280 0.000291 0.000302 0.000313 0.000325 0.
  • 3.3 0.000350 0.000362 0.000376 0.000390 0.000404 0.000419 0.000434 0.000450 0.000467 0.
  • 3.2 0.000501 0.000519 0.000538 0.000557 0.000577 0.000598 0.000619 0.000641 0.000664 0.
  • 3.1 0.000711 0.000736 0.000762 0.000789 0.000816 0.000845 0.000874 0.000904 0.000935 0.
  • 3.0 0.001001 0.001035 0.001070 0.001107 0.001144 0.001183 0.001223 0.001264 0.001306 0.
  • 2.9 0.001395 0.001441 0.001489 0.001538 0.001589 0.001641 0.001695 0.001750 0.001807 0.
  • 2.8 0.001926 0.001988 0.002052 0.002118 0.002186 0.002256 0.002327 0.002401 0.002477 0.
  • 2.7 0.002635 0.002718 0.002803 0.002890 0.002980 0.003072 0.003167 0.003264 0.003364 0.
  • 2.6 0.003573 0.003681 0.003793 0.003907 0.004025 0.004145 0.004269 0.004396 0.004527 0.
  • 2.5 0.004799 0.004940 0.005085 0.005234 0.005386 0.005543 0.005703 0.005868 0.006037 0.
  • 2.4 0.006387 0.006569 0.006756 0.006947 0.007143 0.007344 0.007549 0.007760 0.007976 0.
  • 2.3 0.008424 0.008656 0.008894 0.009137 0.009387 0.009642 0.009903 0.010170 0.010444 0.
  • 2.2 0.011011 0.011304 0.011604 0.011911 0.012224 0.012545 0.012874 0.013209 0.013553 0.
  • 2.1 0.014262 0.014629 0.015003 0.015386 0.015778 0.016177 0.016586 0.017003 0.017429 0.
  • 2.0 0.018309 0.018763 0.019226 0.019699 0.020182 0.020675 0.021178 0.021692 0.022216 0.
  • 1.9 0.023295 0.023852 0.024419 0.024998 0.025588 0.026190 0.026803 0.027429 0.028067 0.
  • 1.8 0.029379 0.030054 0.030742 0.031443 0.032157 0.032884 0.033625 0.034379 0.035148 0.
  • 1.7 0.036727 0.037538 0.038364 0.039204 0.040059 0.040929 0.041815 0.042716 0.043633 0.
  • 1.6 0.045514 0.046479 0.047460 0.048457 0.049471 0.050503 0.051551 0.052616 0.053699 0.
  • 1.5 0.055917 0.057053 0.058208 0.059380 0.060571 0.061780 0.063008 0.064256 0.065522 0.
  • 1.4 0.068112 0.069437 0.070781 0.072145 0.073529 0.074934 0.076359 0.077804 0.079270 0.
  • 1.3 0.082264 0.083793 0.085343 0.086915 0.088508 0.090123 0.091759 0.093418 0.095098 0.
  • 1.2 0.098525 0.100273 0.102042 0.103835 0.105650 0.107488 0.109349 0.111233 0.113140 0.
  • 1.1 0.117023 0.119000 0.121001 0.123024 0.125072 0.127143 0.129238 0.131357 0.133500 0.
  • 1.0 0.137857 0.140071 0.142310 0.144572 0.146859 0.149170 0.151505 0.153864 0.156248 0.
  • 0.9 0.161087 0.163543 0.166023 0.168528 0.171056 0.173609 0.176185 0.178786 0.181411 0.
  • 0.8 0.186733 0.189430 0.192150 0.194894 0.197662 0.200454 0.203269 0.206108 0.208970 0.
  • 0.7 0.214764 0.217695 0.220650 0.223627 0.226627 0.229650 0.232695 0.235762 0.238852 0.
  • 0.6 0.245097 0.248252 0.251429 0.254627 0.257846 0.261086 0.264347 0.267629 0.270931 0.
  • 0.5 0.277595 0.280957 0.284339 0.287740 0.291160 0.294599 0.298056 0.301532 0.305026 0.
  • 0.4 0.312067 0.315614 0.319178 0.322758 0.326355 0.329969 0.333598 0.337243 0.340903 0.
  • 0.3 0.348268 0.351973 0.355691 0.359424 0.363169 0.366928 0.370700 0.374484 0.378281 0.
  • 0.2 0.385908 0.389739 0.393580 0.397432 0.401294 0.405165 0.409046 0.412936 0.416834 0.
  • 0.1 0.424655 0.428576 0.432505 0.436441 0.440382 0.444330 0.448283 0.452242 0.456205 0.
  • 0.0 0.464144 0.468119 0.472097 0.476078 0.480061 0.484047 0.488033 0.492022 0.496011 0.
    • z - APPENDIX A
  •  .995 .990 .975 .950 .900 .500 .100 .050 .025 .010.  - 1 .00 .00 .00 .00 .02 .45 2.71 3.84 5.02 6.63 7. - 2 .01 .02 .05 .10 .21 1.39 4.61 5.99 7.38 9.21 10. - 3 .07 .11 .22 .35 .58 2.37 6.25 7.81 9.35 11.34 12. - 4 .21 .30 .48 .71 1.06 3.36 7.78 9.49 11.14 13.28 14. - 5 .41 .55 .83 1.15 1.61 4.35 9.24 11.07 12.83 15.09 16. - 6 .68 .87 1.24 1.64 2.20 5.35 10.65 12.59 14.45 16.81 18. - 7 .99 1.24 1.69 2.17 2.83 6.35 12.02 14.07 16.01 18.48 20. - 8 1.34 1.65 2.18 2.73 3.49 7.34 13.36 15.51 17.53 20.09 21. - 9 1.73 2.09 2.70 3.33 4.17 8.34 14.68 16.92 19.02 21.67 23.
    • 10 2.16 2.56 3.25 3.94 4.87 9.34 15.99 18.31 20.48 23.21 25.
    • 11 2.60 3.05 3.82 4.57 5.58 10.34 17.28 19.68 21.92 24.72 26.
    • 12 3.07 3.57 4.40 5.23 6.30 11.34 18.55 21.03 23.34 26.22 28.
    • 13 3.57 4.11 5.01 5.89 7.04 12.34 19.81 22.36 24.74 27.69 29.
    • 14 4.07 4.66 5.63 6.57 7.79 13.34 21.06 23.68 26.12 29.14 31.
    • 15 4.60 5.23 6.27 7.26 8.55 14.34 22.31 25.00 27.49 30.58 32.
    • 16 5.14 5.81 6.91 7.96 9.31 15.34 23.54 26.30 28.85 32.00 34.
    • 17 5.70 6.41 7.56 8.67 10.09 16.34 24.77 27.59 30.19 33.41 35.
    • 18 6.26 7.01 8.23 9.39 10.87 17.34 25.99 28.87 31.53 34.81 37.
    • 19 6.84 7.63 8.91 10.12 11.65 18.34 27.20 30.14 32.85 36.19 38.
    • 20 7.43 8.26 9.59 10.85 12.44 19.34 28.41 31.41 34.17 37.57 40.
    • 21 8.03 8.90 10.28 11.59 13.24 20.34 29.62 32.67 35.48 38.93 41.
    • 22 8.64 9.54 10.98 12.34 14.04 21.34 30.81 33.92 36.78 40.29 42.
    • 23 9.26 10.20 11.69 13.09 14.85 22.34 32.01 35.17 38.08 41.64 44.
    • 24 9.89 10.86 12.40 13.85 15.66 23.34 33.20 36.42 39.36 42.98 45.
    • 25 10.52 11.52 13.12 14.61 16.47 24.34 34.28 37.65 40.65 44.31 46.
    • 26 11.16 12.20 13.84 15.38 17.29 25.34 35.56 38.89 41.92 45.64 48.
    • 27 11.81 12.88 14.57 16.15 18.11 26.34 36.74 40.11 43.19 46.96 49.
    • 28 12.46 13.57 15.31 16.93 18.94 27.34 37.92 41.34 44.46 48.28 50.
    • 29 13.12 14.26 16.05 17.71 19.77 28.34 39.09 42.56 45.72 49.59 52.
    • 30 13.79 14.95 16.79 18.49 20.60 29.34 40.26 43.77 46.98 50.89 53.
    • 40 20.71 22.16 24.43 26.51 29.05 39.34 51.81 55.76 59.34 63.69 66.
    • 50 27.99 29.71 32.36 34.76 37.69 49.33 63.17 67.50 71.42 76.15 79.
    • 60 35.53 37.48 40.48 43.19 46.46 59.33 74.40 79.08 83.30 88.38 91.
    • 70 43.28 45.44 48.76 51.74 55.33 69.33 85.53 90.53 95.02 100.42 104.
    • 80 51.17 53.54 57.15 60.39 64.28 79.33 96.58 101.88 106.63 112.33 116.
    • 90 59.20 61.75 65.65 69.13 73.29 89.33 107.57 113.14 118.14 124.12 128.
  • 100 67.33 70.06 74.22 77.93 82.36 99.33 118.50 124.34 129.56 135.81 140. - χ α, ν   degrees of freedom.
  •  .40 .25 .10 .05 .025 .01 .005 .0025 .001.  - 1 .325 1.000 3.078 6.314 12.706 31.821 63.657 127.32 318.31 636. - 2 .289 .816 1.886 2.920 4.303 6.965 9.925 14.089 23.326 31. - 3 .277 .765 1.638 2.353 3.182 4.541 5.841 7.453 10.213 12. - 4 .271 .741 1.533 2.132 2.776 3.747 4.604 5.598 7.173 8. - 5 .267 .727 1.476 2.015 2.571 3.365 4.032 4.773 5.893 6. - 6 .265 .718 1.440 1.943 2.447 3.143 3.707 4.317 5.208 5. - 7 .263 .711 1.415 1.895 2.365 2.998 3.499 4.029 4.785 5. - 8 .262 .706 1.397 1.860 2.306 2.896 3.355 3.833 4.501 5. - 9 .261 .703 1.383 1.833 2.262 2.821 3.250 3.690 4.297 4.
    • 10 .260 .700 1.372 1.812 2.228 2.764 3.169 3.581 4.144 4.
    • 11 .260 .697 1.363 1.796 2.201 2.718 3.106 3.497 4.025 4.
    • 12 .259 .695 1.356 1.782 2.179 2.681 3.055 3.428 3.930 4.
    • 13 .259 .694 1.350 1.771 2.160 2.650 3.012 3.372 3.852 4.
    • 14 .258 .692 1.345 1.761 2.145 2.624 2.977 3.326 3.787 4.
    • 15 .258 .691 1.341 1.753 2.131 2.602 2.947 3.286 3.733 4.
    • 16 .258 .690 1.337 1.746 2.120 2.583 2.921 3.252 3.686 4.
    • 17 .257 .689 1.333 1.740 2.110 2.567 2.898 3.222 3.646 3.
    • 18 .257 .688 1.330 1.734 2.101 2.552 2.878 3.197 3.610 3.
    • 19 .257 .688 1.328 1.729 2.093 2.539 2.861 3.174 3.579 3.
    • 20 .257 .687 1.325 1.725 2.086 2.528 2.845 3.153 3.552 3.
    • 21 .257 .686 1.323 1.721 2.080 2.518 2.831 3.135 3.527 3.
    • 22 .256 .686 1.321 1.717 2.074 2.508 2.819 3.119 3.505 3.
    • 23 .256 .685 1.319 1.714 2.069 2.500 2.807 3.104 3.485 3.
    • 24 .256 .685 1.318 1.711 2.064 2.492 2.797 3.091 3.467 3.
    • 25 .256 .684 1.316 1.708 2.060 2.485 2.787 3.078 3.450 3.
    • 26 .256 .684 1.315 1.706 2.056 2.479 2.779 3.067 3.435 3.
    • 27 .256 .684 1.314 1.703 2.052 2.473 2.771 3.057 3.421 3.
    • 28 .256 .683 1.313 1.701 2.048 2.467 2.763 3.047 3.408 3.
    • 29 .256 .683 1.311 1.699 2.045 2.462 2.756 3.038 3.396 3.
    • 30 .256 .683 1.310 1.697 2.042 2.457 2.750 3.030 3.385 3.
    • 40 .255 .681 1.303 1.684 2.021 2.423 2.704 2.971 3.307 3.
    • 60 .254 .679 1.296 1.671 2.000 2.390 2.660 2.915 3.232 3.
  • 120 .254 .677 1.289 1.658 1.980 2.358 2.617 2.860 3.160 3. -  .253 .674 1.282 1.645 1.960 2.326 2.576 2.807 3.090 3.