Appendix B — Statistical Reference Tables

Status: ported 2026-05-19. Reviewed by editor: pending.

This appendix gathers the small set of reference tables most useful in a descriptive and probabilistic course: the standard-normal CDF, binomial and Poisson probability mass values for the parameter combinations that recur in worked examples and exercises, and a short list of standard-normal critical values. Because the entries below are produced inside R by the distribution functions pnorm(), dbinom(), and dpois(), they are exact to machine precision rather than rounded to fewer decimals as in printed appendices. A reader who wants a value not listed here can simply re-run the corresponding chunk with extra rows or columns. Fuller critical-value tables for the \(t\), \(F\), and \(\chi^{2}\) distributions belong to the inferential setting of TC2 / Econometrics I and are reproduced in that book’s reference appendix.

B.1 B.1 Standard normal CDF — \(\Phi(z)\)

The table reports \(\Phi(z) = P(Z \leq z)\) for the standard normal random variable \(Z \sim N(0, 1)\). Rows index the first decimal of \(z\) (from \(0.0\) to \(3.4\) in steps of \(0.1\)); columns index the second decimal (from \(0.00\) to \(0.09\)). The entry in row \(z = 1.2\) and column \(0.06\), for instance, is \(\Phi(1.26)\). For negative arguments use the symmetry of the standard normal: \[ \Phi(-z) = 1 - \Phi(z), \qquad z > 0. \]

Standard normal CDF \(\Phi(z) = P(Z \leq z)\) for \(z \in [0.00, 3.49]\). Rows: first decimal of \(z\). Columns: second decimal.
z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

The two most-used entries are worth memorising: \(\Phi(1.96) \approx 0.9750\) (so \(P(-1.96 < Z < 1.96) \approx 0.95\)) and \(\Phi(2.576) \approx 0.9950\) (so \(P(-2.576 < Z < 2.576) \approx 0.99\)). The first appears in the section on the normal approximation to the binomial in Chapter 5.

B.2 B.2 Binomial probabilities — \(P(X = k)\)

The tables below tabulate the binomial PMF \[ P(X = k) = \binom{n}{k} p^{k}(1-p)^{n-k}, \qquad k = 0, 1, \ldots, n, \] for the parameter pairs \((n, p)\) used most frequently in Chapter 5. Each table fixes \(n\) and lets \(p\) vary across the columns; rows index \(k\). Entries smaller than \(0.00005\) are displayed as \(0.0000\).

Binomial probabilities \(P(X = k)\) for \(n \in {5, 10}\) and \(p \in {0.1, 0.2, 0.3, 0.5}\).
n k p = 0.1 p = 0.2 p = 0.3 p = 0.5
5 0 0.5905 0.3277 0.1681 0.0312
1 0.3280 0.4096 0.3601 0.1562
2 0.0729 0.2048 0.3087 0.3125
3 0.0081 0.0512 0.1323 0.3125
4 0.0005 0.0064 0.0284 0.1562
5 0.0000 0.0003 0.0024 0.0312
10 0 0.3487 0.1074 0.0282 0.0010
1 0.3874 0.2684 0.1211 0.0098
2 0.1937 0.3020 0.2335 0.0439
3 0.0574 0.2013 0.2668 0.1172
4 0.0112 0.0881 0.2001 0.2051
5 0.0015 0.0264 0.1029 0.2461
6 0.0001 0.0055 0.0368 0.2051
7 0.0000 0.0008 0.0090 0.1172
8 0.0000 0.0001 0.0014 0.0439
9 0.0000 0.0000 0.0001 0.0098
10 0.0000 0.0000 0.0000 0.0010
Binomial probabilities \(P(X = k)\) for \(n \in {15, 20}\) and \(p \in {0.1, 0.2, 0.3, 0.5}\).
n k p = 0.1 p = 0.2 p = 0.3 p = 0.5
15 0 0.2059 0.0352 0.0047 0.0000
1 0.3432 0.1319 0.0305 0.0005
2 0.2669 0.2309 0.0916 0.0032
3 0.1285 0.2501 0.1700 0.0139
4 0.0428 0.1876 0.2186 0.0417
5 0.0105 0.1032 0.2061 0.0916
6 0.0019 0.0430 0.1472 0.1527
7 0.0003 0.0138 0.0811 0.1964
8 0.0000 0.0035 0.0348 0.1964
9 0.0000 0.0007 0.0116 0.1527
10 0.0000 0.0001 0.0030 0.0916
11 0.0000 0.0000 0.0006 0.0417
12 0.0000 0.0000 0.0001 0.0139
13 0.0000 0.0000 0.0000 0.0032
14 0.0000 0.0000 0.0000 0.0005
15 0.0000 0.0000 0.0000 0.0000
20 0 0.1216 0.0115 0.0008 0.0000
1 0.2702 0.0576 0.0068 0.0000
2 0.2852 0.1369 0.0278 0.0002
3 0.1901 0.2054 0.0716 0.0011
4 0.0898 0.2182 0.1304 0.0046
5 0.0319 0.1746 0.1789 0.0148
6 0.0089 0.1091 0.1916 0.0370
7 0.0020 0.0545 0.1643 0.0739
8 0.0004 0.0222 0.1144 0.1201
9 0.0001 0.0074 0.0654 0.1602
10 0.0000 0.0020 0.0308 0.1762
11 0.0000 0.0005 0.0120 0.1602
12 0.0000 0.0001 0.0039 0.1201
13 0.0000 0.0000 0.0010 0.0739
14 0.0000 0.0000 0.0002 0.0370
15 0.0000 0.0000 0.0000 0.0148
16 0.0000 0.0000 0.0000 0.0046
17 0.0000 0.0000 0.0000 0.0011
18 0.0000 0.0000 0.0000 0.0002
19 0.0000 0.0000 0.0000 0.0000
20 0.0000 0.0000 0.0000 0.0000

For values of \(p\) greater than \(0.5\) use the symmetry \(P(X = k \mid n, p) = P(X = n-k \mid n, 1-p)\): a binomial with success probability \(0.7\) and \(n = 10\) is read off the \(p = 0.3\) column with \(k\) replaced by \(10 - k\).

B.3 B.3 Poisson probabilities — \(P(X = k)\)

The table reports the Poisson PMF \[ P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!}, \qquad k = 0, 1, 2, \ldots \] for the rate parameters \(\lambda \in \{0.5, 1, 2, 3, 4, 5, 7, 10\}\) most often seen in exercises. Rows index \(k\) from \(0\) to \(15\), which captures essentially all the mass for \(\lambda \leq 10\) (the upper tail beyond \(k = 15\) accounts for less than \(0.05\%\) even at \(\lambda = 10\)).

Poisson probabilities \(P(X = k)\) for selected rate parameters \(\lambda\). Rows: \(k = 0, 1, \ldots, 15\). Columns: \(\lambda\).
k λ = 0.5 λ = 1.0 λ = 2.0 λ = 3.0 λ = 4.0 λ = 5.0 λ = 7.0 λ = 10.0
0 0.6065 0.3679 0.1353 0.0498 0.0183 0.0067 0.0009 0.0000
1 0.3033 0.3679 0.2707 0.1494 0.0733 0.0337 0.0064 0.0005
2 0.0758 0.1839 0.2707 0.2240 0.1465 0.0842 0.0223 0.0023
3 0.0126 0.0613 0.1804 0.2240 0.1954 0.1404 0.0521 0.0076
4 0.0016 0.0153 0.0902 0.1680 0.1954 0.1755 0.0912 0.0189
5 0.0002 0.0031 0.0361 0.1008 0.1563 0.1755 0.1277 0.0378
6 0.0000 0.0005 0.0120 0.0504 0.1042 0.1462 0.1490 0.0631
7 0.0000 0.0001 0.0034 0.0216 0.0595 0.1044 0.1490 0.0901
8 0.0000 0.0000 0.0009 0.0081 0.0298 0.0653 0.1304 0.1126
9 0.0000 0.0000 0.0002 0.0027 0.0132 0.0363 0.1014 0.1251
10 0.0000 0.0000 0.0000 0.0008 0.0053 0.0181 0.0710 0.1251
11 0.0000 0.0000 0.0000 0.0002 0.0019 0.0082 0.0452 0.1137
12 0.0000 0.0000 0.0000 0.0001 0.0006 0.0034 0.0263 0.0948
13 0.0000 0.0000 0.0000 0.0000 0.0002 0.0013 0.0142 0.0729
14 0.0000 0.0000 0.0000 0.0000 0.0001 0.0005 0.0071 0.0521
15 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0033 0.0347

A useful sanity check is that each column sums to (very nearly) \(1\): for \(\lambda \leq 10\) the truncation at \(k = 15\) loses an essentially negligible amount of mass.

B.4 B.4 Selected standard-normal critical values

For the few occasions in TC1 where a one- or two-sided “\(z\)-value” is referenced — chiefly the normal approximation to the binomial in Chapter 5 — the table below collects the three most-used standard-normal critical values \(z_{1-\alpha}\), defined by \(\Phi(z_{1-\alpha}) = 1 - \alpha\).

Standard-normal critical values \(z_{1-\alpha}\): \(z\) such that \(\Phi(z) = 1 - \alpha\).
α 1 − α z_{1−α}
0.050 0.950 1.6449
0.025 0.975 1.9600
0.010 0.990 2.3263
0.005 0.995 2.5758

The entries reproduce the canonical pair \(z_{0.95} = 1.6449\) and \(z_{0.975} = 1.9600\). Fuller critical-value tables for the \(t\), \(F\), and \(\chi^{2}\) distributions — needed only once hypothesis testing enters the picture — live in the reference appendix of TC2 / Econometrics I.