10.1. Table: Standard Normal Distribution#
The table shows one-sided (right-hand) probabilities \(\alpha\) as function of the critical value \(k_\alpha\), i.e. \(\alpha = P(Z\geq k_\alpha)\). Note that this table gives a different value than the double-tailed illustration as described in Precision and confidence intervals.
To evalute the table for \(k_\alpha\): find the value up to the first decimal is in the first column, then find the column in the first row that most closely matches the second decimal.
Example: \(\alpha\) = 0.0250 for \(k_\alpha\) = 1.96.
from scipy.stats import norm
The cell below is set up to use interactively. To use it, click –> Live Code on the top right corner of this screen and then wait until Python interaction is ready. The method scipy.stats.norm has already been imported as norm.
alpha = 0.0250
k_alpha = norm.ppf(1 - alpha)
print(f"For alpha = {alpha:.4f}, k_alpha = {k_alpha:0.4f}.")
print(f"The probability in the upper (right-hand) tail is {100*alpha:.1f}%.")
print(f"\nIf this k_alpha defined a confidence level, it would be {100*alpha*2:.1f}%,")
print(f" and the bounds would be {k_alpha:.2f} standard deviations from the mean.")
For alpha = 0.0250, k_alpha = 1.9600.
The probability in the upper (right-hand) tail is 2.5%.
If this k_alpha defined a confidence level, it would be 5.0%,
and the bounds would be 1.96 standard deviations from the mean.
Table of Values#
\(k_\alpha\) |
0.00 |
0.01 |
0.02 |
0.03 |
0.04 |
0.05 |
0.06 |
0.07 |
0.08 |
0.09 |
|---|---|---|---|---|---|---|---|---|---|---|
0.0 |
0.5000 |
0.4960 |
0.4920 |
0.4880 |
0.4840 |
0.4801 |
0.4761 |
0.4721 |
0.4681 |
0.4641 |
0.1 |
0.4602 |
0.4562 |
0.4522 |
0.4483 |
0.4443 |
0.4404 |
0.4364 |
0.4325 |
0.4286 |
0.4247 |
0.2 |
0.4207 |
0.4168 |
0.4129 |
0.4090 |
0.4052 |
0.4013 |
0.3974 |
0.3936 |
0.3897 |
0.3859 |
0.3 |
0.3821 |
0.3783 |
0.3745 |
0.3707 |
0.3669 |
0.3632 |
0.3594 |
0.3557 |
0.3520 |
0.3483 |
0.4 |
0.3446 |
0.3409 |
0.3372 |
0.3336 |
0.3300 |
0.3264 |
0.3228 |
0.3192 |
0.3156 |
0.3121 |
0.5 |
0.3085 |
0.3050 |
0.3015 |
0.2981 |
0.2946 |
0.2912 |
0.2877 |
0.2843 |
0.2810 |
0.2776 |
0.6 |
0.2743 |
0.2709 |
0.2676 |
0.2643 |
0.2611 |
0.2578 |
0.2546 |
0.2514 |
0.2483 |
0.2451 |
0.7 |
0.2420 |
0.2389 |
0.2358 |
0.2327 |
0.2296 |
0.2266 |
0.2236 |
0.2206 |
0.2177 |
0.2148 |
0.8 |
0.2119 |
0.2090 |
0.2061 |
0.2033 |
0.2005 |
0.1977 |
0.1949 |
0.1922 |
0.1894 |
0.1867 |
0.9 |
0.1841 |
0.1814 |
0.1788 |
0.1762 |
0.1736 |
0.1711 |
0.1685 |
0.1660 |
0.1635 |
0.1611 |
1.0 |
0.1587 |
0.1562 |
0.1539 |
0.1515 |
0.1492 |
0.1469 |
0.1446 |
0.1423 |
0.1401 |
0.1379 |
1.1 |
0.1357 |
0.1335 |
0.1314 |
0.1292 |
0.1271 |
0.1251 |
0.1230 |
0.1210 |
0.1190 |
0.1170 |
1.2 |
0.1151 |
0.1131 |
0.1112 |
0.1093 |
0.1075 |
0.1056 |
0.1038 |
0.1020 |
0.1003 |
0.0985 |
1.3 |
0.0968 |
0.0951 |
0.0934 |
0.0918 |
0.0901 |
0.0885 |
0.0869 |
0.0853 |
0.0838 |
0.0823 |
1.4 |
0.0808 |
0.0793 |
0.0778 |
0.0764 |
0.0749 |
0.0735 |
0.0721 |
0.0708 |
0.0694 |
0.0681 |
1.5 |
0.0668 |
0.0655 |
0.0643 |
0.0630 |
0.0618 |
0.0606 |
0.0594 |
0.0582 |
0.0571 |
0.0559 |
1.6 |
0.0548 |
0.0537 |
0.0526 |
0.0516 |
0.0505 |
0.0495 |
0.0485 |
0.0475 |
0.0465 |
0.0455 |
1.7 |
0.0446 |
0.0436 |
0.0427 |
0.0418 |
0.0409 |
0.0401 |
0.0392 |
0.0384 |
0.0375 |
0.0367 |
1.8 |
0.0359 |
0.0351 |
0.0344 |
0.0336 |
0.0329 |
0.0322 |
0.0314 |
0.0307 |
0.0301 |
0.0294 |
1.9 |
0.0287 |
0.0281 |
0.0274 |
0.0268 |
0.0262 |
0.0256 |
0.0250 |
0.0244 |
0.0239 |
0.0233 |
2.0 |
0.0228 |
0.0222 |
0.0217 |
0.0212 |
0.0207 |
0.0202 |
0.0197 |
0.0192 |
0.0188 |
0.0183 |
2.1 |
0.0179 |
0.0174 |
0.0170 |
0.0166 |
0.0162 |
0.0158 |
0.0154 |
0.0150 |
0.0146 |
0.0143 |
2.2 |
0.0139 |
0.0136 |
0.0132 |
0.0129 |
0.0125 |
0.0122 |
0.0119 |
0.0116 |
0.0113 |
0.0110 |
2.3 |
0.0107 |
0.0104 |
0.0102 |
0.0099 |
0.0096 |
0.0094 |
0.0091 |
0.0089 |
0.0087 |
0.0084 |
2.4 |
0.0082 |
0.0080 |
0.0078 |
0.0075 |
0.0073 |
0.0071 |
0.0069 |
0.0068 |
0.0066 |
0.0064 |
2.5 |
0.0062 |
0.0060 |
0.0059 |
0.0057 |
0.0055 |
0.0054 |
0.0052 |
0.0051 |
0.0049 |
0.0048 |
2.6 |
0.0047 |
0.0045 |
0.0044 |
0.0043 |
0.0041 |
0.0040 |
0.0039 |
0.0038 |
0.0037 |
0.0036 |
2.7 |
0.0035 |
0.0034 |
0.0033 |
0.0032 |
0.0031 |
0.0030 |
0.0029 |
0.0028 |
0.0027 |
0.0026 |
2.8 |
0.0026 |
0.0025 |
0.0024 |
0.0023 |
0.0023 |
0.0022 |
0.0021 |
0.0021 |
0.0020 |
0.0019 |
2.9 |
0.0019 |
0.0018 |
0.0018 |
0.0017 |
0.0016 |
0.0016 |
0.0015 |
0.0015 |
0.0014 |
0.0014 |
3.0 |
0.0013 |
0.0013 |
0.0013 |
0.0012 |
0.0012 |
0.0011 |
0.0011 |
0.0011 |
0.0010 |
0.0010 |
3.1 |
0.0010 |
0.0009 |
0.0009 |
0.0009 |
0.0008 |
0.0008 |
0.0008 |
0.0008 |
0.0007 |
0.0007 |
3.2 |
0.0007 |
0.0007 |
0.0006 |
0.0006 |
0.0006 |
0.0006 |
0.0006 |
0.0005 |
0.0005 |
0.0005 |
3.3 |
0.0005 |
0.0005 |
0.0005 |
0.0004 |
0.0004 |
0.0004 |
0.0004 |
0.0004 |
0.0004 |
0.0003 |
3.4 |
0.0003 |
0.0003 |
0.0003 |
0.0003 |
0.0003 |
0.0003 |
0.0003 |
0.0003 |
0.0003 |
0.0002 |